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0, |r|∞ ≤ 1}, {w0 : |w0 | = wo = βo }. In (15) 0 X |aT (A)||z(h, w0 )|A , (16) C 0 ≡ sup{h,w0 } P0
A
means only x1 ∈ supp A. where The sum in (16) can be bounded as in the standard free energy bound in [3]. Let us recall the -value from [3]. If (a) the bound for N (x, k), the number of polymers of size |γ| = k passing through x, can be written ec|γ| and (b) the bound for |z(γ)| can be written as e−E|γ| then the -value is ec−E . If < 16 the bound for the sum in (16) is (see [3]) 2 −1 1 5 5 + . (17) 1− H() = 1− 8 1− 1− In our case N (x, k) ≤ (2d)k and using Lemma 1 for each factor in the denominator of (12) gives the bound |z(γ, h, w0 )| ≤ e−(− ln β−β−3|h|∞ )|γ| ,
(18)
where we have used that |h|1 ≤ |h|∞ |γ| and that |h|1 is evaluated on γ. This leads to the -value (19) (β) = e−(| ln β|−β−3|h|∞ −ln 2d) .
EXPONENTIAL DECAY OF TRUNCATED CORRELATION FUNCTIONS
...
435
Thus, if |h|∞ ≤ 1, for all β < βo , βo sufficiently small, we have (β) < (βo ) < 16 and C 0 ≤ H(), which proves Theorem 1. 4. Low Temperature Ising Model We treat explicitly the case of dimension d = 2, the extension for d > 2 is straightforward. We take the + boundary partition function, with Λ ⊂ Z 2 , defined by Eq. (2). As in [2], we obtain the polymer system representation Z+ (Λ, h) =
XY
z(γα , h) =
{γα } α
X
a(A)z(h)A ,
(20)
A
where z(γ, h) = e−2β|γ|
Z− (Intγ, h) , Z+ (Int γ, h)
(21)
where the γ are Peierls contours and |γ| equals the perimeter length. We identify γ with the bonds where si sj = −1 or the midpoint of the bond. We extend z(γ, h) to z(γ, h, w) analytic in w in |w| ≤ wo ≡ e−2βo by replacing e−2β|γ| by w|γ| . Thus the m-point truncated function is Sm =
Z 00 m Y 1 dhj X T a (A)z(h, w)A |w=e−2β , 2 2πi h |hj |=rj j j=1
(22)
A
where 00 means {xi } ⊂ ∪i Int γi ≡ v(A). In words, {xi } are contained in the “volume” enclosed by {γi }. As in the analysis for high temperature, analyticity of ln Z+ (Λ, h) in the h’s follows from [3]. Note that each term in (22) has at least a factor wdp , where for any integer p, 1 ≤ p < m, dp = min dist {xk1 , . . . , xkp } , {xkp+1 , . . . , xkm } , k1 ,...,km
(23)
where the minimum is taken over all permutations {k1 , . . . , km } of {1, . . . , m}. Multiplying and dividing by ( wwo )dp , using Cauchy estimates and the maximum modulus theorem in w we have, for |w| ≤ wo
" # 00 m Y X dp 1 T 0 A w |Sm | ≤ |a (A)||z(h, w )| sup{h,w0 } r wo w=e−2β j=1 j A
≤
m Y 1 0 −2(β−βo )dp . Ce r j=1 j
(24)
In (24) 0
0
C ≡ sup
X
{h,w 0 } A
|aT (A)||z(h, w0 )|A ,
(25)
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G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
P0 where only means that x1 ∈ V (A) and sup is taken over {h : |hi | = ri > 0, |r|1 ≤ 1}, {w0 : |w0 | = wo ≡ e−2βo }. Concerning the bound C 0 we first consider the bounds for z(γ, h, w0 ) (for values 2 x, k) (here x ˆ ∈ Z + 12 and of w0 on the disc of radius e−2β < e−2βo ) and N (ˆ x ˆ ∈ γ). The ratio of partition functions is bounded using Lemma 1, noting that 0 0 − (Λ ,h)/Z− (Λ ,0) 0 0 Z− (Λ0 , h)/Z+ (Λ0 , h) = Z Z+ (Λ0 ,h)/Z+ (Λ0 ,0) and Z+ (Λ , 0) = Z− (Λ , 0). This leads to the bound |z(γ, h, w0 )| ≤ e−2β|γ|e3|hInt γ |1 ≤ e−(2β−3|hInt γ |1 )|γ| since |γ| ≥ 1 (actually |γ| ≥ 4). A bound for N (ˆ x, k) is 4k . Thus the -value we take, with |h|1 ≤ 1, (β) = e−(2β−3|h|1−ln 4) .
(26)
The bound of the sum in (25) differs from the corresponding high temperature ˆ1 ∈ {γi } with x ˆ1 fixed. one, since x1 ∈ V (A). The free energy bound requires only x 0 where C10 is the sum over the linear terms, i.e. Write C 0 = C10 + C>2 sup{h,w0 }
X
|z(γ, h, w0 )| .
γ:x1 ∈V (γ)
For a given size |γ| = k only contours γ will contribute that lie within a square of side length k centered at x1 . Thus for < 1 C10
≤
∞ X k=1
k 2 k =
(1 + ) ≡ L1 () . (1 − )3
(27)
0 is obtained analogously to the “sum over trees” bound for The bound for C≥2 the free energy in Chapter V of [3] with the difference that x1 ∈ V (γ1 ). Summing over ordered sequences of polymers (γ1 , . . . , γn ) with sizes (k1 , . . . , kn ) following [3] the bound is given by
0 C≥2
∞ X (n − 2)! 1 X Q ≤ n! l (dl − 1)! n=2 {dj }
X (k1 ,...,kn )
k12
n Y
kr (dr −1) kr .
(28)
r=1
The sum is over ordered sequences (k1 , . . . , kn ), ki ≥ 1, and differs from the free energy bound by the appearance of the factor k12 as in the case of linear terms only. {. . .} is Cayley’s tree formula for the number of trees with n vertices, dj , Pn 1 ≤ dj ≤ n − 1, j=1 dj = 2(n − 1). Using Lemma V.7.5 of [3] for the {ki } sums with < Lemma V.7.8 for the {dj } sums we obtain the bounds
1 6,
d1 ≤ n − 1 and
EXPONENTIAL DECAY OF TRUNCATED CORRELATION FUNCTIONS
0 C≥2 ≤
...
437
n 5 1 X d1 + 1)d1 ( n−1 4(1 − ) n=2 ∞ X
{dj }
∞ X ≤ n n=2
1 ≤ 8
5 4(1 − )
5 1−
n X
1
{dj }
2 −2 5 1− 1−
≡ L≥2 () .
(29)
Thus, with |h|1 ≤ 1, if β > βo and βo is sufficiently large, we have (β) < (βo ) < 16 and C 0 ≤ L1 () + L≥2 () ≡ L(), which proves Theorem 2. 5. Contour Model The proof of Theorem 3 follows along the lines of the proof of Theorem 2, Sec. 4. Here we extend the contour activities to the complex w-plane and, using assumptions (7) (h-dependent Peierls condition) and (8) (q-phase stability), we find the -value for these modified activities. Thus, z(γ, h), defined by Eq. (6), is extended to z(γ, h, w) by extending ρ(γ, h) to ρ(γ, h, w) =
w|γ| ρ(γ, h) , e−β|γ|
where ρ(γ, h, w) is analytic in w, in particular for |w| ≤ e−βo ≡ wo , and w = e−β is the physical value. Using the Lemma 1 and (8) we bound Y Zm (h) Y Zm (h)/Zm (0) Zm (0) = Zq (h) Zq (h)/Zq (0) Zq (0) m m Q by m e3|hIntγ |1 +cm |γ| . Since for |w| < e−β , the Ineq. (7) gives the bound |ρ(γ, h, w)| P ≤ e−(β−b)+|rγ |1 )|γ| with c = m cm , we have the bound (again for |w0 | < e−β )
|z(γ, h, w0 )| ≤ e−(β−(b+3)|r|1 −c)|γ| . Thus, the -value is 0
= e−(β−(b+3)|r|1 −c−d ) . The rest of the proof is as in Sec. 4.
6. Acknowledgments This work has been partially supported by the Brazilian agencies CNPq, FAPEMIG and PROPES-UFMG. Gast˜ ao A. Braga and Paulo C. Lima thank CAPES for the financial support while visiting the Institute for Advanced Study and Rutgers University, respectively.
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References [1] E. Seiler, “Gauge theories as a problem of constructive quantum field theory and statistical mechanics”, Lecture Notes in Phys., 159 (1982) Berlin, New York, Springer. [2] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, 2nd edition, New York, Springer, 1986. [3] B. Simon, The Statistical Mechanics of Lattice Gases, Princeton Univ. Press, 1993. [4] V. A. Malyshev and R. A. Minlos, Gibbs Random Fields: Cluster Expansions, Boston, Kluwer Academic Publ., 1991. [5] R. Kotecky and D. Preiss, “Cluster expansion for abstract polymer models”, Commun. Math. Phys. 103 (1986) 491–498. [6] Y. Sinai, Theory of Phase Transitions: Rigorous Results, Pergamon Press, 1982. [7] M. Zahradnik, “An alternate version of Pirogov–Sinai theory”, Commun. Math. Phys. 93 (1984) 559–581. [8] C. Borgs and J. Imbrie, “A unified approach to phase diagrams in field theory and statistical mechanics”, Commun. Math. Phys. 123 (1989) 305–328. [9] R. L. Dobrushin, “Estimates of semiinvariants for the Ising Model at low temperatures”, The Erwin Schroedinger Inst. for Math. Phys. preprint ESI 125 (1994). [10] M. L. O’Carroll, “Analyticity properties and a convergent expansion for the inverse correlation length of the low temperature d-dimensional Ising model”, J. Stat. Phys. 34 (1984) 609–614. [11] R. S. Schor, “The particle structure of ν-dimensional Ising model at low temperature”, Commun. Math. Phys. 59 (1978) 213–233. [12] M. Duneau, D. Iagolnitzer and B. Souillard, “Strong cluster properties for classical systems with finite range interactions,” Commun. Math. Phys. 35 (1974) 307–320.
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF THEIR CONTINUOUS FUNCTIONS ELISA ERCOLESSI Dipartimento di Fisica, Universit` a di Bologna and INFM Via Irnerio 46, I-40126, Bologna, Italy
GIOVANNI LANDI The E. Schr¨ odinger International Institute for Mathematical Physics Pasteurgasse 6/7, A-1090 Wien, Austria Dipartimento di Scienze Matematiche, Universit` a di Trieste P. le Europa 1, I-34127, Trieste, Italy and INFN, Sezione di Napoli, Napoli, Italy
PAULO TEOTONIO-SOBRINHO The E. Schr¨ odinger International Institute for Mathematical Physics Pasteurgasse 6/7, A-1090 Wien, Austria Department of Physics, University of Illinois at Chicago 60607-7059 Chicago, IL, USA and Universidade de Sao Paulo, Instituto de Fisica - DFMA Caixa Postal 66318, 05389-970, Sao Paulo, SP, Brasil Received 14 December 1996 Received 15 May 1997 Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure ole spaces of noncommutative C ∗ -algebras. These noncommutative algebras play the same rˆ as the algebra of continuous functions C(M ) on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C ∗ -algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.
1. Introduction It is well known that the standard discretization methods used in quantum physics (where a manifold is replaced by a lattice of points with the discrete topology) are not able to describe any significant topological attribute of the continuum, this being equally the case for both the local and global properties. For example, 439 Reviews in Mathematical Physics, Vol. 10, No. 4 (1998) 439–466 c World Scientific Publishing Company
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there is no nontrivial concept of winding number and hence no way to formulate theories with topological solitons or instantons on these lattices. A new kind of finite approximation to continuum topological spaces has been introduced in [1], with the name of posets or partially ordered sets. As we will see in Sec. 3, posets are also T0 topological spaces and can reproduce important topological properties of the continuum, such as the homology and the homotopy groups, with remarkable fidelity [1, 2]. This ability to capture topological information has been the main motivation for their use in quantum physics in place of the ordinary discrete lattices. In [3], quantum mechanics has been formulated on posets and it has been proved that it is possible to study nontrivial topological configurations, such as θ-states for particles on the poset approximations to a circle. Some promising results have also been obtained in the formulation of solitonic field theories [3] as well as of gaouge field theories [4]. In [5], the poset approximation scheme has been developed in a novel direction. Indeed, it has been observed that posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces (spaces of irreducible representations) of noncommutative C ∗ -algebras. These noncommutative algebras play the same rˆ ole as the algebra of continuous functions C(M ) on a manifold M and can be thought of as algebras of operator valued functions on posets. This naturally leads to the use of noncommutative geometry [6] (see also [7]) as the tool to rewrite quantum theories on posets and gives a remarkable connection between topologically meaningful finite approximations to quantum physics and noncommutative geometry. The duality relation between Hausdorff topological spaces and commutative C ∗ -algebras is provided by the Gel’fand–Naimark theorem. There is no analogue of this theorem in the noncommutative setting. In this article, we will review how it is possible to establish a relation between finite posets and a particular class of noncommutative C ∗ -algebras. For such class of algebras the situation is very similar to the commutative case. We will see that the algebras in question are all approximately finite dimensional (AF) postliminal algebras [8, 9], i.e. C ∗ -algebras that can be approximated in norm by a sequence of finite dimensional matrix algebras and whose irreducible representations are completely characterized by the kernels. This is exactly what makes them of some interest in mathematics: they present virtually all the attributes and complications of other infinite dimensional algebras, but many techniques and results valid in the finite dimensional case can be used in their study. Thus, for example, a complete classification of AF C ∗ -algebras is available [10]. These algebras were first extensively studied by Bratteli [8], who also introduced a diagrammatic representation which is very useful for the study of their algebraic properties. In particular we will see how to use the Bratteli diagram of an AF algebra to construct its structure space. Then we will see how, given any finite poset P , it is possible to construct the Bratteli diagram of an AF algebra whose structure space is the given poset P . Being noncommutative, this AF C ∗ -algebras is far from being unique. Indeed there is a whole family of AF algebras that have
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P as structure space and that can be classified by means of results due to Behncke and Leptin [11]. In this article, we will not present the classification of AF C ∗ -algebras that can be formulated in terms of their algebraic K-theory [10]. In view of their relation with posets, this would represent also a first step in the construction of bundles and characteristic classes over noncommutative lattices. This is indeed the content of [12], to which we refer the interested reader for a detailed analysis of the K-theory of AF algebras. This article is organized as follows. In Sec. 2 we review some elementary algebraic concepts as well as the Gel’fand Naimark theorem in order to clarify the notation and the terminology used in the sequel. In Sec. 3 we briefly describe the topological approximation of continuous spaces that leads to partially ordered sets (posets). In Sec. 4, AF C ∗ -algebras are introduced and the connection between Bratteli diagrams and posets is discussed in detail. Finally, in Sec. 5 we present the classification theorem of the AF C ∗ -algebras that have a poset as structure space due to Behncke and Leptin. Several interesting examples will be examined throughout the article. 2. C∗ -algebras and Structure Spaces Let us start with some elementary algebraic preliminaries [13, 14] that will be also useful to establish notation. In the sequel, A will always denote a C ∗ -algebra over the field of complex numbers C. We remind that this means that A is equipped with a norm of algebra k · k : A → C (with respect to which A is complete) and an involution ∗ : A → A, satisfying the identity: (2.1) ka∗ ak = kak2 , ∀ a ∈ A . The following are examples of commutative and noncommutative C ∗ -algebras which will be used in the article: (1) the (noncommutative) algebra M(n, C) of n × n complex matrices T , with T ∗ given by the hermitian conjugate of T and the squared norm k T k2 being equal to the largest eigenvalue of T ∗ T ; (2) the (noncommutative) algebra B(H) of bounded operators B on a separable (infinite-dimensional) Hilbert space H as well as its subalgebra K(H) of compact operators. Now ∗ is given by the adjoint and the norm is the operator norm: kBk = supkξk≤1 kBξk (ξ ∈ H); (3) the (commutative) algebra C(M ) of continuous functions on a compact Hausdorff topological space M , with * denoting complex conjugation and the norm given by the supremum norm, k f k∞ = supx∈M |f (x)|. If M is not compact but only locally compact, then one can consider the algebra C0 (M ) of functions vanishing at infinity. Notice that K(H) and C0 (M ) (with M only locally compact) are examples of C ∗ -algebras without unit I, in contrast to M(n, C) and B(H).
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2.1. Commutative C∗ -algebras: the Gel’fand Naimark Theorem In the third example above we have seen how it is possible to associate a commutative C ∗ -algebra with (without) unit, namely C(M ) (C0 (M )), to any Hausdorff compact (locally compact) topological space M . Vice versa, given any commutative C ∗ -algebra C with (without) unit, it is possible to construct a Hausdorff compact (locally compact) topological space M such that C is isometrically ∗-isomorphic to the algebra of continuous functions C(M ) (C0 (M )). This is precisely the content of the Gel’fand–Naimark theorem [14] that will be discussed in this paragraph. For simplicity we will consider only the case when C is a commutative C ∗ -algebra with unit. Given such a C, we let Cb denote the structure space of C, namely the space of equivalence classes of irreducible ∗ -representations (IRR’s) of C. The trivial IRR given by C → {0} is not included in M and will therefore be ignored here and hereafter. Since the C ∗ -algebra C is commutative, every IRR is one-dimensional, i.e. is a (non-zero) linear functional φ : C → C satisfying φ(ab) = φ(a)φ(b) and b The space Cb is made φ(a∗ ) = φ(a), ∀a, b ∈ C. It follows that φ(I) = 1, ∀φ ∈ C. into a topological space by endowing it with the Gel’fand topology, namely with the topology of pointwise convergence on C. Then C can be proved to be a Hausdorff compact topological space. For a commutative C ∗ -algebra, two-irreducible representations are unitarily equivalent if and only if they have the same kernel. Thus one can consider also the space of kernels of IRR’s, the so called primitive spectrum PrimC. Now, these kernels are maximal ideals of C and, vice versa, any maximal ideal is the kernel of b then Ker(φ) is of an irreducible representation [14]. Indeed, suppose that φ ∈ C, codimension 1 and so is a maximal ideal of C. Conversely, suppose that I is a maximal ideal of C, then the natural representation of C on C/I is irreducible, hence one-dimensional. It follows that C/I ∼ = C, so that the quotient homomorphism b Clearly, I =Ker(φ). Thus PrimC C → C/I can be identified with an element φ ∈ C. can be thought of as the space of maximal ideals. As such, PrimC is equipped with the Jacobson or hull kernel topology, that will be described in the next paragraph. The map that to each class of unitary representations associates its kernel gives a map Cb → PrimC, which turns out to be a homeomorphism of the two topological spaces so that we may equivalently talk of the structure space or of the primitive spectrum of a commutative C ∗ -algebra. If c ∈ C, the Gel’fand transform cˆ of c is the complex-valued function on Cb given by cˆ(φ) = φ(c) , ∀φ ∈ Cb . (2.2) It is clear that cˆ is continuous for each c. We thus get the interpretation of elements b The Gel’fand–Naimark theorem states in C as C-valued continuous functions on C. b that all continuous functions on C are indeed of the form (2.2) for some c ∈ C [14]: Proposition 2.1. Let C be a commutative C ∗ -algebra with unit. Then the Gel’fand transform map c 7→ cˆ is an isometric ∗-isomorphism of the C ∗ -algebra C b (equipped with the supremum norm k · k∞ ). onto the C ∗ -algebra C(C)
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Suppose now that M is a compact Hausdorff topological space. We have a natural C ∗ -algebra, C(M ), associated to it. It is then natural to ask what is the \) and M itself. It turns out that this two relation between the Gel’fand space C(M spaces can be identified both setwise and topologically. We notice first that each \) through the evaluation map: m ∈ M gives a complex homomorphism φm ∈ C(M φm : C(M ) → C ,
φm (f ) = f (m) .
(2.3)
Let Im denote the kernel of φm , namely the maximal ideal of C(M ) consisting of all functions vanishing at m. We have the following theorem [14]: Proposition 2.2. The map Φ : m 7→ φm given by (2.3) is a homeomorphism of \), namely M ∼ \). Moreover, every maximal ideal of C(M ) is of M onto C(M = C(M the form Im for some m ∈ M . In conclusion, the previous two theorems set up a one-to-one correspondence between the ∗-isomorphism classes of commutative C ∗ -algebras and the homeomorphism classes of locally compact Hausdorff spaces. If C has a unit, then Cˆ and PrimC are compact. 2.2. Noncommutative algebras and associated spaces The scheme described in the previous section cannot be directly generalized to noncommutative C ∗ -algebras. There is more than one candidate for the analogue of the topological space M . In particular, since non-equivalent unitary transformations may now have the same kernel, we have to distinguish even setwise between: (1) the structure space Ab of A or the space of all unitary equivalence classes of irreducible ∗ -representations;a (2) the primitive spectrum PrimA of A or the space of kernels of irreducible ∗ -representations. Any element of PrimA is automatically a two-sided ∗ ideal of A. One can define a natural topology on both Ab and PrimA. While for a commutative C ∗ -algebra the resulting topological spaces are homeomorphic, this is no longer true in the noncommutative case. For instance, in Sec. (4.1) we will describe a C ∗ -algebra A associated to the Penrose tiling of the plane [6], whose structure space Ab consists of an infinite set of points, whereas PrimA consists of a single point. The topology one puts on Ab is called regional topology [14] and is a generalization of the pointwise convergence we have used in the previous paragraph, to which it a b A is often referred to as the spectrum of the algebra A (see for example [13]). Here we prefer to
call it structure space (as it is done for example in [14]) to avoid any confusion with the concept of primitive spectrum.
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reduces in the commutative case. This topology is constructed by defining a basis of neighborhoods for the points (classes of representations) of Ab as follows. Given b let us denote with HT the Hilbert space of the representation T . Then an a T ∈ A, open neighborhood of T is identified by a finite sequence ξ1 , ξ2 , . . . , ξn of vectors in HT , a positive number and a finite not void set F ⊂ A by means of: U (T ; ; ξ1 , ξ2 , . . . , ξn ; F ) =: {T 0 ∈ Ab : ∃ ξ10 , ξ20 , . . . , ξn0 ∈ HT 0 with |(ξi0 , ξj0 )HT 0 − (ξi , ξj )HT | < , |(T 0 (a)ξi0 , ξj0 )HT 0 − (T (a)ξi , ξj )HT | < for i, j = 1, 2, . . . , n
and ∀a ∈ F } .
(2.4)
On PrimA we instead define a closure operation as follows [13, 14]. Given any subset W of PrimA, the closure W of W is by definition the set of all elements in PrimA containing the intersection of the elements of W , namely \ W = {I ∈ Prim A : W ⊂ I} . (2.5) This “closure operation” satisfies the Kuratowski axioms [15] and thus defines a topology on PrimA, which is called Jacobson or hull-kernel topology. With respect to this topology we have: Proposition 2.3. Let W be a subset of PrimA. Then W is closed if and only if W is exactly the set of primitive ideals containing some subset of A. Proof. If W is closed, by 2.5, W is the set of primitive ideals containing W ⊂ I. Conversely, let V ⊆ A. If W is the set of primitive ideals of A containing T V , then V ⊆ W ⊂ I, for all I ∈ W , so that W ⊂ W , and W = W . T
In general Ab and PrimA fail to be Hausdorff (or T2 ). Recall [15] that a topological space is called T0 if for any two distinct points of the space there is a neighborhood of one of the two points which does not contain the other. It is called T1 if any point of the space is closed. It is called T2 if there exist disjoint neighborhoods of any two points. Whereas nothing can be said concerning the separation properties b it turns out that PrimA is always a T0 space and that it is T1 if and only if all of A, primitive ideals in A are maximal, as it is established by the following propositions [13, 14]. Proposition 2.4. The space PrimA is a T0 space. Proof. Suppose I1 and I2 are two distinct points of PrimA so that say I1 6⊂ I2 . Then the set of those I ∈ PrimA which contain I1 is a closed subset W (by Proposition 2.3) such that I1 ∈ W and I2 6∈ W . Then its complement W c is an open set containing I2 and not I1 . Proposition 2.5. Let I ∈ PrimA. Then the point {I} is closed in PrimA if and only if I is maximal among primitive ideals.
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Proof. Indeed {I} is just the set of primitive ideals of A containing I. As in the commutative case, both Ab and PrimA are locally compact topological spaces. In addition, if A has a unit, then they are compact. Notice that, in general, Ab being compact does not imply that A has a unit. For instance, the algebra K(H) of compact operators on an infinite dimensional Hilbert space H has no unit but its structure space consists of a single point. Let us now come to a comparison between the space Ab and PrimA. There is a canonical surjection of Ab onto PrimA, given by the map that to each IRR π associates its kernel kerπ. The pull-back of the Jacobson topology from PrimA to Ab defines another topology on the latter that turns out to be equivalent to the regional topology defined above [14]. But Ab and PrimA are homeomorphic only under the hypotheses stated below [14]. Proposition 2.6. Let A be a C*-algebra, then the following conditions are equivalent : (i) Ab is a T0 space. (ii) Two irreducible representations of Ab with the same kernel are equivalent. (iii) The canonical map Ab −→ PrimA is a homeomorphism. Proof. By construction, a subset S ∈ Ab will be closed if and only if it is of the form {π ∈ Ab : kerπ ∈ W } for some W closed in PrimA . As a consequence, given b the representation π1 will be in the any two (classes of) representations π1 , π2 ∈ A, closure of π2 if and only if ker π1 is in the closure of ker π2 , or, by Proposition 2.5 if and only if kerπ2 ⊂ kerπ1 . In turn, π1 and π2 are one in the closure of the other if and only if kerπ2 = kerπ1 . Therefore, π1 and π2 will not be distinguished by the topology of Ab if and only if they have the same kernel. On the other side, if Ab is T0 one is able to distinguish points. It follows that (i) implies (ii), namely, that if Aˆ is a T0 space, two representations with the same kernel must be equivalent. The other implications are obvious. 3. Noncommutative Lattices For convenience, we will review in this section the content of [1, 2, 3], where it is shown how it is possible to approximate a continuum topological space by means of a finite or countable set of points P [1] which, being equipped with a partial order relation, is a partially ordered set or a poset. As explained there, these approximating spaces are able to reproduce important topological properties of the continuum. Moreover, in Sec. 4.1 we will see that any of these spaces can be identified with the space Ab = PrimA of primitive ideals of some (noncommutative) AF algebra A, which thus plays the rˆ ole of the algebra of continuous functions on P [5]. This fact will make any poset a truly noncommutative space [1], hence also the name noncommutative lattice. This is the reason why, in this article, we will consider only a special class of algebras, namely postliminal approximately finite (AF) algebras. In Sec. 4 we will
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see in detail that AF algebras are approximated in norm by direct sums of finite dimensional matrices [8, 9]. As for postliminal we refer to [13] for the exact definition. For what concerns this article, we need just to know that, as a consequence of general theorems, this implies that Ab and PrimA are homeomorphic. In other words, in the following we will have to deal only with structure spaces (or primitive spectrum spaces) which are T0 locally compact topological spaces. 3.1. The finite topological approximation Let M be a continuum topological space. Experiments are never so accurate that they can detect events associated with points of M ; rather they only detect events as occurring in certain sets Oλ . It is therefore natural to identify any two points x, y of M if they can never be separated or distinguished by the sets Oλ . Let us assume that each Oλ is open and that the family {Oλ } covers M : [ Oλ . (3.1) M= λ
We also assume that {Oλ } is a topology for M [15]. This implies that both 0λ ∪ 0µ and 0λ ∩ Oµ are in U if Oλ,µ ∈ U. This hypothesis is physically consistent because experiments can isolate events in Oλ ∪ Oµ and Oλ ∩ Oµ if they can do so in Oλ and Oµ separately, the former by detecting an event in either Oλ or Oµ , and the latter by detecting it in both Oλ and Oµ . Given x and y in M , we write x ∼ y if every set Oλ containing either point x or y contains the other too: x ∼ y means x ∈ Oλ ⇔ y ∈ Oλ for every Oλ .
(3.2)
Then ∼ is an equivalence relation, and it is reasonable to replace M by M/ ∼ ≡ P (M ) to reflect the coarseness of observations. We assume that the number of sets Oλ is finite when M is compact so that P (M ) is an approximation to M by a finite set in this case. When M is not compact, we assume instead that each point has a neighborhood intersected by only finitely many Oλ , so that P (M ) is a “finitary” approximation to M [1]. In the notation we employ, if P (M ) has N points, we sometimes denote it by PN (M ). The space P (M ) inherits the quotient topology from M [15], i.e. a set in P (M ) is declared to be open if its inverse image for Φ is open in M , Φ being the map from M to P (M ) obtained by identifying equivalent points. The topology generated by these open sets is the finest one compatible with the continuity of Φ. Let us illustrate these considerations for a cover of M = S 1 by four open sets as in Fig. 1(a). In that figure, O1 , O3 ⊂ O2 ∩ O4 . Figure 1(b) shows the corresponding discrete space P4 (S 1 ), the points xi being images of sets in S 1 . The map Φ : S 1 → P4 (S 1 ) is given by O1 → x1 ,
O2 \ [O2 ∩ O4 ] → x2 ,
O3 → x3 ,
O4 \ [O2 ∩ O4 ] → x4 .
(3.3)
''$$ &&%%
ss s s
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
O2
O1
O3
...
447
x2
Φ
-
x1
O4
x3
x4
Fig. 1. The covering of S 1 that gives rise to the poset P4 (S 1 ).
The quotient topology for P4 (S 1 ) can be read off from Fig. 1, the open sets being {x1 } , {x3 } , {x1 , x2 , x3 } , {x1 , x4 , x3 } ,
(3.4)
and their unions and intersections (an arbitrary number of the latter being allowed as P4 (S 1 ) is finite). Notice that our assumptions allow us to isolate events in certain sets of the form Oλ \ [Oλ ∩ Oµ ] which may not be open. This means that there are in general points in P (M ) coming from sets which are not open in M and therefore are not open in the quotient topology. This implies that in general P (M ) is neither Hausdorff nor T1 . However, it can be shown [1] that it is always a T0 space. For example, iven the points x1 and x2 of P4 (S 1 ), the open set {x1 } contains x1 and not x2 , but there is no open set containing x2 and not x1 . We will see now how the topological properties of P (M ) can also be encoded in a combinatorial structure, namely a partial order relation, that can be defined on it. Since P (M ) is finite (finitary), its topology is generated by the smallest open neighborhoods Ox of its points x. It is possible to introduce a partial order relation [2, 16] by declaring that: x y ⇔ Ox ⊂ Oy .
(3.5)
In this way P (M ) becomes a partially ordered set or a poset. Later, we will write x ≺ y to indicate that x y and x 6= y. A point x ∈ P such that there exists no y ∈ P with x ≺ y(x y) is said to be maximal (minimal). In addition, a set {x1 , x2 , . . . , xk } of points in P is said to be a chain if xj+1 covers xj (j = 1, . . . , k − 1). A chain is maximal if x1 and xk are respectively a minimal and a maximal point. It is easy to read the topology of P (M ) once the partial order is given. It is not difficult to check that Ox = {y ∈ P (M ) : y x} . Indeed, one can even prove a stronger result [1, 2], namely that any finite set P on which a partial order is defined can be made into a finite T0 topological space by declaring that the smallest open neighborhood Ox containing x is given exactly by the above set.
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Throughout this article, we will use “finite poset” and “finite T0 space” interchangeably. It is convenient to graphically represent a poset by a diagram, the Hasse diagram, constructed by arranging its points at different levels and connecting them using the following rules [1, 16]: (1) if x ≺ y, then x is at a lower level than y; (2) if x ≺ y and there is no z such that x ≺ z ≺ y, then x is at a level immediately below y and these two points are connected by a line called a link. Let us consider P4 (S 1 ) again. The partial order reads x1 x2 , x1 x4 , x3 x2 , x3 x4 ,
(3.6)
where we have omitted writing the relations xj xj . The corresponding Hasse diagram is shown in Fig. 2. In the language of partially ordered sets, the smallest open set Ox containing a point x ∈ P (M ) consists of all y preceding x : Ox = {y ∈ P (M ) : y x}. In the x4
x1
s s s s s
@@
@@ @
@
@
s s s s s
x2
@x
3
Fig. 2. The Hasse diagram for the circle poset P4 (S 1 ).
x6
x4
x1
@@
@
@ @@
@@
@
@ @@
x5
@@
x2
@@
x3
Fig. 3. The Hasse diagram for the sphere poset P6 (S 2 ).
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
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449
Hasse diagram, it consists of x and all points we encounter as we travel along links from x to the bottom. In Fig. 2, this rule gives {x1 , x2 , x3 } as the smallest open set containing x2 , just as in (3.4). As one example of a three-level poset, consider the Hasse diagram of Fig. 3 for a finite approximation P6 (S 2 ) of the two-dimensional sphere S 2 derived in [1]. Its open sets are generated by {x1 }, {x3 }, {x1 , x2 , x3 }, {x1 , x4 , x3 } , {x1 , x2 , x3 , x4 , x5 }, {x1 , x2 , x3 , x4 , x6 } ,
(3.7)
by taking unions and intersections. One of the most remarkable properties of a poset is its ability to accurately reproduce the homology and the homotopy groups of the Hausdorff topological space it approximates. For example, as for S 1 , the fundamental group of PN (S 1 ) is Z whenever N ≥ 4 [1]. Similarly, as for S 2 , π1 (P6 (S 2 )) = {0} and π2 (P6 (S 2 )) = Z. This has been widely discussed in our previous work [3, 5], where we argued that global topological information relevant for quantum physics can be captured by such discrete approximations. Furthermore, the topological space being approximated can be recovered by considering a sequence of finer and finer coverings, the appropriated framework being that of projective systems of topological spaces. We refer to [1, 17] for details. In this article we are however mostly concerned with the algebraic properties of a poset, i.e. with the fact that any finite poset can be regarded as the structure space of a C ∗ -algebra. This will be extensively discussed and proved in the following sections, but let us first illustrate a simple example. Consider the following C ∗ algebra: (3.8) A = {λ1 I1 + λ2 I2 + k12 : λj ∈ C, k12 ∈ K12 } L acting on the direct sum of two Hilbert spaces H = H1 H2 and generated by multiples of the identity I1 on H1 , multiples of the identity I2 on H2 and compact operators K12 on the whole Hilbert space H. This algebra admits only three classes of irreducible representations, two finite dimensional ones and an infinite dimensional one: (1) π1 : λ1 I1 + λ2 I2 + k12 7→ λ1 , (2) π2 : λ1 I1 + λ2 I2 + k12 7→ λ2 , (3) ρ : λ1 I1 + λ2 I2 + k12 7→ λ1 I1 + λ2 I2 + k12 . Hence the primitive spectrum consists of only three points p1 = kerπ1 , p2 = kerπ2 , q = kerρ, corresponding respectively to the three representations given above. This space has to be given the Jacobson topology as explained in the previous section. This is easily done if one notices that, the space being finite, this amounts to give a partial order relation on the set {p1 , p2 , q} [14]. Indeed one can show that, on any finite primitive spectrum PrimA of a C ∗ -algebra A, the Jacobson topology is equivalent to the following partial order relation: pj ≺ pk ⇔ kerπj ⊂ kerπk ,
(3.9)
s
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p1
@
@ @@
s
p2
@
q
Fig. 4. The
W
poset, primitive spectrum of A = CI1 + CI2 + K12 .
where pj is the point in PrimA corresponding to the IRR πj of A. Thus in our example, since kerρ ⊂ kerπ1 and kerρ ⊂ kerπ2 , the set {p1 , p2 , q} is equipped with the order relations q ≺ p1 , q ≺ p2 and therefore corresponds to the poset of Fig. 4, W which will be referred to as the poset from now on. 4. AF Algebras 4.1. Bratteli diagrams A C ∗ -algebra A is said to be approximately finite dimensional (AF) [8, 9] if there exists an increasing sequence I0
I1
I2
I3
In−1
In
A0 ,→ A1 ,→ A2 ,→ A3 ,→ · · · ,→ An ,→ · · ·
(4.10)
of finite dimensional subalgebras of A, such that A is the norm closure of ∪n An . Here the maps In are injective ∗ -homomorphisms. In other words, A is the direct limit in the category of C ∗ -algebras with morphisms given by *-algebra maps (not S isometries) of the sequence (An )n∈N . As a set, n An is made of coherent sequences, [ An = {a = (an )n∈N , an ∈ An |∃N0 : an+1 = In (an ), ∀ n > N0 } . (4.11) n
Now the sequence (kan kAn )n∈N is eventually decreasing, since kan+1 k ≤ kan k (the maps In are norm decreasing) and therefore convergent. One writes for the norm k(an )N k = lim kan kAn . n→∞
(4.12)
Since the maps In are injective, the expression (4.12) gives directly a true norm and not simply a seminorm and there is no need to divide out by the zero norm elements. Each subalgebra An , being a finite dimensional C ∗ -algebra, is a matrix algebra L n (n) (dk , C) where M(n) (dk , C) is the and therefore can be written as An = N k=1 M algebra of dk × dk matrices with complex coefficients. Given any two such matrix L 1 L 2 (1) (2) (dj , C) and A2 = N (dk , C) with A1 ,→ A2 , one algebras A1 = N j=1 M k=1 M can always choose suitable bases in A1 and A2 such that A1 is identified with a subalgebra of A2 of the following form [8]: N2 N1 M M Nkj M(1) (dj , C) . (4.13) A1 ' k=1
j=1
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
...
451
Here, for any nonnegative integers p and q, the symbol p M(q, C) stands for M(q, C)⊗ Ip . In (4.13), the coefficients Nkj represent the multiplicity of the partial embedding of M(1) (dj , C) in M(2) (dk , C) for each k and j and satisfy the condition N1 X
Nkj dj = dk .
(4.14)
j=1
A useful way to represent the algebras A1 , A2 and the embedding A1 ,→ A2 is by means of a diagram, the Bratteli diagram [8], which can be constructed out of the dimensions, dj (j = 1, . . . , N1 ) and dk (k = 1, . . . , N2 ), of the diagonal blocks of the two algebras and the numbers Nkj that describe the partial embeddings. To construct the diagram, we draw two horizontal rows of vertices, the top (bottom) one representing A1 (A2 ) and consisting of N1 (N2 ) vertices, labeled by the corresponding dimensions d1 , . . . , dN1 (d1 , . . . , dN2 ). Then for each j = 1, . . . , N1 and k = 1, . . . , N2 , we draw Nkj edges between dj and dk . We will also write (1) (2) (1) (2) dj &Nkj dk to denote the fact that M(dj , C) is embedded in M(dk , C) with multiplicity Nkj . By repeating the procedure at each level, we obtain a semi-infinite diagram denoted by D(A) which completely defines A up to isomorphisms. Notice that the diagram D(A) depends not only on A but also on the particular sequence {An }n∈N which generates A. However, it is possible to show [8] that all diagrams corresponding to AF algebras which are isomorphic to A can be obtain from the chosen D(A) by means of an algorithm. As an example of an AF algebra, let us consider the subalgebra A of the algebra B(H) of bounded operators on H = H1 ⊕ H2 given in (3.8). This C ∗ -algebra can be obtained as the direct limit of the following sequence of finite dimensional algebras: A0 = M(1, C) A1 = M(1, C) ⊕ M(1, C) A2 = M(1, C) ⊕ M(2, C) ⊕ M(1, C) .. . An = M(1, C) ⊕ M(2n − 2, C) ⊕ M(1, C) .. . where, for n ≥ 1, An is embedded in An+1 as the subalgebra M(1, C) ⊕ [M(1, C) ⊕ M(2n − 2, C) ⊕ M(1, C)] ⊕ M(1, C):
λ1 an = 0 0
0 m2n−2×2n−2 0
λ1
0 0 0 ,→ 0 λ2 0 0
0
0
0
λ1 0
0 m2n−2×2n−2
0 0
0 0
0 0
λ2 0
0
. 0 λ2 0 0
(4.15)
452
ss ss
s ss s
ss ss
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1
@
@
@1 @ @ 1 @ @2 1 @ 1 @ @4 1 @ 1 @ @6 1 1
.. .
.. .
Fig. 5. The Bratteli diagram corresponding to A = CI1 + CI2 + K12 .
It is therefore described by the diagram of Fig. 5. As a second example, consider the C ∗ -algebra of the Penrose tiling. This is an example of an AF algebra which is not postliminal, since this algebra admits an infinite number of nonequivalent representations all with the same kernel. At each level, the finite dimensional algebra is given by [6] An = M(dn , C) ⊕ M(d0n , C) , with inclusion An ,→ An+1 : A 0 A 0 ,→ 0 B 0 B 0 0
0 0 ; A ∈ M(dn , C) , A
ss s ss ss s s
n ≥ 1,
B ∈ M(d0n , C) ,
1
@
@ HH @ 1 HHH 2H HHHH 1 3 HH HH 2 HHH H 5H H 3 1
.. .
Fig. 6. The Bratteli diagram of the Penrose tiling.
(4.16)
(4.17)
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
...
453
so that dn+1 = dn + d0n and d0n+1 = dn . The corresponding Bratteli diagram is shown in Fig. 6. To conclude this section we remark that an AF algebra is commutative if and only if all its factors M(n) (dk , C) are one dimensional, i.e. they are just C. Thus the corresponding diagram has the property that for each M(n) (dk , C) with n ≥ 1 there is exactly one M(n−1) (dj , C) and M(n−1) (dj , C) &pkj M(n) (dk , C) with pkj = 1. An interesting example is given in Fig. 7, which corresponds to the AF C ∗ -algebra of continuous functions on the Cantor set [10].
s
s s ss s s s s s s s s s s HH HH 1
1 ZZ ZZ 1 1
JJ
JJ
J@1
1 J@1 1
..@ .. ..@ .. .
.
.
.
HHH HZ1 ZZ Z1 1
J
JJ
J
J@1 1
1J 1
..@ ..@ ..@ .. .
.
.
.
Fig. 7. The Bratteli diagram corresponding to the AF C ∗ -algebra of continuous functions on the Cantor set.
4.2. From Bratteli diagrams to posets The Bratteli diagram D(A) of an AF algebra A is useful not only because it gives the finite approximations of the algebra explicitly, but also because it is possible to read the ideals and the primitive ideals of the algebra (hence the topological properties of PrimA) out of it very easily. Indeed one can show that the following proposition holds [8]: Proposition 4.1. (1) There is a one-to-one correspondence between the proper ideals I of A and the subsets Λ = ΛI of the Bratteli diagram satisfying the following two properties: (i) if M(n) (dk , C) ∈ Λ and M(n) (dk , C) & M(n+1) (dj , C) then necessarily M(n+1) (dj , C) belongs to Λ; (ii) if all factors M(n+1) (dj , C) (j = {1, 2, . . . , Nn+1 }), for which M(n) (dk , C) & M(n+1) (dj , C), belong to Λ, then M(n) (dk , C) ∈ Λ. (2) A proper ideal I of A is primitive if and only if the associated subdiagram ΛI satisfies: (iii) ∀n there exists an M(m) (dj , C), with m > n, not belonging to ΛI such that, for all k ∈ {1, 2, . . . , Nn } with M(n) (dk , C) not in ΛI , one can find
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a sequence M(n) (dk , C) & M(n+1) (dα , C) & M(n+2) (dβ , C) & · · · & M(m) (dj , C). For example, consider the diagram of Fig. 5, representing the AF C ∗ -algebra A = CI1 + CI2 + K12 already discussed in Sec. 3.1. This algebra contains only three nontrivial ideals, whose diagrams are represented in Figs. 8(a), (b), (c). In these pictures the points belonging to the ideals are marked with a “♣”. It is not difficult to check that only I1 and I2 are primitive ideals, since I3 does not satisfy property (iii) above.
ss ss s
@I @@ ♣ @ @@ @@ ♣ ♣ @ @@ ♣ ♣ @ @♣ ♣ .. .
1
.. .
ss sss ss ss ss s s s
@I @@ ♣ @ @@ ♣ @@ ♣ @ ♣ @@ ♣ ♣ @ @♣
(a)
@I @@
2
.. .
.. .
(b)
3
@ @@ @@ ♣ @ @@ ♣ @ @♣ .. .
.. .
(c)
Fig. 8. The representation of the ideals of A = CI1 + CI2 + K12 in the corresponding Bratteli diagram.
We remark the following here: (1) The whole A is an ideal, which by definition is not primitive since the trivial representation A → 0 is not irreducible. (2) The set {0} ⊂ A is an ideal, which is primitive if and only if A has one irreducible faithful representation. This can also be understood from the Bratteli diagram in the following way. The set {0} is not a subdiagram of D(A), being represented by the element 0 of the matrix algebra of each finite level, so that there is at least one element a ∈ A not belonging to the ideal {0} at any level. Thus to check if {0} is primitive, i.e. to check property (iii) above, we have to examine whether all the points at a given level, say n, can be connected to a single point at a level m > n. For example this is the case for the diagram of Fig. 5 and not for that of Fig. 7. Proposition 4.1 above allows us to understand the topological properties of PrimA at once. This becomes particularly simple if the algebra admits only a finite number of nonequivalent irreducible representations. In this case PrimA is a T0 topological space with only a finite number of points, hence a finite poset P . To reconstruct the latter we just need to draw the Bratteli diagram D(A) and find
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
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the subdiagrams that, according to properties (i), (ii), (iii), correspond to primitive ideals. Then P has so many points as the number of primitive ideals and the partial order relation in P that determines the T0 topology is simply given by the inclusion relations that exist among the primitive ideals. As an example consider again Fig. 5. We have seen that the corresponding AF algebra has only three primitive ideals: the {0} ideal and the ideals I1 , I2 W represented in Fig. 8(a), (b). Clearly {0} ⊂ I1 , I2 so that PrimA is the poset of Fig. 4. Figure 7 leads to another interesting topological space. As we have mentioned, such a diagram corresponds to a commutative AF algebra C and hence to a Hausdorff PrimC, which is homeomorphic to the Cantor set. 4.3. From posets to Bratteli diagrams In the preceding subsection we have described the properties of the Bratteli diagram D(A) of an AF algebra A and in particular we have seen how, out of it, it is possible to read the primitive spectrum of A, in particular when the latter is a finite poset. In the following we will see that, under some rather mild hypotheses which are always verified in the cases of interest to us, it is possible to reverse the construction and thus build the AF algebra that corresponds to a given (finite) T0 topological space. Such a reconstruction rests on the following theorem of Bratteli and Elliott [18, 19], which specifies a class of topological spaces which are the primitive spectra of AF algebras: Proposition 4.2. A topological space Y is the primitive spectrum PrimA of an AF algebra A if it has the following properties: (i) Y is T0 ; (ii) Y contains at most a countable number of closed sets; (iii) if {Fn }n∈Λ , Λ being any direct set, is a decreasing sequence of closed subsets of Y, then ∩n Fn is an element in {Fn }n∈Λ ; (iv) if F ⊂ Y is a closed set which is not the union of two proper closed subsets, then F is the closure of a one-point set. It is not difficult to check that all the above conditions hold true if Y is a T0 topological space with a finite number of points, so that we have the corollary: Corollary. A finite poset P is the primitive spectrum PrimA for some AF algebra A. Here we will not report the proof of Proposition 4.2, which can be found in [18]. Also a more general characterization of spaces arising as the primitive spectrum of a separable AF algebra has been given in [19]. Here, starting from the techniques used in such a proof, we want to show only how one can explicitly find an AF algebra A whose primitive spectrum is a given finite poset P . First we will give the general construction and then discuss an example.
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Let {K1 , K2 , K3 , . . .} be the collection of all closed sets in P , where K1 = P . To construct the nth level of the Bratteli diagram D(A), we consider the subcollection of closed sets Kn ≡ {K1 , K2 , . . . , Kn } and denote with Kn0 the smallest collection of closed sets in P that contains Kn and is closed under union and intersection. The collection Kn determines a partition of the topological space P , by taking intersections and complements of the sets Kj ∈ Kn (j = 1, . . . , n). We denote with Y (n, 1), Y (n, 2), . . ., Y (n, kn ) the sets of such partition. Also, we write F (n, j) for the smallest closed set which contains Y (n, j) and belongs to the subcollection Kn0 . Then we can construct a Bratteli diagram following the rules: (1) the nth level of D(A) has kn points, one for each set Y (n, j); (2) the point at the level n of the diagram corresponding to Y (n, α) is linked to the point at the level n + 1 corresponding to Y (n + 1, β) if and only if Y (n, α) ∩ F (n + 1, β) 6= ∅. In this case, the multiplicity of the embedding is always 1. To illustrate this construction, let us consider the {p1 , p2 , q}. Now there are four closed sets:
W
poset of Fig. 4: P =
K1 = {p1 , p2 , q} , K2 = {p1 } , K3 = {p2 } , K4 = {p1 , p2 } . Thus it is not difficult to check that: K1 = {K1 }
K10 = {K1 }
Y (2, 1) = {p1 , p2 , q} ⊂ F (1, 1) = K1
K2 = {K1 , K2 }
K20 = {K1 , K2 }
Y (2, 1) = {p1 } Y (2, 2) = {p2 , q}
K3 = {K1 , K2 , K3 }
K30 = {K1 , K2 , K3 , K4 } Y (3, 1) = {p1 } Y (3, 2) = {q} Y (3, 3) = {p2 }
⊂ F (3, 1) = K2 ⊂ F (3, 2) = K1 ⊂ F (3, 3) = K3
K4 = {K1 , K2 , K3 , K4 } K40 = {K1 , K2 , K3 , K4 } Y (4, 1) = {p1 } Y (4, 2) = {q} Y (4, 3) = {p2 } .. .
⊂ F (4, 1) = K2 ⊂ F (4, 2) = K1 ⊂ F (4, 3) = K3
⊂ F (2, 1) = K2 ⊂ F (2, 2) = K1
Notice that, since P has only a finite number of points and hence a finite number of closed sets, the partition of P we have to consider at each level n repeats itself after a certain point (n = 3 in this case). Figure 9 shows the corresponding diagram, obtained through rules (1) and (2) above. Recalling then that the first matrix algebra that gives origin to an AF algebra is C and using the fact that all the embeddings have multiplicity one, we eventually obtain the sequence of finite dimensional algebras shown by the Bratteli diagram of Fig. 5. As we have said previously, such a diagram corresponds to the AF algebra A = CI1 + CI1 + K12 . It is a general fact that the Bratteli diagram describing any finite poset “stabilizes”, i.e. repeats itself, after a certain level n0 , when the family Kn0 of closed sets
ss ss
s ss s
ss ss
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
Y21 Y31 Y41 Y51
Y11
@@
@@ @ @@ Y @ @@ Y @ @Y .. .
32
42
51
.. .
...
457
@Y
22
Y33 Y43 Y53
Fig. 9. The construction of the Bratteli diagram of the AF algebra corresponding to the of Fig. 4.
W
poset
we choose is such that it determines a partition of the poset which distinguishes each point of the poset itself. In particular, this is the case if we choose n0 in such a manner that Kn0 contains all closed sets. Then, each Y (n0 , j) will contain a single point of the poset and F (n0 +1, j) will be the smallest closed set containing Y (n0 , j). It is only this stable part of the diagram which is relevant for the inductive limit and hence for the determination of the AF algebra it represents. Indeed, diagrams (or sequences of finite dimensional algebras) that differ only for a finite numbers of initial levels give different finite approximations to the same AF algebra [8, 9]. To conclude this section, we want to describe the AF algebras whose primitive spectra are the poset approximations of the circle, P4 (S 1 ), and of the sphere, P6 (S 2 ). As for P4 (S 1 ), given in Fig. 2, the Bratteli diagram repeats itself for n > n0 = 4 and the stable partition is given by Y (n0 , 1) = {x2 }
F (n0 + 1, 1) = {x2 }
Y (n0 , 2) = {x1 } Y (n0 , 3) = {x3 }
F (n0 + 1, 2) = {x1 , x2 , x4 } F (n0 + 1, 3) = {x2 , x3 , x4 }
Y (n0 , 4) = {x4 }
F (n0 + 1, 4) = {x4 } .
(4.18)
The corresponding Bratteli diagram is in Fig. 10. The set {0} is not an ideal. The limit algebra A turns out to be a subalgebra of bounded operators on the Hilbert space H = H1 ⊕ · · · ⊕ H4 , with Hi , i = 1, . . ., 4 infinite dimensional Hilbert spaces: (4.19) A = CI13 ⊕ CI24 ⊕ K12 ⊕ K34 . Here Iij and Kij denote the identity operator and the algebra of compact operators on Hi ⊕ Hj respectively. For the poset P6 (S 2 ) for the two-dimensional sphere, given in Fig. 3, n0 = 6 and the stable partition is given by
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Y (n0 , 1) = {x5 } Y (n0 , 2) = {x2 }
F (n0 + 1, 1) = {x5 } F (n0 + 1, 2) = {x2 , x5 , x6 }
Y (n0 , 3) = {x1 } Y (n0 , 4) = {x3 }
F (n0 + 1, 3) = {x1 , x2 , x4 , x5 , x6 , } F (n0 + 1, 4) = {x2 , x3 , x4 , x5 , x6 }
Y (n0 , 5) = {x4 } Y (n0 , 6) = {x6 }
F (n0 + 1, 5) = {x4 , x5 , x6 } F (n0 + 1, 6) = {x6 } .
(4.20)
ss ss ss ss ssss
The corresponding Bratteli diagram is in Fig. 11.
The set {0} is not an ideal. The inductive limit is a subalgebra of bounded operators on the Hilbert space H = (H1 ⊕ H3 ) ⊗ (H5 ⊕ H6 ) ⊕ (H2 ⊕ H4 ) ⊗ (H7 ⊕ H8 ) .. .
H@H@HH @HHH H@H@ H @HHH @ H @ .. .
Fig. 10. The stable part of the Bratteli diagram for the circle poset P4 (S 1 ).
s s s s s s s s s s s s s s s s s s .. .
H PPHPHP XXXXX @ H @ @H H H@ HP X P H X P X @X H P X @ X P H X H XX @ HP @ H X X H P X @ HP HP P @ X H XHXXHXX P @@ HHHP @ P PPPHHXXXX H @ H PPHPHPH XXXXX @ H @ H PHP XX @ H @ X P H X H P X @H P X @ H X P H X H P X @@HHP @ H X PX X H P X H P @ X H X P @ HHH P @PPHXPXHXHXXX @@ HH@ PPHPHPHXXXXX H @ H@H PHPHP XXXX @ X P H X H P X @@HP PPXX @@HH HX .. . Fig. 11. The stable part of the Bratteli diagram for the sphere poset P6 (S 2 ).
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with Hi , i = 1, . . ., 8 infinite dimensional Hilbert spaces, given by: A = CIH1 ⊗(H5 ⊕H6 )⊕H2 ⊗(H7 ⊕H8 ) ⊕ CIH3 ⊗(H5 ⊕H6 )⊕H4 ⊗(H7 ⊕H8 ) ⊕ (KH1 ⊕H3 ⊗ CIH5 ⊕H6 ) ⊕ (KH2 ⊕H4 ⊗ CIH7 ⊕H8 ) ⊕ KH5 ⊗(H1 ⊕H3 )⊕H7 ⊗(H2 ⊕H4 ) ⊕ KH6 ⊗(H1 ⊕H3 )⊕H8 ⊗(H2 ⊕H4 ) . 5. The Behncke Leptin Construction Given a poset P , there is always an AF algebra A such that Aˆ = P . A particular procedure to find such an algebra has been described in the previous section, but it is known that there exists more than one C ∗ -algebra A whose primitive spectrum is P . For example, if P consists of a single point, we can take for A any of the C ∗ -algebras M(n, C) of all n × n matrices valued in C. It is natural then to ask what are all the algebras associated to a given finite T0 topological space P . This problem was solved by Behncke and Leptin in 1973. In [11] they give a complete classification of all separable C ∗ -algebras A with finite primitive spectra. Such classification requires the definition of a function d on P , called defector, valued in IN = {∞, 0, 1, 2, . . .}. Given P and d, the Behncke– ˆ d) = P . Leptin construction gives a separable C ∗ -algebra A(P, d) such that A(P, Furthermore, any separable C ∗ -algebra A satisfying Aˆ = P is isomorphic to A(P, d) for some d. A defector d on the poset P is an IN -valued function on P such that d(x) > 0 if x is maximal.
(5.22)
Two defectors d and d0 are declared to be equal if there exists an automorphism ϕ of P such that d0 = d ◦ ϕ. They are called immediately equivalent if d(x) = d(x0 ) for all x ∈ P with the exception of at most one nonmaximal y ∈ P , such that d(y) = d0 (y) + d0 (z) or d0 (y) = d(y) + d(z) for some z coveringc y, if d(z) = d0 (z) < ∞. In the case d(z) = ∞, d(y) and d0 (y) may be arbitrary. Then two defectors are defined to be equivalent (d ∼ d0 ) if there exists a finite sequence of immediately equivalent defectors connecting them. We will start by describing the Behncke–Leptin construction for a special class of posets called forests. Then we will give the generalization for an arbitrary finite poset. 5.1. The Behncke Leptin construction for a forest A forest is a poset F such that {x, y, z ∈ F, x z, y z} ⇒ {x y or y x} . c We
say that y covers x if x ≺ y and there is no z such that x ≺ z ≺ y.
(5.23)
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E. ERCOLESSI, G. LANDI and P. TEOTONIO-SOBRINHO
Given a forest F and a defector d on F , the Behncke–Leptin construction consists of the following steps. First we introduce a Hilbert space H(F, d) associated to the whole forest F . Second, for each point x ∈ F , we introduce a subspace H(x) ⊆ H(F, d) and a set of operators Rx acting on H(x). Actually, Rx can be thought of as acting on the whole H(F, d) by defining its action on the complement of H(x) to be zero. Then the C ∗ -algebra A associated to the forest is the one generated by the Rx ’s as x varies in F . Now we explain how to determine H(F, d), H(x) and Rx . The Hilbert space H(F, d) can be obtained using an auxiliary forest F 0 constructed from F in the following way. The forest F 0 contains a point x(1) for each maximal point x ∈ F (1) (2) and a pair of points xi and xi for each non maximal point xi ∈ F . Then on (2) (1) F 0 we introduce a partial order by declaring that xi is covered by both xj and (2)
s
s s
s
xj if and only if xi is covered by xj . Figure 12 shows an example of F and the corresponding F 0 . x3
@
@@
s s
(1)
x4
@@ x
x3
@
(1) 1
x1
ss ss
@@ @@ x A x AA AA x x (1) 2
2
(1)
x4
(2) 2
(2) 1
F0
F
Fig. 12. An example of a forest F and the auxiliary forest F 0 .
In F 0 we consider the maximal chains Cα = {x1 be seen to be necessarily of the form
(p1 )
(2)
(2)
(2)
(p2 )
, x2
(p )
, . . . , xk k }, which can
(1)
Cα = {x1 , x2 , . . . , xk−1 , xk } .
(5.24)
For example, in F 0 of Fig. 12, the maximal chains are {x1 }, {x1 , x2 }, {x1 , x2 , (1) (2) (2) (1) x3 }, {x1 , x2 , x4 }. To each maximal chain Cα we associate the Hilbert space (1)
h(Cα ) = lx1 ⊗ lx2 ⊗ . . . ⊗ lxk−1 ⊗ Cd(xk ) ,
(2)
(1)
(2)
(2)
(5.25)
where d(xk ) is the value of the defector d at the point xk ∈ F and lxi can be realized as the Hilbert space `2 of sequences (f1 , f2 , . . .) of complex numbers with P 2 n |fn | < ∞. We then define the total Hilbert space H(F, d) associated to F to be M h(Cα ) , (5.26) H(F, d) = α
where we sum over all maximal chains Cα in F 0 .
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In a similar way, we introduce the subspaces H(xi ) associated to a single point xi ∈ F by M (p) h(Cβ ) for all Cβ such that xi ∈ Cβ . (5.27) H(xi ) = β
Notice that if we consider the subforest Fx of F given by Fx = {y ∈ F |x y} ,
(5.28)
we can construct the Hilbert space H(Fx , dx ), where dx is the restriction of d to Fx . An important property of H(x) defined in (5.27) is that it satisfies H(x) = Hx ⊗ H(Fx , dx ) where Hx =
O
lxi for all xi ≺ x
(5.29)
(5.30)
i
and Hx = C if x is a minimal point. Now we are ready to define the C ∗ -algebra A(F, d). First, let us introduce the algebra of operators Rx , acting on H(x), given by Rx = CIHx ⊗ K(H(Fx , dx )) ,
(5.31)
IHx being the identity operator on Hx and K(H(Fx , dx )) being the algebra of compact operators on H(Fx , dx ). In other words, Rx acts as multiples of the identity on the Hilbert space Hx determined by the points xi ≺ x which precede x, as in (5.30), and as compact operators on the Hilbert space H(Fx , dx ) determined by the points xj x which follow x. Then A(F, d) is the algebra of operators on H(F, d) generated by all Rx as x varies in F . The algebras Rx , with x ∈ F , satisfy: Rx Ry ⊂ Rx if x y and Rx Ry = 0 if x and y are incomparable .
(5.32)
One of the major results of [11] is the following theorem, which establishes that the primitive spectrum of the C ∗ -algebra A(F, d) constructed according to the rules given above is homeomorphic to the forest F : Proposition 5.1. Let F be a finite forest with defector d and A(F, d) the algebra of operators on H(F, d) defined as above. Then we have: N (i) if E is a closed subset of F with complement U, then IE = x∈U Rx is a N closed two-sided ideal of A(F, d), and AE = x∈E Rx is a closed subalgebra of A(F, d); (ii) every two-sided ideal of A(F, d) is of the form IE for some closed E ⊂ F ˆ d) = F . and IE is primitive iff E = {x}. In particular, A(F,
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E. ERCOLESSI, G. LANDI and P. TEOTONIO-SOBRINHO
s
s
Let us illustrate the Behncke–Leptin construction for a very simple forest, W namely the poset of Fig. 4. The correspondent associated forest P 0 is illustrated in Fig. 13.
ss
(1)
p1
@@ @
(1)
p2
@@
q (1)
q (2)
Fig. 13. The forest associated to the
W
poset.
We consider a generic defector d. From the diagram of P 0 in Fig. 13 we can write down all its maximal chains: (1)
(1)
{q (2) , p1 }, {q (2) , p2 }, {q (1) } and following (5.25) and (5.26) we see that H(F, d) is given by H(F, d) = lq ⊗ Cd(p1 ) ⊕ lq ⊗ Cd(p2 ) ⊕ Cd(q) .
(5.33)
The subspaces H(xi ) can also be determined from the diagram of P 0 : H(p1 ) = lq ⊗ Cd(p1 ) H(p2 ) = lq ⊗ Cd(p2 ) H(q) = H(F, d) .
(5.34)
Notice that the factorization expressed in (5.29) is satisfied, where now Hp1 = lq , H(Fp1 , dp1 ) = Cd(p1 ) Hp2 = lq , H(Fp2 , dp2 ) = Cd(p2 ) Hq = C, H(Fq , dq ) = H(F, d) .
(5.35)
The C ∗ -algebra A(F, d) is generated by all Rx , x ∈ F . The latter reads Rp1 = CIHp1 ⊗ K(Cd(p1 ) ) Rp2 = CIHp2 ⊗ K(Cd(p2 ) ) Rq = K(H(F, d)) .
(5.36)
Notice that for the defector d(p1 ) = d(p2 ) = 1 and d(q) = 0 we get H(F, d) = W H1 ⊕ H2 and A = CI1 + CI2 + K12 and thus recover the algebra we got for the poset via the Bratteli construction in Sec. 4.1.
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5.2. The Behncke Leptin construction for posets To generalize the procedure of the last section to an arbitrary poset P with defector d, we have first to introduce a forest P , uniquely determined by P . Let P be a finite poset. A rope r of P is a (not necessarily maximal) chain in P starting from a minimal element and ending at some x ∈ P . The set P of all ropes of P ordered by inclusion is a poset. One can show that P is in fact a forest. Let ϕ : P → P denote the surjective map which assigns to each rope r ∈ P its end point ϕ(r) ∈ P . Following [11], we will call the pair (P , ϕ) the covering forest of P . An example is given in Fig. 14, which shows the covering forest of the circle poset P4 (S 1 ) of Fig. 2.
s
{x1 , x2 }
@@ @@
s
s s
{x1 , x4 }
{x3 , x2 }
@@ @@
@
{x1 }
s
s
{x3 , x4 }
@
{x3 }
Fig. 14. The covering forest of the circle poset P4 (S 1 ).
Given a defector d on P we define a defector d on P in a natural way via the pull-back: d = d ◦ ϕ. (5.37) Then, since P is a forest, we can construct the algebra A(P , d) following Sec. 5.1. Finally, to identify the C ∗ -algebra A(P, d) associated to the poset P and the defector d, we proceed to realize A(P, d) as a subalgebra of A(P , d). In order to do so, we need to point out a simple property of the covering forest (P , ϕ). Let r, s ∈ P be in the inverse image ϕ−1 (x) of x ∈ P . Then, the subforest (P )r (see (5.28)) is naturally isomorphic to (P )s . Indeed, (P )r and (P )s consist of all extensions of the rope r and s respectively. By hypothesis, r and s have the same end point x ∈ P , so that (P )r ∼ (P )s . Thus K H((P )r , dr ) ' K H((P )s , ds ) ≡ Kx , so that the algebras Rs , Rr ∈ A(P , d) are given by Rr = CIHr ⊗ Kx , Rs = CIHs ⊗ Kx . For each x ∈ P we define the algebra Ax M Ax = r∈ϕ−1 (x)
Rr
(5.38)
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and a subalgebra Rx ⊂ Ax given by all elements a ∈ Ax of the form a = (λr1 IHr1 ⊗ k) + (λr2 IHr2 ⊗ k) + · · · + (λrn IHrn ⊗ k) , where ri ∈ ϕ−1 (x), λrj ∈ C and k ∈ Kx . Thus Rx = {a ∈ Ax |a =
M
(λr IHr ⊗ k), λr ∈ C and k ∈ Kx } .
(5.39)
r∈ϕ−1 (x)
The C ∗ -algebra A(P, d) that satisfies ˆ d) = P A(P,
(5.40)
is then generated by all Rx with x ∈ P . There is an intuitive interpretation for (5.39). The poset P can be obtained from P by identifying any two ropes r and s that have the same ending point. Equation (5.39) simply expresses this identification at an algebraic level. For example, for the circle poset P4 (S 1 ) these rules give the following algebras, acting on H = H1 ⊕ H2 ⊕ H3 ⊕ H4 : Ax4 = CI1 + CI3 Ax2 = CI2 + CI4 Ax1 = CI1 + CI2 + K12 Ax3 = CI3 + CI4 + K34 ,
(5.41)
if one chooses the defector d(x1 ) = d(x2 ) = 1, d(x3 ) = d(x4 ) = 0. Thus the algebra associated to P4 (S 1 ) is A = CI1 + CI2 + CI3 + CI4 + K12 + K34 .
(5.42)
As before this is the algebra one gets for P4 (S 1 ) by means of the Bratteli construction explained in Sec. 4.3. Equivalent defectors give rise to isomorphic C ∗ -algebras, whereas by choosing different non-equivalent defectors one can construct non-isomorphic C ∗ -algebras that all have P as primitive spectrum. In this way one can obtain all C ∗ -algebras A whose (finite) spectrum Aˆ is homeomorphic to the poset P , as it is established in [11], which we quote to conclude this section: Proposition 5.2. (i) Every separable C ∗ -algebra A with finite dual Aˆ = P is isomorphic to some A(P, d). (ii) A(P, d) is isomorphic to A(P, d0 ) if and only if d and d0 are equivalent.
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6. Final Remarks In this article, we have seen how a finite poset is truly a “noncommutative space” or “noncommutative lattice”, since it can be described as the primitive spectrum of a noncommutative C ∗ -algebra A, which turns out to be always a postliminal AF algebra. We have also seen that this correspondence is not one-to-one, more than one non-isomorphic C ∗ -algebra leading to the same poset. This relation between posets and C ∗ -algebras was used in [17] to give a dualization of the approximation method for topological spaces introduced in [1]. In our previous work [5] we have showed how it is possible to construct a quantum theory on posets, by making use of the corresponding C ∗ -algebra. We have also seen how important topological properties of the continuum, such as homotopy, can be captured by the poset approximation and manifest themselves in the corresponding quantum mechanics. We are thus naturally led to examine how one can construct further geometric structures on posets, as is suggested by Connes’ noncommutative geometry [6]. First of all, we are interested in the construction of bundles and characteristic classes over a poset and, as a first step in this direction, one should examine the K-theory of these noncommutative lattices. This is the topic discussed in [12], where we present a study of the algebraic K-theory of AF algebras associated to a poset. Then one would like to construct bundles, and notably nontrivial bundles, over a poset, and consider, for instance, the analogue of the monopole bundle over the lattice approximating the two-dimensional sphere and of nontrivial “topological charges”. Work in this direction is in progress. Acknowledgements This work was initiated while the authors were at Syracuse University. We thank A. P. Balachandran, G. Bimonte, F. Lizzi e G. Sparano for many fruitful discussions and useful advice. The final version was written while G. L. and P. T-S were at ESI in Vienna. They would like to thank G. Marmo and P. Michor for the invitation and all people at the Institute for the warm and friendly atmosphere. We thank the “Istituto Italiano per gli Studi Filosofici” in Napoli for partial support. The work of P. T-S. was also supported by the Department of Energy, U.S.A. under contract number DE-FG-02-84ER40173. The work of G. L. was partially supported by the Italian “Ministero dell’ Universit`a e della Ricerca Scientifica”. References [1] R. D. Sorkin, Int. J. Theor. Phys. 30 (1991) 923. [2] P. S. Aleksandrov, Combinatorial Topology, Vols. 1-3, Greylock, 1960. [3] A. P. Balachandran, G. Bimonte, E. Ercolessi and P. Teotonio-Sobrinho, Nucl. Phys. B418 (1994) 923. [4] A. P. Balachandran, G. Bimonte, G. Landi, F. Lizzi and P. Teotonio-Sobrinho, “Lattice Gauge Fields and noncommutative geometry” (preprint ESI 299, 1995, hepth/9604012), to appear in J. Geom. Phys. [5] A. P. Balachandran, G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. Teotonio-Sobrinho, Nucl. Phys. B 37C Proc. Suppl., 20 (1995); J. Geom. Phys. 18
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(1996) 163. [6] A. Connes, Noncommutative Geometry, Academic Press, 1994; G. Landi, An introduction to Noncommutative Spaces and their Geometries, Springer-Verlag, 1997. [7] J. C. V´ arilly and J. M. Gracia-Bond´ia, J. Geom. Phys. 12 (1993) 223. [8] O. Bratteli, Trans. Amer. Math. Soc. 171 (1972) 195. [9] K. R. Goodearl, Notes on Real and Complex C ∗ -algebras , Shiva Publishing Limited. [10] G. A. Elliott, J. Alg. 38 (1976) 29. E. G. Effros, Dimension and C ∗ -algebras, Amer. Math. Soc., 1981. [11] H. Behncke and H. Leptin, J. Functional Analysis 14 (1973) 253; 16 (1974) 241. [12] E. Ercolessi, G. Landi and P. Teotonio-Sobrinho, “K-theory of noncommutative lattices” (preprint, 1995); (9-alg/9607017). [13] J. Dixmier, C ∗ -algebras , North-Holland, 1982. [14] J. M. G. Fell and R. S. Doran, Representations of ∗ -Algebras, Locally Compact Groups and Banach ∗ -Algebraic Bundles, Academic Press, 1988. [15] J. L. Kelley, General Topology, Springer-Verlag, 1955. J. Hocking, G. Young, Topology, Dover, 1988. [16] R. P. Stanley, Enumerative Combinatorics, Vol. 1, Wordsworth and Brooks/Cole Advanced Books and Software, 1986. [17] G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. Teotonio-Sobrinho, “Lattices and their continuum limits”, J. Geom. Phys. 20 (1996) 318; (hepth/9507147). G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. TeotonioSobrinho, “Noncommutative lattices and their continuum limits”, J. Geom. Phys. 20 (1996) 329 (hep-th/9507148). [18] O. Bratteli, J. Functional Analysis 16 (1974) 192. [19] O. Bratteli and G. A. Elliott, J. Functional Analysis 30 (1979) 74.
ON THE SUPER-UNITARITY OF DISCRETE SERIES REPRESENTATIONS OF ORTHOSYMPLECTIC LIE SUPERALGEBRAS AMINE M. EL GRADECHI D´ epartement de Math´ ematiques, Facult´ e des sciences Jean Perrin Universit´ e d’Artois, rue Jean Souvraz S.P. 18, 62307 Lens, France and U.R.A. C.N.R.S. 0751 D E-mail : [email protected] E-mail : [email protected] Received 5 April 1997 Revised 2 July 1997 1991 Mathematical Subject Classification: 17A70, 22E43, 22E45, 46C05, 47A05, 47A67, 47B25 We investigate the notion of super-unitarity from a functional analytic point of view. For this purpose we consider examples of explicit realizations of a certain type of irreducible representations of low rank orthosymplectic Lie superalgebras which are super-unitary by construction. These are the so-called superholomorphic discrete series representations of osp(1/2, R) and osp(2/2, R) which we recently constructed using a Z2 –graded extension of the orbit method. It turns out here that super-unitarity of these representations is a consequence of the self-adjointness of two pairs of anticommuting operators which act in the Hilbert sum of two Hilbert spaces each of which carrying a holomorphic discrete series representation of su(1, 1) such that the difference of the respective lowest weights is 12 . At an intermediate stage, we show that the generators of the considered orthosymplectic Lie superalgebras can be realized either as matrix-valued first order differential operators or as first order differential superoperators. Even though the former realization is less convenient than the latter from the computational point of view, it has the advantage of avoiding the use of anticommuting Grassmann variables, and is moreover important for our analysis of super-unitarity. The latter emphasizes the fundamental role played by the atypical (or degenerate) superholomorphic discrete series representations of osp(2/2, R) for the super-unitarity of the other representations considered in this work, and shows that the anticommuting (unbounded) self-adjoint operators mentioned above anticommute in a proper sense, thus connecting our work with the analysis of supersymmetric quantum mechanics. Keywords: Orthosymplectic Lie superalgebra, discrete series representation, superunitarity, first order differential operator, self-adjoint operator, anticommuting self-adjoint operators.
1. Introduction Our recent successful extension of geometric quantization to certain coadjoint orbits of low rank non-compact orthosymplectic Lie supergroups, has led to explicit constructions of infinite-dimensional super-unitary irreducible representations of the corresponding Lie superalgebras [1, 2]. In analogy with the non-graded 467 Reviews in Mathematical Physics, Vol. 10, No. 4 (1998) 467–497 c World Scientific Publishing Company
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situation, we coined the obtained representations superholomorphic discrete series representations. These are concrete realizations of the abstract discrete series representations recently studied, from different points of view using purely algebraic methods, by Furutsu and Hirai [3], Nishiyama [4], and Jakobsen [5]. Up to our knowledge, except for the superholomorphic realizations considered here, the only known explicit realizations of super-unitary representations of orthosymplectic Lie superalgebras are the so-called oscillator representations previously obtained by Nishiyama [6] (see also [7]); these are very special realizations. In the present work, we investigate from the functional analytic and operatorial theoretic points of view the structure of the superholomorphic discrete series representations of the Lie superalgebras osp(1/2, R) and osp(2/2, R). In particular, we present here an original interpretation of the notion of super-unitarity. Our interpretation relies on simple Hilbert space theoretic considerations, even though super-unitarity is consistently defined on a super-Hilbert space. The results of our previous contributions [1, 2] (see also [8]) and those described here put Kostant’s program within reach. Initiated in [9], this program which is partly at the origin of our interest in this subject is aimed at developing a harmonic analysis on Lie supergroups and on their homogeneous spaces (see [10] for an explicit statement). Quite remarkably, our present analysis of the super-unitarity of the superholomorphic discrete series representations under consideration combines in a unique way both classical and modern notions and techniques in functional analysis and operator theory. Indeed, on one hand this analysis relies on methods analogous to those developed by Bargmann in his celebrated work on the holomorphic realizations of unitary irreducible representations of two different Lie algebras: the bosonic C.C.R. (or Weyl–Heisenberg Lie algebra) [11] and the su(1, 1) Lie algebra [12]. This last realization was the first holomorphic discrete series representation to be constructed (see [13] for a modern perspective). Methods relevant to its study naturally intervene in our present work since the superholomorphic discrete series representations of osp(1/2, R) and osp(2/2, R) decompose into a direct sum of such su(1, 1) representations [2, 4], su(1, 1) ' sp(2, R) being a subalgebra of osp(1/2, R) and osp(2/2, R). Moreover, methods pertaining to the study of the holomorphic realization of the C.C.R. appear here to be of relevance in the analysis of the properties of the operators representing the odd generators of osp(1/2, R) and osp(2/2, R). We recall that the holomorphic realization of the C.C.R. was originally devised by Fock [14], his partial results were subsequently completed and generalized by Dirac [15], Bargmann [11], and Segal [16]. (The description in [11] is sufficient for our purpose.) On the other hand, the recently introduced notion of anticommuting (unbounded) self-adjoint operators plays here an important role as it turns out that the superunitarity of the considered representations is a consequence of the self-adjointness of two pairs of anticommuting operators acting in the Hilbert sum of two Hilbert spaces each of which carrying a holomorphic discrete series representation of su(1, 1) such that their respective lowest weights differ by 12 . The notion of anticommuting self-adjoint operators was first introduced by Vasilescu [17], it was then further
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developed by Samoilenko [18] and Pedersen [19]. Several interesting applications of this theory were considered by Arai [20] (see also references therein), these range from supersymmetric quantum mechanics to the analysis of operators of Dirac’s type. Hence our study shows that representation theory of orthosymplectic Lie superalgebras provides a new field of applications for this theory. In fact, we will show how one of Arai’s characterizations of proper anticommutativity of self-adjoint operators apply to the above pairs of operators. As already mentioned before, our considerations are restricted here to the superholomorphic discrete series representations of osp(1/2, R) and osp(2/2, R) as obtained by geometric quantization in [1, 2]. In the case of osp(2/2, R) both the typical and the atypical (or degenerate) superholomorphic discrete series representations are studied [2]. (It is well known that only the former have non-graded analogs.) It turns out from our analysis that the latter play a fundamental role regarding super-unitarity of all the other representations considered in this work. Indeed, the operators mentioned in the previous paragraph belong to the odd part of the Lie superalgebra osp(2/2, R) in the atypical superholomorphic discrete series representations. Our analysis of super-unitarity requires a preliminary step which consists in rewriting the results of [1, 2] in a more appropriate form. Indeed, thanks once again to the special structure of the carrier spaces of the superholomorphic discrete series representations of osp(1/2, R) and osp(2/2, R), the first order differential superoperators representing the generators of these Lie superalgebras obtained in [1, 2] can be rewritten in the form of matrix-valued first order differential operators. The superoperator realization is expressed in terms of anticommuting variables that belong to a complex Grassmann algebra, while in the alternative realization the matrix form replaces the dependence in such variables. Very much in the spirit of the present journal, this paper gathers original results (Secs. 4 and 5), an overview of the central theme which is representation theory of orthosymplectic Lie superalgebras (Sec. 3), mathematical preliminaries intended to make the content as self contained as possible (Sec. 2) and an up to date (though not exhaustive) bibliography. It is organized as follows. In Sec. 2 we set our notations by giving the definitions of the main notions used throughout. Section 3 starts with an overview of the recent progress made in the construction of both abstract and explicit realizations of discrete series representations of orthosymplectic Lie superalgebras. After that, we give a description of the superholomorphic discrete series representations of the Lie superalgebras osp(1/2, R) and osp(2/2, R), and we display the first order differential superoperators representing their generators as obtained by geometric quantization. Finally, we derive the alternative realization of the latter in terms of matrix-valued first order differential operators. Section 4 is devoted to a rigorous Hilbert space analysis of the notion of super-unitarity which is shown to follow from the self-adjointness of two pairs of (naively) anticommuting operators. In Sec. 5 we show that the latter anticommute in a proper sense. Concluding remarks and future directions of investigation are presented in Sec. 6.
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2. Preliminaries In this section we give basic definitions and we introduce the main notations for the ingredients needed throughout. More precisely, we give the definitions of an orthosymplectic Lie superalgebra, a Grassmann algebra, super-unitarity and anticommuting self-adjoint operators, together with a brief account on the holomorphic discrete series representations of su(1,1). 2.1. Orthosymplectic Lie superalgebras Let V = V0 ⊕ V1 be a direct sum of two vector spaces over R or C ; V will be called a Z2 -graded vector space. A homogeneous vector a ∈ V is a vector which belongs either to V0 or to V1 . If a ∈ Vj , then its degree is (a) = j, for j ∈ Z2 . Note that in what follows we implicitly assume that those elements which appear in an equation together with their degree of homogeneity are homogeneous elements of the Z2 -graded vector space they belong to. Moreover, “super”, “Z2 –graded” or simply “graded” are used interchangeably throughout. Definition 2.1. A Lie superalgebra is a Z2 -graded algebra (g = g0 ⊕ g1 , [·, ·]), i.e. [gi , gj ] ⊂ g(i+j)mod 2 , such that: (i) [a, b] = (−1)(a)(b) [b, a] , ∀ a, b ∈ g , (ii) (−1)(a)(c)[a, [b, c]]+(−1)(b)(a)[b, [c, a]]+(−1)(c)(b)[c, [a, b]] = 0 , ∀ a, b, c ∈ g . Starting from an associative Z2 -graded algebra (g = g0 ⊕ g1 , ·), i.e. gi · gj ⊂ g(i+j)mod 2 , one can equip g with a Lie superalgebra structure using the Lie superbracket: (2.1) [a, b] = ab − (−1)(a)(b) ba , ∀ a, b ∈ g . The restriction of this superbracket to g0 × g0 and g0 × g1 is a commutator, while its restriction to g1 × g1 is an anticommutator. We denote the latter [·, ·]− and [·, ·]+ , respectively. Now, let V = V0 ⊕ V1 be a real Z2 -graded vector space, and let V0 and V1 be, respectively, m and 2n-dimensional. The algebra gl(V, R) of linear operators on V is naturally a Z2 -graded algebra. Indeed, gl(V, R) = gl(V, R)0 ⊕ gl(V, R)1 , where gl(V, R)i = A ∈ gl(V ) | AVj ⊂ V(i+j)mod 2 for i = 0, 1. We designate by gl(m/2n, R) the matrix representation of gl(V, R) in a basis of V , ordered in such a way that its first m elements form a basis of V0 . Let (·, ·) be the non-degenerate bilinear form on V , defined by: u, v ∈ V ,
(u, v) = t uBv , where
Im
B= 0
0
0 0 −In
0
In . 0
(2.2)
(2.3)
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Definition 2.2. The orthosymplectic Lie superalgebra osp(m/2n, R) is the Z2 –graded subalgebra of gl (m/2n, R) given by: osp(m/2n, R) = {X ∈ gl(m/2n, R) (Xu, v) + (−1)(X)(u) (u, Xv) = 0 ,
∀ u, v ∈ V } .
(2.4)
In what follows osp(m/2n, R) will be simply denoted osp(m/2n). For more detail about these (and other) Lie superalgebras we refer the reader to the basic references [21, 22, 23]. In the remaining of this paper we will only consider the low rank cases, namely osp(1/2) and osp(2/2). We now display their defining relations in a specific basis of the superalgebra. Let {K0 , K± , F± } be the Cartan–Weyl basis of osp(1/2). The defining relations of the latter are as follows: [K0 , K± ]− = ±K± , [K0 , F± ]− = ± 12 F± ,
[K+ , K− ]− = −2K0 ,
[K± , F± ]− = 0 ,
[F± , F± ]+ = K±
[K± , F∓ ]− = ∓F± ,
and [F+ , F− ]+ = K0 .
(2.5) (2.6) (2.7)
As a Z2 -graded algebra osp(1/2) = osp(1/2)0 ⊕ osp(1/2)1 , where osp(1/2)0 is the simple Lie algebra sp(2, R) ' su(1, 1), with its usual Cartan–Weyl basis {K0 , K± }, and osp(1/2)1 is a 2-dimensional irreducible su(1, 1)-module. Similarly, let {B, K0 , K± ; V± , W± } be the Cartan–Weyl basis of osp(2/2). The defining relations of the latter are as follows: [K0 , K± ]− = ±K± ,
[K+ , K− ]− = −2K0 ,
(2.8a)
[B, K± ]− = 0,
[B, K0 ]− = 0 ,
(2.8b)
[K0 , V± ]− = ± 12 V± ,
[K0 , W± ]− = ± 12 W± ,
(2.8c)
[K± , V± ]− = 0,
[K± , W± ]− = 0 ,
(2.8d)
[K± , V∓ ]− = ∓V± ,
[K± , W∓ ]− = ∓W± ,
(2.8e)
[B, V± ]− = 12 V± ,
[B, W± ]− = − 21 W± ,
(2.8f)
[V± , V± ]+ = 0,
[W± , W± ]+ = 0 ,
(2.8g)
[V± , V∓ ]+ = 0,
[W± , W∓ ]+ = 0 ,
(2.8h)
[V± , W± ]+ = K± ,
[V± , W∓ ]+ = K0 ∓ B .
(2.8i)
As a Z2 -graded algebra osp(2/2) = osp(2/2)0 ⊕ osp(2/2)1 , where osp(2/2)0 = so(2) ⊕ su(1, 1), with its Cartan–Weyl basis {K0 , K± , B}, and osp(2/2)1 is the direct sum of two irreducible 2-dimensional su(1, 1)-modules spanned, respectively, by {V+ , V− } and {W+ , W− }.
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2.2. Grassmann algebras Definition 2.3. A real (complex) Grassmann algebra BN is a 2N -dimensional associative unital algebra over R (C) generated by I, ξ1 , . . . , ξN which satisfy the relations: I ξi = ξi I = ξi ,
and ξi ξj = −ξj ξi ,
∀ i, j = 1, . . . , N .
(2.9)
Clearly, all the generators are nilpotent, i.e. ξi2 = 0, ∀ i = 1, . . . , N . The example of a complex Grassmann algebra we will be using throughout is V that of the complex exterior algebra over CN , i.e. BN ≡ CN . As a Z2 -graded V V 0 1 0 1 ⊕ BN , where BN = ⊕r even r CN and BN = ⊕r odd r CN . algebra, BN = BN 0 1 (ξ1 ∈ BN ) Hence, any element ξ ∈ BN decomposes as ξ = ξ0 + ξ1 , where ξ0 ∈ BN is called the even (odd) component of ξ. Moreover, the component of ξ ∈ BN that V0 N C ≡ C is called the body of ξ and the remaining part is called its belongs to soul. Grassmann algebras were introduced in physics in order to provide physical models with anticommuting variables that are used in the mathematical description of observables which obey to Fermi statistics. Usual analysis has been extended to accommodate (super)functions of both commuting and anticommuting variables [24]. Because of the nilpotent character of the anticommuting variables, such functions are straightforward generalizations of usual functions. For instance, let z ∈ C 1 , then the superfunction and θ ∈ BN f (z, θ) ≡ f0 (z) + θ f1 (z) ,
(2.10)
where f0 and f1 are usual functions on C. In particular, a notion of integration was introduced. This is a functional procedure, simply defined by the following rules: Z Z 1 . (2.11) θ dθ = 1 and dθ = 0 , ∀θ ∈ BN More about this and other graded extensions of usual analytical, algebraic and geometric notions can be found in [9, 24]. 2.3. Super-unitarity In order to extend representation theory to the Z2 –graded context, one needs to define extensions of the notions of a Hermitian structure, a Hilbert space and unitarity. So far, different definitions of such extensions have been proposed. Here we only consider those we used in [1, 2]. They are based on the general definition of a graded Hermitian structure given in [25]. They moreover agree with the definition of super-unitarity used in [4, 6]. In [1, 2] we went one step further. We explicitly defined the notion of a super-Hilbert space. This was a consequence of our explicit construction of irreducible super-unitary representations of osp(1/2) and osp(2/2). Below, the definitions are given in the following order: super-Hermitian structure, super-Hilbert space and then super-unitarity.
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Definition 2.4. Let V = V0 ⊕ V1 be a Z2 -graded complex vector space. A super-Hermitian structure on V is a sesquilinear form hh·, ·ii : V × V → C, such that hhu, vii = (−1)(u)(v) hhv, uii ,
∀ u, v ∈ V .
(2.12)
Such a form is said to be an even super-Hermitian form, if hhu, vii = 6 0 only when u and v have the same degree. In the present work we will consider the following type of even super-Hermitian forms: hhu, vii = hu0 , v0 i0 + i hu1 , v1 i1 ,
(2.13)
where h·, ·ii is a Hermitian form on Vi , (i = 0, 1), and u = u0 + u1 , v = v0 + v1 ∈ V . Such an even super-Hermitian form is said to be positive definite if both h·, ·i0 and h·, ·i1 are positive definite. Definition 2.5. A super-Hilbert space is a Z2 -graded complex vector space V equipped with a non-degenerate even positive definite super-Hermitian structure hh·, ·ii, such that (V0 , h·, ·i0 ) and (V1 , h·, ·i1 ) are Hilbert spaces. In particular, v = v0 + v1 ∈ V is said to be super square integrable if Re(hhv, vii) = kv0 k20 < ∞ and Im(hhv, vii) = kv1 k21 < ∞, where “Re” and “Im” designate, respectively, the real and the imaginary parts, and k · ki is the L2 -norm on Vi (for i = 0, 1). For g a given Lie superalgebra, assume that V is a g-module equipped with a positive definite super-Hermitian form. Definition 2.6. The representation ρ of g in V is said to be super-unitary if hhρ(X)u, vii = (−1)(X)(u) hhu, ρ(X)vii ,
∀ u, v ∈ V and ∀ X ∈ g, .
(2.14)
This simply means that ρ(X) has to be super-Hermitian (or super symmetric). This equation differs by a sign from the one used in [4], simply because there ρ(X) is required to be super-skew-Hermitian. Note also that if ρ(X) is an unbounded linear differential superoperator on a super-Hilbert space, then (2.14) only makes sense for u and v in the domain of ρ(X). Moreover, for X ∈ g0 such that ρ(X) is unbounded, (2.14) does not imply that ρ(X) is self-adjoint. Hence, by Stone’s theorem [26] a self-adjoint extension of ρ(X) needs to be found in order to be able to associate to X a one parameter group of unitaries on V . We are mentioning these subteleties because all the (super)operators that appear throughout are unbounded and hence domain considerations are mandatory. Let us also mention that the choice of sign made in (2.13) (i.e. “+i” instead of “−i”) is the one that allows super-unitarization of the lowest weight modules of the Lie superalgebras considered in this work. Super-unitarization of the highest weight modules requires the opposite sign. For more details we refer to [4, 2]; in the second reference a geometric interpretation of this fact is given. Finally, it is important to note that even though the notion of a super-Hilbert space given in Definition 2.5 is the most appropriate from the Z2 -grading point of view, it is nevertheless possible to use a more conventional concept [22]. More
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precisely, one can define a super-Hilbert space as in Definition 2.5, but with the super-Hermitian structure hh·, ·ii replaced with a usual Hermitian one h·, ·i (i.e. replacing the ‘i’ in (2.13) by ‘1’). The super-Hilbert space turns then into a Hilbert sum. Accordingly, one needs to modify Definition 2.6 in the following way: (2.14) has to be replaced by hρ(X)u, vi = (−i)(X) hu, ρ(X)vi. Our notion of a super-unitary representation becomes then the notion of a star representation introduced in the physics literature (see [5, 22] for more details). This equivalence will be used in Sec. 4 in the investigation, by means of pure Hilbert space analytic methods, of properties of certain odd operators in terms of which the super-unitarity of the superholomorphic diecrete series representations of osp(1/2) and osp(2/2) is expressed. 2.4. Holomorphic discrete series of su(1,1) Here we give a brief description of the holomorphic discrete series representations of SU(1,1). More precisely, we will concentrate our attention on their infinitesimal realizations, namely the holomorphic discrete series of su(1, 1) that are derived representations of SU(1, 1). (For more details we refer the reader to Bargmann’s original work [12] and to Knapp’s book [13].) These representations are lowest weight unitary irreducible representations, denoted by D(k), where the lowest weight k is such that: k ∈ 12 N and k > 12 . They are explicitly realized in the Hilbert space Hk defined by: n o (2.15) Hk = ψ : D(1) → C, ψ holomorphic kψk2k < ∞ , where D(1) = {z ∈ C | |z| < 1} is the unit disc, and k · kk is the L2 -norm associated with the inner product h , ik given by: Z 2k − 1 dz d¯ z φ(z) ψ(z) , ∀ φ, ψ ∈ Hk . (2.16) hφ, ψik = 2 )2−2k π (1 − |z| (1) D The normalization is chosen here in such a way that kψk2k = 1 for ψ(z) = 1. The Hilbert space Hk is separable. The following set of holomorphic functions on D(1) is a complete orthonormal basis of Hk : ) ( 1/2 Γ(m + 2k) (k) m z , m∈N . (2.17) um (z) = m! Γ(2k) (k)
The superscript (k) in um (z) refers to the discrete series D(k). The generators K0 and K± of su(1, 1) which satisfy (2.5), are represented by the following first order differential operators on D(1) : b + = z 2 d + 2kz , K dz
b0 = z d + k , K dz
b− = d . K dz
(2.18)
These are unbounded linear operators on Hk . On a properly defined dense domain b 0 is self-adjoint, while K b ± is the adjoint of K b ∓ [12]. in Hk , one can show that K More precisely, one has
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b 0 ψik = hK b 0 φ, ψik hφ, K
b ± ψik = hK b ∓ φ, ψik , and hφ, K
(
where Uk =
∀ φ, ψ ∈ Uk ⊂ Hk , (2.19) )
∞ X 2 ψ ∈ Hk m2 |hu(k) m , ψik | < ∞
,
(2.20)
m=0
and moreover DK0 = DK † = DK± = DK † = Uk . Here DA denotes the domain of ∓
0
the operator A in Hk while A† denotes the adjoint of A. The above realizations of the discrete series representations of su(1, 1) can be explicitly constructed using the general method of geometric quantization [27, 28] which associates these representations to the elliptic coadjoint orbits of SU(1, 1). The latter are K¨ ahler manifolds represented by the unit disc D(1) ∼ = SU(1, 1)/U(1) z. equipped with its SU(1, 1)-invariant K¨ ahler form ω = −2ik (1 − |z|2 )−2 dz ∧ d¯ 2.5. Anticommuting self-adjoint operators In this section we give Pedersen’s definition of the anticommutativity of two non necessarily bounded self-adjoint operators in a Hilbert space [19]. Then, following Arai [20], we give one characterization of such a notion that will be useful in Sec. 5. Proofs of the results cited below can be found in [20]. Definition 2.7. Two (unbounded) self-adjoint operators A and B in a Hilbert space H are said to be anticommuting if eitA B ⊂ Be−itA ,
∀t ∈ R.
(2.21)
As shown in [19] this definition is symmetric in A and B. A characterization of the anticommutativity of self-adjoint operators devised by Arai [20, Theorem 6.3] in the context of supersymmetric quantum mechanics is now given. Proposition 2.8 [20]. Let Q1 and Q2 be self-adjoint operators in a Hilbert space H with inner product (· , ·) such that Q21 = Q22
and
(Q1 ψ, Q2 φ) + (Q2 ψ, Q1 φ) = 0 ,
ψ, φ ∈ DQ1 ∩ DQ2 .
(2.22)
Then Q1 and Q2 anticommute. 3. Superholomorphic Discrete Series and First Order Differential Operators The first part of this section is devoted to a qualitative review of the recent progress made in the extension of the theory of discrete series representations to orthosymplectic Lie superalgebras. We first describe different abstract constructions of such representations, then their explicit realizations through geometric methods. The second part of this section contains a detailed description of the structure of the superholomorphic realization of the discrete series representations of osp(1/2) and osp(2/2), and a derivation of their alternative matrix realization.
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3.1. An overview The abstract representation theory of Lie superalgebras started with Kac’s seminal work [21] and culminated recently in Jakobsen’s systematic classification of all unitarizable highest weight modules of basic classical Lie superalgebras [5]. Its gradual development was strewed by several significant contributions in which both mathematical physicists and pure mathematicians investigated specific examples. The list of these contributions is too long to be included here. For a good account on the evolution of the theory of finite and infinite-dimensional representations of Lie superalgebras and a list of references, we refer the interested reader to [5, 29]. Here we concentrate our presentation on the representation theory of the two orthosymplectic Lie superalgebras of interest to us. Abstract irreducible representations of osp(1/2) and osp(2/2) have been considered in [3] and [4, 5] from two different points of view. In the first reference the authors develop a formalism aimed at classifying the irreducible representations of a Lie superalgebra g = g0 ⊕ g1 starting from the known irreducible representations of its even Lie subalgebra g0 . They define a notion of super-unitarity which allows them to decide which of the irreducible representations found are superunitarizable. The example of osp(1/2) is then fully considered. It turns out that the only non-trivial irreducible representations of osp(1/2) which are super-unitarizable are those which are irreducible extensions of the discrete series representations of su(1,1) ' osp(1/2)0 . It seems then well justified to call the obtained super-unitary irreducible representations discrete series representations of osp(1/2). (Note that the first abstract construction of the latter was obtained in [30] using the shift operator technique which was developed in the mathematical physics literature. Moreover, [30] contains the first oscillator representation of an ortosymplectic Lie superalgebra. A generalization of this type of representations to all orthoymplectic Lie superalgebras has been obtained in [6, 7].) In [4] a more conventional approach is adopted. More precisely, a generalization of the standard procedure of induction from a parabolic sub-(super)algebra (with a compact reductive part) is considered, and is subsequently applied to osp(2/2). Provided that a notion of super-unitarity is defined, such a method leads, as in the non-graded case, to only a subset of the set of all super-unitary irreducible representations of the considered Lie superalgebra. For osp(2/2), only discrete series representations are obtained in this way [4]. More precisely, the representations so obtained are identified as elements of the discrete series because they share with the usual discrete series representations several of their known properties [4]. These include the facts that the eigenvalues of the Laplace–Casimir operator are discretely distributed and that the representation space is an extremal weight module. Note that this method has recently been extended by Jakobsen to all basic classical Lie superalgebras and has led to a classification of all their unitarizable highest weight modules [5]. The different approaches described here are based on equivalent notions of super-unitarity. It would be interesting to probe their equivalence in general. It is worth mentioning in connection with what precedes that the point that is of importance to us in this work is that a discrete series representation of an
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orthosymplectic Lie superalgebra decomposes into a direct sum of discrete series representations of the even part of the superalgebra (see [4] for the proof). This can be viewed as the reverse procedure to the extension technique of [3] applied to discrete series. In the case of osp(2/2), there appear two types of super-unitary irreducible representations that belong to the discrete series [4] (see also [2]). We coined them in [2] the typical and the atypical discrete series. We borrowed this terminology from the theory of finite-dimensional irreducible representations of Complex Basic Classical (CBC) Lie superalgebras [21, 23]. In order to explain the situation in that context, let us recall a known result: any finite-dimensional reducible representation of a complex semi-simple Lie algebra is completely reducible. This result does not hold in the graded case. Indeed, there are CBC Lie superalgebras which admit reducible but not completely reducible finite-diemensional representations. The irreducible quotients obtained from the latter are precisely the so-called atypical representations. They have no counterparts in the non-graded case. On the other hand, the typical representations are the irreducible summands of the direct sum decomposition of a completely reducible representation. (Notice that apart from osp(1/2n,C) all CBC Lie superalgebras admit both types of representations.) As shown in [2, 4], the same situation holds for the infinite-dimensional representations of osp(2/2) which is a real form with non-compact even part of the CBC Lie superalgebra of type I osp(2/2,C). According to [4], the typical discrete series representations are “generic” lowest weight super-unitary irreducible modules constructed through parabolic induction. For some “specific” values of the lowest weight, there appears in the preceding osp(2/2)-module a primitive vector which breaks the irreduciblity by generating a submodule. The atypical discrete series representations appear then as the super-unitary irreducible quotients. (Note that the abstract atypical discrete series representations of orthosymplectic Lie superalgebras were previously coined in the mathematical physics literature shortened multiplets or shortened representations [31].) As explained in the introduction, our contribution to this field originated from our desire to pursue Kostant’s program [9, 10]. We recall that harmonic analysis is based on explicit unitary irreducible representations of Lie (super)groups or Lie (super)algebras. In the non-graded context, Kirillov’s orbit method [32] which is nothing but geometric quantization [33] applied to a special type of symplectic manifolds, allows one to associate in an explicit and constructive way unitary irreducible representations of a Lie group G to those of its coadjoint orbits which admit an invariant polarization (see [34] for an up-to-date general description of the orbit method). The obtained representation space is the space of those L2 -sections of a certain complex line bundle-with-connection over the orbit which are moreover covariantly constant along the polarization. In order to construct explicit representations of Lie supergroups, one needs to extend the orbit method to Z2 –graded coadjoint orbits, or more generally, geometric quantization to supersymplectic supermanifolds. This last point was partially achieved (only prequantization) by
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Kostant in [9]. The extension of the full procedure was lacking the crucial notion of a polarization. Using Z2 –graded extensions of coherent states techniques [35] and the available abstract representation theory, we were able in [8, 1, 2] to overcome this difficulty by producing in a systematic way naturally polarized Z2 –graded coadjoint orbits. The complete orbit method was then successfully applied to examples of this kind. More precisely, our approach in [8, 1, 2] consists of two main steps. In the first one, we identify the coadjoint orbits associated to the abstract discrete series representations of osp(1/2) and osp(2/2). This is done using a Z2 –graded extension of Berezin’s dequantization procedure [35]: a method based on the notion of coherent states. It yields a coordinatization of the coadjoint orbits, an explicit expression of their invariant supersymplectic forms, and a locally equivariant moment map. A detailed geometric study of these results leads then to a natural definition of a super-K¨ ahler supermanifold, the considered orbits being particular examples of this notion. The second step consists in applying Kostant’s prequantization [9] to the obtained supersymplectic supermanifolds, and then completing the quantization procedure using the invariant super-K¨ ahler polarizations uncovered in the first step. As a result we obtain superholomorphic discrete series representations as realizations of the abstract discrete series representations we started from. In particular, the generators of the Lie superalgebras are represented by first order superoperators acting in a super-Hilbert space. For the considered examples, namely osp(1/2) and osp(2/2), it turns out, as expected, that the obtained representations are Z2 –graded extensions of the known holomorphic discrete series of su(1,1). For a detailed description of the above mentioned geometric constructions we refer to [8, 1, 2]. A summary of the main results is displayed in the first parts of the following subsections. In the second part of each of them we derive the alternative matrix realization of the considered representations of osp(1/2) and osp(2/2). 3.2. Discrete series representations of osp(1/2) The defining relations of osp(1/2) are given in (2.5)–(2.7). This is a rank one Lie superalgebra. Very much like its Lie subalgebra su(1, 1) (see (2.5)), its Cartan subalgebra is generated by K0 . Its so-called superholomorphic discrete series representations are lowest weight super-unitary irreducible representations, denoted by V (τ ), where the lowest weight τ is such that: τ ∈ 12 N and τ > 12 (as for k in Sec. 2.4). Let O(D(1) ) denotes the space of holomorphic functions on the unit disc D(1) , then O(D(1) ) ⊗ B1 is the holomorphic superstructure sheaf of the super unit disc D(1|1) . The latter is a natural graded extension of the unit disc D(1) . It is a realization of the OSp(1/2)-coadjoint orbit OSp(1/2)/U(1) which extends the K¨ ahler elliptic SU(1, 1)-coadjoint orbit SU(1, 1)/U(1) (see [1] for more details). Sections of O(D(1) ) ⊗ B1 are called superholomorphic functions on D(1|1) . These are functions Ψ(z, θ), where z ∈ D(1) and θ is the complex anticommuting variable generating B1 .
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The super-Hilbert space carrying V (τ ) is: n o Vτ = Ψ ∈ O(D(1) ) ⊗ B1 Re (hhΨ, Ψiiτ ) < ∞ and Im (hhΨ, Ψiiτ ) < ∞ , (3.1) where the super-Hermitian structure hh·, ·iiτ is given by: Z i dz d¯ z dθ dθ¯ hhΦ, Ψiiτ = 2 ¯ 1−2τ Φ(z, θ) Ψ(z, θ) , π D(1|1) (1 − |z| − iθθ)
∀Ψ, Φ ∈ Vτ .
(3.2)
Here θ¯ designates the complex conjugate of θ. In Vτ the osp(1/2) generators (2.5)–(2.7) are represented by the following first order differential superoperators, b0 = z ∂ + θ ∂ + τ , K ∂z 2 ∂θ
(3.3)
b + = z 2 ∂ + zθ ∂ + 2τ z , K ∂z ∂θ ∂ i ∂ b −z + 2τ θ , F+ = − √ zθ ∂z ∂θ 2
b− = ∂ , K ∂z i Fb− = − √ 2
(3.4) ∂ ∂ − . θ ∂z ∂θ
(3.5)
The super-unitarity of V (τ ) is expressed by the following equalities, b 0 Ψiiτ , b 0 Φ, Ψiiτ = hhΦ, K hhK
b ± Φ, Ψiiτ = hhΦ, K b ∓ Ψiiτ , hhK
hhFb± Φ, Ψiiτ = i (−1)(Φ) hhΦ, Fb∓ Ψiiτ .
(3.6) (3.7)
These equations differ from those in (2.14) because we are working in the Cartan– Weyl basis. Note also that the notation used here for osp(1/2) differs √ slightly from that in [8, 1]. Indeed, the anticommuting variable θ in [8, 1] is 2 times the one used here. As observed in [1], and in agreement with the description of the abstract discrete series in [3], the representation V (τ ) decomposes in the following way in terms of the discrete series D(k) (see Sec. 2.4) of su(1, 1) ⊂ osp(1/2): V (τ ) = D(k = τ ) ⊕ D(k = τ + 12 ) .
(3.8)
Concretely, writing Ψ(z, θ) = ψ0 (z) + θ
√ 2τ ψ1 (z) ,
∀ Ψ ∈ Vτ ,
(3.9)
and performing Berezin’s integration (see (2.11)) over θ and θ¯ in (3.2), one obtains [1] (3.10) hhΦ, Ψiiτ = hφ0 , ψ0 ik=τ + ihφ1 , ψ1 ik=τ + 12 , where h·, ·ik is given in (2.16). Combining this result with (3.1), (2.16) and (2.13) one immediately sees that as a vector space Vτ = Hk=τ ⊕ Hk=τ + 12 .
(3.11)
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A. M. EL GRADECHI
(This reflects at the same time the Z2 -gradation of Vτ .) One can then represent any Ψ ∈ Vτ (of the form (3.9)) as a 2-components column vector t (ψ0 , ψ1 ), with ψ0 ∈ Hτ and ψ1 ∈ Hτ + 12 . One can easily show that in this alternative realization of Vτ , the first order differential superoperators given in (3.3)–(3.5) are represented by the following matrix-valued first order differential operators: e0 = K
e+ = K
d z dz +τ
!
0 d +τ + z dz
0
,
1 2
!
d z 2 dz + 2τ z
0
0
d + 2(τ + 12 )z z 2 dz
−i Fe+ = √ 2 τ
0
−2τ z
d z dz + 2τ
0
(3.12)
! ,
,
e− = K −i Fe− = √ 2 τ
d dz
0
0
d dz
! ,
0
−2τ
d dz
0
(3.13) ! . (3.14)
The main interesting feature of this new realization of osp(1/2) generators lies in its independance on anticommuting variables. It shows that when dealing with super structures, at the representation theoretic level, it is not necessary to introduce anticommuting variables. As a consequence of (3.6) and (3.10), super-unitarity of V (τ ) on the even b ± , reflects usual unitarity of D(k = τ ) and D(k = τ + 12 ). b 0 and K superoperators K However, for the odd superoperators Fb± , super-unitarity expresses interesting new features that involve simultaneously both D(k = τ ) and D(k = τ + 12 ). This point will be discussed in detail in Sec. 4. 3.3. Typical discrete series representations of osp(2/2) The defining relations of osp(2/2) are given in (2.8a)–(2.8i). This is a rank two Lie superalgebra. Its Cartan subalgebra is generated by K0 and B. Its so-called typical superholomorphic discrete series representations are lowest weight superunitary irreducible representations, denoted by V (τ, b), where the lowest weight (τ, b) (associated with the pair (K0 , B)) is such that: τ ∈ 12 N, τ > 12 , b ∈ 12 Z and |b| < τ . (The atypical superholomorphic discrete series representations which will be considered in the next subsection appear for the limiting values b = ±τ of the last inequality.) Let O(D(1) ) denotes as before the space of holomorphic functions on the unit disc D , then O(D(1) ) ⊗ B2 is the holomorphic superstructure sheaf of the super unit disc D(1|2) . The latter is another natural graded extension of the unit disc D(1) ; it is a realization of the OSp(2/2)-coadjoint orbit OSp(2/2)/U(1)×U(1) which extends the K¨ ahler elliptic SU(1, 1)-coadjoint orbit SU(1, 1)/U(1) (see [2] for more details). Sections of O(D(1) ) ⊗ B2 are called superholomorphic functions on D(1|2) . These are functions Ψ(z, θ, χ), where z ∈ D(1) and θ, χ are the complex anticommuting variables generating B2 . (1)
ORTHOSYMPLECTIC LIE SUPERALGEBRAS
481
The super-Hilbert space carrying V (τ, b) is: n o Vτ,b = Ψ ∈ O(D(1) ) ⊗ B2 Re (hhΨ, Ψiiτ,b ) < ∞ and Im (hhΨ, Ψiiτ,b ) < ∞ , (3.15) where the super-Hermitian structure hh·, ·iiτ,b is given by: Z 2τ dz d¯ z dθ dθ¯ dχ dχ ¯ hhΦ, Ψiiτ,b = π(b2 − τ 2 ) D(1|2) 2 2 ¯ χχ ¯ θθ − b −τ Φ(z, θ, χ) Ψ(z, θ, χ) e 2τ 1−|z|2 × −b−τ , ¯ b−τ 1 − |z|2 − iχχ ¯ 1 − |z|2 − iθθ
(3.16)
¯ designate the complex conjugates of θ and χ, respec∀ Ψ, Φ ∈ Vτ,b . Here θ¯ and χ tively. In Vτ,b the osp(2/2) generators (2.8a)–(2.8i) are represented by the following first order differential superoperators, b = θ ∂ − χ ∂ +b, B 2 ∂θ 2 ∂χ
(3.17)
b0 = z ∂ + θ ∂ + χ ∂ + τ , K ∂z 2 ∂θ 2 ∂χ
(3.18)
b + = z 2 ∂ + zθ ∂ + zχ ∂ + 2τ z , K ∂z ∂θ ∂χ
(3.19)
b− = ∂ , K ∂z
(3.20)
∂ ∂ + i (z + κ− χθ) − 2iτ κ− θ , Vb+ = −iκ− zθ ∂z ∂χ
(3.21)
c− = −iκ+ χ ∂ + i ∂ , W ∂z ∂θ
(3.22)
∂ ∂ +i , Vb− = −iκ− θ ∂z ∂χ
(3.23)
c+ = −iκ+ zχ ∂ + i (z − κ+ χθ) ∂ − 2iτ κ+ χ , W ∂z ∂θ where κ± = equalities:
τ ±b 2τ .
(3.24)
The super-unitarity of V (τ, b) is expressed through the following b τ,b , b Ψiiτ,b = hhΦ, BΨii hhBΦ,
b 0 Ψiiτ,b , b 0 Φ, Ψiiτ,b = hhΦ, K hhK c± Φ, Ψiiτ,b = i (−1)(Φ) hhΦ, Vb∓ Ψiiτ,b , hhW
b ± Φ, Ψiiτ,b = hhΦ, K b ∓ Ψiiτ,b , hhK
(3.25) (3.26)
c∓ Ψiiτ,b . hhVb± Φ, Ψiiτ,b = i (−1)(Φ) hhΦ, W (3.27)
482
A. M. EL GRADECHI
Note that our notation for the Cartan–Weyl basis of osp(2/2) is misleading. Indeed, the pairs V+ versus V− and W+ versus W− do not correspond to the root space decomposition: “positive root” versus its opposite “negative root”. This is clearly reflected in the above equations. Our notation stresses the role played by the pairs (V+ , V− ) and (W+ , W− ) as basis of the two irreducible su(1, 1)-modules intervening in the definition of osp(2/2) as a Lie superalgebra (see (2.8a)–(2.8i)). As for osp(1/2) the representation V (τ, b) decomposes in the following way in terms of discrete series D(k) of su(1, 1) ⊂ osp(1/2) [2, 4], V (τ, b) = D(k = τ ) ⊕ 2 · D(k = τ + 12 ) ⊕ D(k = τ + 1) ,
(3.28)
where the factor 2 in front of D(k = τ + 12 ) indicates that this representation has multiplicity 2 in the decomposition. Concretely, writing p p p Ψ(z, θ, χ) = ψ1 (z)+θ 2τ κ− ψ2 (z)+χ 2τ κ+ ψ3 (z)+χθ 2τ (2τ + 1)κ+ κ− ψ4 (z) , (3.29) ¯ in ∀ Ψ ∈ Vτ,b , and performing Berezin’s integration (see (2.11)) over θ, χ, θ¯ and χ (3.16) one obtains [2], hhΦ, Ψiiτ,b = hφ1 , ψ1 ik=τ + ihφ2 , ψ2 ik=τ + 12 + ihφ3 , ψ3 ik=τ + 12 + hφ4 , ψ4 ik=τ +1 , (3.30) where h·, ·ik is given in (2.16). Hence, one has Vτ,b = Hk=τ ⊕ 2 · Hk=τ + 12 ⊕ Hk=τ +1 .
(3.31)
Here the Z2 -gradation is less transparent than in the case of osp(1/2): the even subspace of Vτ,b is the direct sum of the first and the last Hilbert spaces of the above direct sum decomposition, the odd subspace consists of the two copies of Hk=τ + 12 . Now, one can represent any Ψ ∈ Vτ,b (of the form (3.29)) as a 4-components column vector t (ψ1 , ψ2 , ψ3 , ψ4 ), with ψ1 ∈ Hτ , ψ2 , ψ3 ∈ Hτ + 12 and ψ4 ∈ Hτ +1 . The first order differential superoperators given in (3.17)–(3.24) can then be represented by the following matrix-valued first order differential operators: b 0 e B = 0 0
0 b+
0 1 2
0
0
b−
0
0
1 2
0 0 , 0 b
(3.32)
d z dz + τ 0 e0 = K 0
0
0 d z dz
+τ + 0 0
0 1 2
0
0 d z dz
+τ + 0
0 1 2
0 d +τ +1 z dz
,
(3.33)
483
ORTHOSYMPLECTIC LIE SUPERALGEBRAS
d z 2 dz + 2τ z 0 e+ = K 0 0
0 d z 2 dz
0
0
0
0
0
d + 2(τ + 12 )z z 2 dz
0
0
0
d + 2(τ + 1)z z 2 dz
+ 2(τ +
1 2 )z
(3.34)
d
dz 0 e K− = 0 0
0
0
d dz
0
0
d dz
0
0
0 0 , 0
(3.35)
d dz
0
q −i κ− z d + 2τ 0 2τ dz Ve+ = 0 0 0 0
√ i 2τ κ+ z
0
q i
κ− 2τ +1
0
p 0 i (2τ + 1)κ+ z , 0 0 d 0 z dz + 2τ + 1 (3.36)
0 q0 f− = W −i κ+ d 2τ dz 0
√ i 2τ κ− 0
0 0
0
0
−i
q
0 q
−i κ− e 2τ V− = 0 0
d dz
κ+ d 2τ +1 dz
0
√ i 2τ κ+
0
0
0
q 0
0
i
κ− d 2τ +1 dz
0
p , −i (2τ + 1)κ− 0 0 0
(3.37)
0
p i (2τ + 1)κ+ , 0 0 √ i 2τ κ− z
0 0 0 q f+ = W κ+ d 0 −i 2τ z dz + 2τ q + d + 2τ + 1 z dz 0 −i 2τκ+1
(3.38)
0 0 0 0
0 p . −i (2τ + 1)κ− z 0 0
(3.39)
484
A. M. EL GRADECHI
We see here that compared to its superoperator counterpart, the matrix realization starts to become less convenient for practical uses. This situation worsens for higher rank orthosymplectic Lie superalgebras. For instance, the generic discrete series representations of osp(N/2) will admit a matrix realization in terms of 2N × 2N matrices. 3.4. Atypical discrete series representations of osp(2/2) When the lowest weight (τ, b) of the previous section is such that b = ±τ , a primitive vector (i.e. a null vector generating a submodule) occurs in Vτ,b=±τ viewed simply as an osp(2/2)-module (not as a super-Hilbert space). The quotient module is irreducible and can be super-unitarized, leading thus to the so-called atypical superholomorphic discrete series representations of osp(2/2) [2, 4]. We denote the latter A± (τ ) for b = ±τ , respectively. As in [2], here we only consider the case b = −τ , the other situation is perfectly symmetric. The explicit realization of A− (τ ) given below follows [2]. Most of the results of this section can be obtained as the limit when b → −τ of those of the previous one. When this limit doesn’t exist (as it is the case for the inner product defining the super-Hilbert space) one has to rederive the needed expressions. However, this can be avoided using the similarities between the geometry of the coadjoint orbits to which are associated the atypical superholomorphic discrete series of osp(2/2), and the geometry of the OSp(1/2) coadjoint orbits of Sec. 3.2. It turns out that the super-Hilbert space carrying A− (τ ) is exactly the super-Hilbert space Vτ of (3.1) equipped with the same super-Hermitian structure (3.2). This is a consequence of the fact that the super unit disc D(1|1) , already encountered in Sec. 3.2, is also a realization of the OSp(2/2)-coadjoint orbit OSp(2/2)/U(1/1) whose geometric quantization leads to A± (τ ) (see [2] for more details). In Vτ the osp(2/2) generators (2.8a)–(2.8i) are represented by the following first order differential superoperators, b0 = z ∂ + θ ∂ + τ , K ∂z 2 ∂θ
b = θ ∂ −τ, B 2 ∂θ
(3.40)
b + = z 2 ∂ + zθ ∂ + 2τ z , K ∂z ∂θ
b− = ∂ , K ∂z
(3.41)
∂ − 2iτ θ , Vb+ = −izθ ∂z
c− = i ∂ , W ∂θ
(3.42)
∂ , Vb− = −iθ ∂z
c+ = iz ∂ . W ∂θ
(3.43)
The super-unitarity of A− (τ ) is reflected by the same equations as in the previous section, namely (3.25)–(3.27), rewritten now in terms of the superHermitian structure (3.2) instead of (3.16). Using the same arguments as in Sec. 3.2 which were based on (3.9)–(3.11), one obtains the alternative realization of the above operators in terms of the following matrix-valued first order differential operators:
485
ORTHOSYMPLECTIC LIE SUPERALGEBRAS
e0 = K
e+ = K
d z dz +τ
!
0 d +τ + z dz
0
1 2
d z 2 dz + 2τ z
0
0
d + (2τ + 1)z z 2 dz
−i Ve+ = √ 2τ −i Ve− = √ 2τ
0
0
d z dz + 2τ
0
0
0
d dz
0
!
! ,
,
e =− B
, ! ,
e− = K
τ
0
0
τ−
d dz
0
0
d dz
! , (3.44)
1 2
! ,
√ f− = i 2τ W
0 1
√ f+ = i 2τ W
0
z
0
0
(3.45)
!
0 0
,
(3.46)
.
(3.47)
!
4. Super-unitarity Here we analyze the notion of super-unitarity defined in Sec. 2.3 in the light of the explicit representations described in the preceding section. In Sec. 4.1 we prove a few propositions that lead to a rigorous functional analytic interpretation of super-unitarity. We then discuss the latter and its consequences in Sec. 4.2. 4.1. A Hilbert space analysis Since the superholomorphic discrete series representations of the Lie superalgebras considered here decompose into direct sums of holomorphic discrete series representations of su(1, 1), it is not hard to see that the restriction of super-unitarity of the former to the even part of the Lie superalgebras simply reflects usual unitarity of the latter (separately within each direct summand). However, for the odd part of these Lie superalgebras super-unitarity reveals “something” new worth to be analyzed. Note that these two behaviors are encoded in the form of the matrixvalued operators of Sections. 3.2, 3.3 and 3.4: the even operators are diagonal while the odd ones are not. We start by studying the case of osp(1/2) using results from Sec. 3.2: (a) Even part of osp(1/2) — From (3.12)–(3.13) one immediately sees that (3.6) expresses simply the fact that the holomorphic discrete series representations D(k = τ ) and D(k = τ + 12 ) of su(1, 1) are unitary (see Sec. 2.4). In other words, the first order operators representing K0 (resp. K+ and K− ) within the superholomorphic discrete series representation V (τ ) = D(k = τ ) ⊕ D(k = τ + 12 ) of osp(1/2) is selfadjoint (resp. are each other adjoints) on Uk=τ ⊕ Uk=τ + 12 ⊂ Vτ = Hk=τ ⊕ Hk=τ + 12 , where Uk is defined by (2.20). As mentioned in Sec. 3.2, the previous direct sums are not Hilbert sums, but simply vector spaces direct sums. Note however that when V (τ ) is restricted to su(1, 1) = osp(1/2)0 , (3.6) makes also sense in Vτ = Hk=τ ⊕ Hk=τ + 12 considered now as a Hilbert sum. This is a direct consequence of the fact that the even superoperators are transparent to the super-Hilbert space structure of Vτ .
486
A. M. EL GRADECHI
(b) Odd part of osp(1/2) — From (3.7), with Fb+ in the left-hand side and Fb− in the right hand side, and with Ψ, Φ ∈ Vτ of the form (3.9), a straightforward computation leads to the equalities: √ 1 d φ1 , ψ0 2τ hzφ1 , ψ0 ik=τ = √ , (4.1) dz 2τ k=τ + 1 2
√ 1 2τ hφ0 , ψ1 ik=τ = √ 2τ
d + 2τ φ0 , ψ1 . z dz k=τ + 1
(4.2)
2
The inner products appearing here are those introduced in Sec. 2.4 for the holomorphic discrete series representations of su(1, 1). Note that the operators involved in (4.1)–(4.2) are unbounded. The necessary domain considerations together with further analysis of (4.1)–(4.2) will be considered shortly. Before that we briefly discuss points (a) and (b) above in the case of the typical and the atypical superholomorphic discrete series representations of osp(2/2). Similarly to (a), one easily sees that for both typical and atypical superholomorphic discrete series representations of Secs. 3.3 and 3.4, the restriction of super-unitarity to the even part of osp(2/2) simply reflects unitarity of the holomorphic discrete series representations of su(1, 1) ⊂ osp(2/2) which appear in the direct sum decompositions of V (τ, b) and A± (τ ), respectively. Hence, once again nothing new arises from the even part of the Lie superalgebra. However, as in (b) above, the restriction of super-unitarity to the odd part of osp(2/2), for both typical and atypical superholomorphic discrete series representations, turns out to lead to exactly the two equalities exhibited above. More precisely, in the atypical case one obtains (4.1)–(4.2), while in the typical case one obtains two pairs of equalities, namely (4.1)–(4.2) and their shifted version where τ is replaced by τ + 12 . These facts confer to (4.1)–(4.2) a fundamental role. The rest of this section is devoted to a rigorous and detailed analysis of their validity. Before that, let us verify their formal validity by comparing both sides of (4.1)–(4.2) using results from the theory of holomorphic discrete series representations of su(1, 1) as described in Sec. 2.4. We start by displaying easy to prove and useful identities involving the complete orthonormal basis (2.17) of Hk and the operators intervening in (4.1)–(4.2): Proposition 4.1. √ √ (k=τ + 12 ) (k=τ ) 2τ z um = m + 1 um+1 , 1 1 d (k=τ ) √ (k=τ + 1 ) √ um = m um−1 2 , √ dz 2τ 2τ
(k)
√ √ (k=τ + 12 ) (k=τ ) 2τ um = m + 2τ um ,
(4.3)
√ d (k=τ + 12 ) (k=τ ) + 2τ um = m + 2τ um . z dz (4.4)
(The z-dependance in um (z) has been suppressed for convenience.) Now, using Proposition 4.1, the expansions
487
ORTHOSYMPLECTIC LIE SUPERALGEBRAS
Hk=τ 3 ψ0 (z) =
∞ X
(k=τ ) (k=τ ) hum , ψ0 ik=τ um (z) ,
(4.5)
m=0
Hk=τ + 12 3 ψ1 (z) =
∞ X
(k=τ + 12 )
hum
(k=τ + 12 )
, ψ1 ik=τ + 12 um
(z) ,
(4.6)
m=0
and their analogs for φ0 and φ1 , one can evaluate both sides of (4.1) and (4.2), respectively obtaining the following formal equalities: ∞ X √ √ (k=τ + 12 ) (k=τ ) 2τ hzφ1 , ψ0 ik=τ = m + 1 hum , φ1 ik=τ + 1 hum+1 , ψ0 ik=τ 2
m=0
1 = √ 2τ √
2τ hφ0 , ψ1 ik=τ =
d φ1 , ψ0 , dz k=τ + 1
∞ X √ m=0
1 = √ 2τ
(4.7)
2
(k=τ )
m + 2τ hum
(k=τ + 12 )
, φ0 ik=τ hum
, ψ1 ik=τ + 12
d + 2τ φ0 , ψ1 z . dz k=τ + 1
(4.8)
2
In what follows we provide the necessary domain considerations that will establish the above formal results on a firm ground. More precisely, we will seek for a pure Hilbert space theoretic interpretation of (4.1)–(4.2), since they both involve two holomorphic discrete series representations of su(1, 1), namely D(k = τ ) and D(k = τ + 12 ), without any reference to super-Hilbert spaces (despite their origin). More precisely, even though (4.1)–(4.2) are consequences of the decomposition (3.11) (or (3.31)) which is not a Hilbert sum but simply a direct sum of Hilbert spaces forming a super-Hilbert space, it is nevertheless possible to rewrite them as equations involving operators acting in a Hilbert sum. This is not surprising, it is in fact in a perfect agreement with the remark we made at the end of Sec. 2.3. Concretely, consider the Hilbert sum Wτ = Hk=τ ⊕ Hk=τ + 12 ,
(4.9)
with the inner product of Φ = (φ0 , φ1 ) and Ψ = (ψ0 , ψ1 ) ∈ Wτ given by: (Φ, Ψ)τ = hφ0 , ψ0 ik=τ + hφ1 , ψ1 ik=τ + 12 .
(4.10)
We denote the corresponding norm by k · kτ (there will be no possible confusion with the norms in Hk=τ and Hk=τ + 12 , since these are denoted k · kk=τ and k · kk=τ + 12 , respectively). Consider now the following two pairs of linear operators acting in Wτ ; ! ! √ 0 0 0 z 1 , (4.11) , Q− = 2τ Q+ = √ d 2τ dz 0 0 0 P
+
1 = √ 2τ
0
0
d z dz + 2τ
0
! ,
P
−
√ = 2τ
0
1
0
0
! .
(4.12)
488
A. M. EL GRADECHI
It is not hard to see that the equalities (4.1) and (4.2) can be respectively rewritten in the following form: (Q− Φ, Ψ)τ = (Φ, Q+ Ψ)τ ,
(4.13)
(P − Φ, Ψ)τ = (Φ, P + Ψ)τ .
(4.14)
The first (resp. second) equation is only valid for Ψ ∈ DQ+ (resp. Ψ ∈ DP + ) and Φ ∈ DQ− (resp. Φ ∈ DP − ). Proposition 4.2. The domains DQ± and DP ± are such that : DQ+ = DP +
and DQ− = DP − .
(4.15)
Proof. We start with the first equality in (4.15). Let Ψ = (ψ0 , ψ1 ) ∈ Wτ . From (4.9), (4.10) and (4.11) one immediately sees that kQ
+
Ψk2τ
2 d 1
ψ0 = 2τ dz k=τ + 1
and kP
+
Ψk2τ
2
2
d 1
z + 2τ ψ0 = .
2τ dz k=τ + 12 (4.16)
Using (4.4), (4.5) and (4.16) one obtains kQ+ Ψk2τ =
∞ X
(k=τ ) m|hum , ψ0 ik=τ |2 ,
∞ X
kP + Ψk2τ =
m=0
(k=τ ) (m + 2τ )|hum , ψ0 ik=τ |2 .
m=0
(4.17) Hence, kP + Ψk2τ = kQ+ Ψk2τ + 2τ kψ0 k2k=τ
(4.18)
which can be rewritten in the form:
2 2
d 1 1
z d + 2τ ψ0
ψ0 = + 2τ kψ0 k2k=τ .
2τ dz 2τ dz k=τ + 1 k=τ + 1 2
(4.19)
2
This equality is to be interpreted in the following way [11]: either both sides are infinite or they are both finite and then have the same finite value. This proves the first part of (4.15). The second part can be proven similarly. Indeed, repeating the same computations for Q− and P − , one arrives to the following: kQ− Ψk2τ =
∞ X
(k=τ + 12 )
(m + 1)|hum
, ψ1 ik=τ + 12 |2 ,
m=0
kP
−
Ψk2τ
=
∞ X
(k=τ + 12 )
(m + 2τ )|hum
, ψ1 ik=τ + 12 |2 .
(4.20)
m=0
Hence, kP − Ψk2τ = kQ− Ψk2τ + (2τ − 1)kψ1 k2k=τ + 1 2
(4.21)
489
ORTHOSYMPLECTIC LIE SUPERALGEBRAS
which can be rewritten in the form: 2τ kψ1 k2k=τ = 2τ kzψ1 k2k=τ + (2τ − 1)kψ1 k2k=τ .
(4.22)
The second part of (4.15) follows from the same arguments used for the first part. An exact characterization of the above domains can be deduced from the proof just given. Indeed, from (4.17) and (4.20) one can easily see that the following takes place: Proposition 4.3. DQ+ = DP + = Kk=τ ⊕ Hk=τ + 12 ⊂ Wτ and DQ− = DP − = Hk=τ ⊕ Kk=τ + 12 ⊂ Wτ , where ( ) ∞ X (k) 2 m|hum , ψik | < ∞ (4.23) Kk = ψ ∈ Hk m=0
is a dense domain in Hk . Moreover, one has: Proposition 4.4. Q+ and Q− (resp. P + and P − ) are each other adjoints, i.e. † † resp. P ± = P ∓ . (4.24) Q± = Q∓
Proof. Recall that (4.13) and (4.14) (and thus (4.1) and (4.2)) are rigorously true provided that Ψ and Φ belong to the appropriate domains (see Proposition 4.2 and 4.3). Hence, in order to prove (4.24) one only needs to prove that D(Q± )† = DQ∓ (resp. D(P ± )† = DP ∓ ). It is not hard to see that these are straightforward consequences of (4.7) (resp. (4.8)). Finally, from each of the two pairs, (Q+ , Q− ) and (P + , P − ), one can construct another pair of operators, namely Q1 = Q+ + Q− , Q2 = i(Q+ − Q− ) and P1 = P + + P − , P2 = i(P + − P − ) (4.25) which are such that: Proposition 4.5. (a) DQi = DPi = DQ+ ∩ DQ− = Kk=τ ⊕ Kk=τ + 12 ⊂ Wτ , for i = 1, 2, and (b) Qi and Pi for i = 1, 2 are self-adjoint on their common dense domain given in (a). Proof. (a) Direct computations based on (4.11), (4.12) and (4.25) lead to: kQ1 Ψk2τ = kQ+ Ψk2τ + kQ− Ψk2τ = kQ2 Ψk2τ ,
(4.26)
kP1 Ψk2τ = kP + Ψk2τ + kP − Ψk2τ = kP2 Ψk2τ .
(4.27)
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Arguments analogous to those used in proving Proposition 4.2 (see the reasoning following (4.19)), show that indeed DQi = DQ+ ∩ DQ− = DPi , for i = 1, 2. The explicit form of the intersection follows from Proposition 4.3. Part (b) is a direct consequence of Proposition 4.4, (4.25) and part (a). Observe that the first equality in Proposition 4.5(a) can be easily seen as a direct consequence of the formula kP1 Ψk2τ = kQ1 Ψk2τ + 2τ kψ0 k2k=τ + (2τ − 1)kψ1 k2k=τ + 1
(4.28)
2
which follows from (4.18), (4.21), (4.26) and (4.27). 4.2. Discussion and consequences Proposition 4.5 provides an alternative interpretation of the fundamental relations (4.1) and (4.2). More precisely, the validity of the latter is equivalent to the self-adjointness of the operators Qi and Pi . Further properties of these operators will be investigated in the next section. Now, we conclude this section by closing the loop. More precisely, since (4.1) and (4.2) are the fundamental equations at the origin of the super-unitarity of the considered representations (and not only of their restrictions to the odd part of the Lie superalgebras as we will shortly show), we should be able to write the matrix-valued first order operators of Secs. 3.2, 3.3 and 3.4 in terms of the Q’s and the P ’s studied in the present section. In an increasing order of difficulty, one finds: A. Atypical discrete series of osp(2/2) — Comparing (3.46)–(3.47) with (4.13)– (4.14) one immediately sees that: f− = iP − , W
Ve+ = −iP + ,
Ve− = −iQ+
f+ = iQ− . and W
(4.29)
B. Discrete series of osp(1/2) — Here also the expressions are not hard to find. Indeed, comparing (3.14) with (4.13)–(4.14) one gets: −i −i and Fe− = √ Q+ − P − . (4.30) Fe+ = √ P + − Q− 2 2 C. Typical discrete series of osp(2/2) — In this case one needs to introduce two copies of the Q’s and P ’s. More precisely, if we denote those used above Q± (τ ) and ± ± ± P(τ ) , we need now to consider also Q(τ + 1 ) and P(τ + 1 ) (a simple shift of τ by 12 ). 2 2 One can then rewrite (3.36)–(3.39) in the following form: ! ! + 0 0 P(τ Q− ) (τ ) √ √ −iϕ −1 , + i κ+ e Rϕ Rϕ Ve+ = −i κ− + − 0 −P(τ + 1 ) 0 −Q(τ + 1 ) 2
2
(4.31) f− = W
√ i κ−
!
− P(τ )
0
0
− −P(τ +1) 2
√ − i κ+ eiϕ Rϕ
!
Q+ (τ )
0
0
−Q+ (τ + 1 )
−1 , Rϕ
2
(4.32)
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√ Ve− = −i κ−
!
Q+ (τ )
0
0
−Q+ (τ + 1 )
√ + i κ+ e−iϕ Rϕ
!
− P(τ )
0
0
− −P(τ +1)
2
−1 , Rϕ
2
(4.33) f+ = W
√ i κ−
!
Q− (τ )
0
0
−Q− (τ + 1 )
√ − i κ+ eiϕ Rϕ
2
!
+ P(τ )
0
0
+ −P(τ +1)
−1 , Rϕ
2
(4.34) where
1 0 Rϕ = 0 0
0
0
0
−eiϕ
e−iϕ
0
0
0
0 0 , 0 1
0 ≤ ϕ < 2π .
(4.35)
The action of Rϕ on t (ψ1 , ψ2 , ψ3 , ψ4 ) ∈ Vτ,b (given by (3.31)) simply interchanges ψ2 and ψ3 (up to a phase factor). This is a well-defined action, since ψ2 and ψ3 belong to the same Hilbert space Hk=τ + 12 . The above formulae give the impression that we have a one parameter family of matrix-valued operators. In fact, the parameter ϕ which is clearly absent in (3.36)–(3.39), is artificial. (Moreover, note that one can use a different ϕ for each of the above formulae.) The appearance of this parameter is reminiscent of an SU(2) symmetry hidden in the structure of the typical discrete series representations of osp(2/2). A discussion of this interesting point is beyond the scope of the present work, we will come back to it in a forthcoming publication [36]. Finally, let us discuss further on points A, B and C, and some of their consequences: I. As already mentioned at the end of Sec. 2.3, instead of working in a super-Hilbert space one can directly work in an associated Hilbert sum. For the matrix-valued operators of Secs. 3.2, 3.3 and 3.4 this correspondance is obvious from the relations obtained in A, B and C above. Note however that the superoperator version in case C is less obvious to obtain. II. From A, B and C, and Propositions 4.2–4.5, one can determine the domains of all the matrix-valued operators considered in this work. Indeed, for the odd operators one finds: II.A — DVe = Kk=τ ⊕ Hk=τ + 12 ⊂ Vτ and DW e ± = Hk=τ ⊕ Kk=τ + 12 ⊂ Vτ , where ± now Vτ is the super-Hilbert space of Sec. 3.2. This follows directly from A and Proposition 4.3. II.B — DFe = DQ+ ∩ DQ− = Kk=τ ⊕ Kk=τ + 12 ⊂ Vτ . This follows from B and ± Proposition 4.5.
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II.C — DVe = Kk=τ ⊕ Hk=τ + 12 ⊕ Kk=τ + 12 ⊕ Kk=τ +1 ⊂ Vτ,b and DW e± = ± Kk=τ ⊕ Kk=τ + 12 ⊕ Hk=τ + 12 ⊕ Kk=τ +1 ⊂ Vτ,b . This follows directly from C and Proposition 4.3. Here one needs to be careful about the ordering of the Hilbert spaces that occur with multiplicity 2 in the decomposition (3.31), since Rϕ interchanges their roles. Domains of the even operators follow from II.A, II.B and II.C, and the oddodd part of the defining relations of the Lie superalgebras under consideration, namely (2.7) and (2.8i). Straightforward computations based on the evaluation of domains of anticommutators lead to results in perfect agreement with the domain considerations of Sec. 2.4. Let n us for example considerothe first relation in (2.7). One e finds that DK e± = D(Fe± )2 = Ψ ∈ DFe± | F± Ψ ∈ DFe± , where the second equality is a definition. Using II.B one gets: DK e± = Uk=τ ⊕ Uk=τ + 12 ⊂ Kk=τ ⊕ Kk=τ + 12 ⊂ Vτ , where Uk was defined in (2.20). III. Similar arguments to those used in the preceding paragraph show that superunitarity of the superholomorphic discrete series representations considered here requires only the validity of (2.14) for X in the odd part of the corresponding Lie superalgebras. This originates from the fact that one can choose for both osp(1/2) and osp(2/2) a system of simple roots which is purely odd. In fact, for osp(1/2) this is the only possible choice. This is not so for osp(2/2) (see [2]). IV. Now, we come back to the fundamental role played by the relations (4.1)–(4.2) (or equivalently (4.13)–(4.14)) uncovered at the beginning of the present section. The expressions found in A, B and C not only confirm this fundamental role, but also show that (4.1)–(4.2) simply expresses super-unitarity of the atypical superholomorphic discrete series representations of osp(2/2). This fact follows from A and III. Hence, super-unitarity of the other two superholomorphic discrete series representations follows from the super-unitarity of the atypical superholomorphic discrete series representations of osp(2/2). This can be viewed as the quantum theoretical counterpart of a similar fact observed in [2] at the level of classical theory. Indeed, the supergeometry of the OSp(1/2) coadjoint orbits and of the OSp(2/2) typical coadjoint orbits, whose quantization leads, respectively, to the superholomorphic discrete series representations of Secs. 3.2 and 3.3, turns out to be completely determined by the supergeometry of the OSp(2/2) atypical coadjoint orbits (see [2] for more details). These observations deserve further investigations; we will come back to this point in a forthcoming publication. 5. Anticommutativity We have just shown that the self-adjointness of the two pairs of operators (Q1 , Q2 ) and (P1 , P2 ) ensures the super-unitarity of the superholomorphic discrete series representations considered in this work. Here we provide a functional analytic and operatorial theoretic analysis of special algebraic properties of these operators. This reveals an interesting connection between representation theory of
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orthosymplectic Lie superalgebras and that of the superalgebra underlying N = 2 supersymmetric quantum mechanics (SSQM). One can easily check that Q1 and Q2 (resp. P1 and P2 ) anticommute in a naive sense. Do they anticommute in the proper sense defined in Sec. 2.5? Proposition 5.1. Q1 and Q2 (resp. P1 and P2 ) are self-adjoint anticommuting operators in Wτ . Proof. The self-adjointness was already proven in Proposition 4.5. The proper anticommutativity follows from Proposition 2.8. Indeed, straightforward computations show that Q1 and Q2 (resp. P1 and P2 ) satisfy both conditions in (2.22) on their common dense domain Kk=τ ⊕ Kk=τ + 12 ⊂ Wτ . Arai’s characterization of the anticommutativity of self-adjoint operators we just used was originally devised in the context of supersymmetric quantum mechanics or more precisely in the context of the representation theory of the superalgebra underlying SSQM (see [20] for a precise definiton of SSQM). Our result shows then that super-unitarity of the superholomorphic discrete series representations of osp(1/2) and osp(2/2) follows from the super-unitarity of a specific holomorphic representation of the superalgebras underlying two N = 2 SSQM. The latter are essentially generated by the triplets (Q1 , Q2 , H1 ) and (P1 , P2 , H2 ), where (Q1 , Q2 ) (resp. (P1 , P2 )) play the role of supercharges while H1 = Q21 = Q22 (resp. H2 = P12 = P22 ) plays the role of the supersymmetric Hamiltonian. In the Hilbert space Wτ , the latter are represented by the following self-adjoint matrix-valued differential operators: ! ! d d 0 + 2τ 0 z dz z dz and H2 = . (5.1) H1 = d d +1 + 2τ 0 z dz 0 z dz There are other interesting connections of our results with another of Arai’s characterizations of proper anticommutativity, we will report on them in a forthcoming publication. 6. Conclusion Our present study of the functional analytic and the operatorial theoretic meaning of super-unitarity of the superholomorphic discrete series representations of osp(1/2, R) and osp(2/2, R) shows that super-unitarity is a consequence of the selfadjointness of two pairs of anticommuting operators (Q1 , Q2 ) and (P1 , P2 ) which act in the Hilbert sum of two Hilbert spaces each of which carrying a holomorphic discrete series representation of su(1, 1) such that the respective lowest weights differ by 1/2. Our analysis exhibits the fundamental role played by the atypical superholomorphic discrete series representations of osp(2/2, R) regarding super-unitarity of the other discrete series of osp(1/2, R) and osp(2/2, R). Direct consequences of these results were discussed in detail at the end of Sec. 4. Now, we would like to discuss possible generalizations of our work.
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(a) Higher rank orthosymplectic Lie superalgebras — It would be interesting to generalize our results to orthosymplectic Lie superalgebras of higher rank. This requires applying to the latter the entire program carried out for osp(1/2, R) and osp(2/2, R) in our previous work [8, 1, 2] as well as in the present contribution. Such a generalization would proceed in two stages involving, first, a classical mechanical (or geometric) part which consists in determining all the super-K¨ahler coadjoint orbits of the Lie supergroup corresponding to the considered Lie superalgebra, and second, a quantum mechanical (or a representation theoretic) part which consists in applying geometric quantization to these orbits. So far, only the first stage has been partly achieved in [37]. Indeed, let us recall that the N = 1 and N = 2 super unit discs studied in [8, 1, 2] are Z2 –graded extensions of the usual unit disc which is the simplest of the Cartan domains of type II. Cartan domains are irreducible symmetric Hermitian spaces of non–compact type which are K¨ ahler homogeneous spaces for simple Lie groups [38]. They are particular coadjoint orbits of the latter. The type II Cartan domains correspond to the homogeneous spaces Sp(2n, R)/U(n). (The unit disc occurs for n = 1.) The super-K¨ ahler structure of their Z2 –graded extensions was determined in [37], except for its geometric interpretation ` a la Rothstein as in [8, 2]. Note that in [37] another type of quantization was applied to these Cartan superdomains. This is the so-called non-perturbative quantization. Unlike geometric quantization, this procedure is not aimed at producing irreducible representations of the considered Lie (super)algebras. Hence, the analysis carried out here cannot be applied to the results obtained in [37], unless the second stage of our program (geometric quantization) is extended to the Cartan superdomains of type II. This would lead to an explicit construction of superholomorphic discrete series representations of OSp(N/2n, R) which would be concrete realizations (other than the oscillator representations considered in [6, 7]) of some of the abstract representations constructed in [5]. Finally, it would be interesting to apply our program to the Z2 –graded extensions of the other types of Cartan domains which are also described in [37]. (b) Matrix-valued operators versus superoperators, and Clifford algebras — Kostant made the following very interesting observation in [9]: applying prequantization to the simplest supersymplectic supermanifold which is just a supermanifold built over a zero-dimensional manifold, leads to a representation of a Clifford algebra which is a quantization of the exterior algebra defining the supermanifold. If one considers a less trivial supersymplectic supermanifold (by definition the latter is built over a symplectic manifold), and succeeds in applying geometric quantization to it, one would then naturally expect the outcome of this procedure to be in the form of a nontrivial combination of the quantization of the base symplectic manifold and the representation of a Clifford algebra. This is very much the case for the examples treated in [1, 2]. Spin structures and Clifford algebras appear at different stages of our program: they appear at the algebraic, the geometric and the representationtheoretic levels. For instance, the matrix realizations derived in Secs. 3.2, 3.3,
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and 3.4 give a flavor of this fact (see the comment at the end of Sec. 3.3). The matrix and the superoperator realizations correspond to two different realizations of a representation of a Clifford algebra. An understanding of these underlying structures will certainly help answering the questions raised above in (a) in an efficient way. (c) Algebraic quantization versus geometric quantization — Recently, we studied the interplay between two types of quantizations: algebraic quantization and geometric quantization [39]. We showed that these two procedures are complementary when used in the construction of holomorphic discrete series representations. It would be interesting to extend these results to the Z2 –graded context by exploring and devising the Z2 –graded algebraic ingredients needed for a superalgebraic quantization. Acknowledgements The author is indebted to S. T. Ali for very stimulating conversations. He thanks C. Duval, G. G. Emch, J. Harnad, G. Tuynman, P. Winternitz, and T. Wurzbacher for their interest in his work and for their much appreciated encouragements. This work was supported by a CRM–ISM fellowship, it was initiated while the author was a postdoctoral fellow at both the Centre de Recherches Math´ematiques of Universit´e de Montr´eal, and the Department of Mathematics and Statistics of Concordia University. The author thanks these institutions for their hospitality. References [1] A. M. El Gradechi, “Geometric quantization of an OSp(1/2) coadjoint orbit”, Lett. Math. Phys. 35 (1995) 13–26. [2] A. M. El Gradechi and L. M. Nieto, “Supercoherent states, super-K¨ ahler geometry and geometric quantization”, Commun. Math. Phys. 175 (1996) 521–563. [3] H. Furutsu and T. Hirai, “Representations of Lie superalgebras. I. Extensions of representations of the even part”, J. Math. Kyoto Univ. 28 (1988) 695–749. [4] K. Nishiyama, “Characters and super-characters of discrete series representations for orthosymplectic Lie superalgebras”, J. Algebra 141 (1991) 399–419. [5] H. P. Jakobsen, “The full set of unitarizable highest weight modules of basic classical Lie superalgebras”, Mem. Amer. Math. Soc. 111 (1994) (532). [6] K. Nishiyama, “Oscillator representations for orthosymplectic algebras”, J. Algebra 129 (1990) 231–262; “Decomposing oscillator representations of osp(2n/n; R) by a super dual pair osp(2/1; R)×so(n), Compositio Math. 80 (1991) 137–149; “Super dual pairs and highest weight modules of orthosymplectic algebras”, Adv. Math. 104 (1994) 66–89. [7] H. Furutsu and K. Nishiyama, “Realization of irreducible unitary representations of osp(M/N;R) on Fock spaces”, in The Proceedings of Fuji-Kawaguchiko Conference on Representation Theory of Lie Groups and Lie Algebras, eds. T. Kawazoe et al. 1–21, World Scientific, Singapore, 1992. [8] A. M. El Gradechi, “On the supersymplectic homogeneous superspace underlying the OSp(1/2) coherent states”, J. Math. Phys. 34 (1993) 5951–5963. [9] B. Kostant, “Graded manifolds, graded Lie theory and prequantization”, in Lecture Notes in Math., Vol. 570, Springer-Verlag, Berlin, 1977, 177–306.
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[10] B. Kostant, “Harmonic analysis on graded (or super) Lie groups”, in Lecture Notes in Phy., Vol. 79, 47–50, Springer-Verlag, Berlin, 1978. [11] V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform – Part I”, Commun. Pure Appl. Math. 14 (1961) 187–214. [12] V. Bargmann, “Irreducible unitary representations of the Lorentz group”, Ann. Math. 48 (1947) 568–640. [13] A. W. Knapp, Representation Theory of Semisimple Groups – An Overview Based on Examples, Princeton Univ. Press, Princeton, New Jersey, 1986. [14] V. Fock, “Verallgemeinerung und l¨ osung der Diracschen statistischen gleichung”, Z. Physik 49 (1928) 339–357. [15] P. A. M. Dirac, “La seconde quantification”, Ann. Inst. H. Poincar´ e 11 (1949) 15–47. [16] I. E. Segal, “Mathematical characterization of the physical vacuum for a linear Bose– Einstein field”, Illinois J. Math. 6 (1962) 500–523. [17] F.-H. Vasilescu, “Anticommuting self-adjoint operators”, Rev. Roumaine Math. Pures Appl. 28 (1983) 77–91. [18] Yu. S. Samoilenko, Spectral Theory of Families of Self-Adjoint Operators, Kluwer Academic Publishers, Dordrecht, 1991. [19] S. Pedersen, “Anticommuting Selfadjoint Operators”, J. Funct. Anal. 89 (1990) 428– 443. [20] A. Arai, “Analysis on Anticommuting Self-Adjoint Operators”, in Advanced Studies in Pure Math., Vol. 23, 1–15, North-Holland, Amsterdam, 1994. [21] V. Kac, “Representations of classical Lie superalgebras”, in Lecture Notes in Math., Vol. 676, 597–626, Springer-Verlag, Berlin, 1978. [22] M. Scheunert, “The Theory of Lie Superalgebras – An Introduction”, in Lecture Notes in Math., Vol. 716, Springer-Verlag, Berlin, 1979. [23] J. F. Cornwell, Group Theory in Physics Vol. 3 – Supersymmetries and Infinite Dimensional Algebras, Academic Press, London, 1989. [24] F. A. Berezin, Introduction to Superanalysis, Reidel, Dordrecht, 1987. [25] S. Sternberg and J. Wolf, “Hermitian Lie algebras and metaplectic representations”, Trans. Amer. Math. Soc. 238 (1978) 1–43. [26] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I, Academic Press, New York, 1972. [27] P. Renouard, Vari´et´es symplectiques et quantification, Thesis, Orsay, 1969. [28] S. De Bi`evre and A. M. El Gradechi, “Quantum mechanics and coherent states on the anti-de Sitter spacetime and their Poincar´e contraction”, Ann. Inst. H. Poincar´ e 57 (1992) 403–428. [29] H. Furutsu and K. Nishiyama, “Classification of irreducible super-unitary representations of su(p,q/n)”, Commun. Math. Phys. 141 (1991) 475–502. [30] J. W. B. Hughes, “Representations of Osp(2, 1) and the metaplectic representation”, J. Math. Phys. 22 (1981) 245–250. [31] J. Van der Jeugt, “Representations of N=2 extended supergravity and unitarity conditions in Osp(N,4)”, J. Math. Phys. 28 (1987) 758–764. ´ ements de la th´eorie des repr´esentations, Editions ´ [32] A. A. Kirillov, El´ Mir, Moscou, 1974. [33] N. M. J. Woodhouse, Geometric Quantization, Clarendon Press, Oxford, 1980. [34] D. A. Vogan, “The orbit method and unitary representations for reductive Lie groups”, in Perspectives in Math., Vol. 17, Academic Press, 1997. [35] F. A. Berezin, “General concept of quantization”, Commun. Math. Phys. 40 (1975) 153–174. [36] A. M. El Gradechi, in preparation. [37] D. Borthwick, S. Klimek, A. Lesniewski, and M. Rinaldi, “Matrix Cartan superdomains, super Toeplitz operators, and quantization”, J. Funct. Anal. 127 (1995) 456–510.
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[38] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. [39] S. T. Ali, A. M. El Gradechi, and G. G. Emch, “Modular algebras in geometric quantization”, J. Math. Phys. 35 (1994) 6237–6243.
ON NAMBU POISSON MANIFOLDS NOBUTADA NAKANISHI Department of Mathematics Gifu Keizai University 5-50 Kitagata-cho, Ogaki-city Gifu, 503, Japan E-mail : [email protected] Received 12 May 1997
1. Introduction In 1973, Y. Nambu [6] gave a generalization of classical Hamiltonian mechanics. Originally he considered his mechanics on R3 . The equation of motion of an observable f ∈ C ∞ (R3 ) is defined by df = {H1 , H2 , f } , dt where H1 , H2 ∈ C ∞ (R3 ) are two Hamiltonians. The bracket in the right-hand side is precisely defined by ∂(H1 , H2 , f ) , {H1 , H2 , f } = ∂(x, y, z) where (x, y, z) are the standard coordinates on R3 . About twenty years later, from the viewpoint of the generalization of classical Poisson brackets, Takhtajan [7] introduced so-called Nambu–Poisson brackets. Let M be a C ∞ -manifold. Then a Nambu–Poisson bracket is an n-linear skew-symmetric mapping from n-copies of C ∞ (M ) into C ∞ (M ), which satisfies the Leibniz rule and the Fundamental Identity: {f1 , . . . , fn−1 , {g1 , . . . , gn }} = {{f1 , . . . , fn−1 , g1 }, g2 , . . . , gn } + {g1 , {f1 , . . . , fn−1 , g2 }, g3 , . . . , gn } + · · · + {g1 , . . . , gn−1 , {f1 , . . . , fn−1 , gn }} for all f1 , . . . , fn−1 , g1 , . . . , gn ∈ C ∞ (M ). We should note that (f1 , . . . , fn−1 ) acts on {g1 , . . . , gn } as a derivation. If n = 2, we have usual Poisson manifolds. But if n ≥ 3, there appear some aspects which are different from the case of usual Poisson manifolds. More precisely, Nambu–Poisson structure should be more rigid than usual Poisson structure. (For example, see Theorem 5.5.) P. Gautheron [3] also proved the same result as ours in a completely different way. Using the Fundamental Identity, we know that the flow of the equation of motion induces an automorphism of a Nambu–Poisson bracket. 499 Reviews in Mathematical Physics, Vol. 10, No. 4 (1998) 499–510 c World Scientific Publishing Company
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For each Nambu–Poisson bracket, there corresponds the n-tensor (or the Vn T M by {f1 , . . . , fn } = η(df1 , . . . , dfn ). Such an n-tensor n-vector) η : M → η must, of course, satisfy the Fundamental Identity. We call this n-tensor Nambu– Poisson tensor . After geometrical formulations have been done by Takhtajan’s work, there can be found several papers on Nambu–Poisson geometry. (See, for example, [1, 2, 3].) In this paper, we shall define a kind of Poisson bracket on some function space, and construct the basic theory similar to the classical Poisson geometry. In particular, we shall also study the normal form of Nambu–Poisson tensors. 2. Nambu–Poisson Manifolds Let M be an m-dimensional C ∞ -manifold and denote by F the algebra of C ∞ functions on M . We shall define a Nambu–Poisson bracket and a Nambu–Poisson manifold following Takhtajan’s formalism [7]. Definition 2.1. A Nambu–Poisson bracket of order n, m ≥ n, on M is an n-linear skew-symmetric map from F n to F such that (1) (Leibniz rule) {f1 , . . . , fn−1 , g1 · g2 } = {f1 , . . . , fn−1 , g1 } · g2 + g1 · {f1 , . . . , fn−1 , g2 } , (2) (Fundamental Identity) {f1 , . . . , fn−1 , {g1 , . . . , gn }} = {{f1, . . . , fn−1 , g1 }, g2 , . . . , gn } + {g1 , {f1 , . . . , fn−1 , g2 }, g3 , . . . , gn } + · · · + {g1 , . . . , gn−1 , {f1 , . . . , fn−1 , gn }} for all f1 , . . . , fn−1 , g1 , . . . , gn ∈ F. Vn
To each Nambu–Poisson bracket, there corresponds an n-vector field η : M → T M such that {f1 , . . . , fn } = η(df1 , . . . , dfn ) ,
which satisfies the Fundamental Identity. Then η is called a Nambu–Poisson tensor of order n. We should remark that the Fundamental Identity implies strong constraints on n-tensor η [7]. Definition 2.2. Let η be a Nambu–Poisson tensor of order n on M . Then the pair (M, η) is called a Nambu–Poisson manifold. Let Rn = (x1 , . . . , xn ) be the n-dimensional Euclidean space. Then we can define a Nambu–Poisson bracket of order n by {f1 , . . . , fn } =
∂(f1 , . . . , fn ) , ∂(x1 , . . . , xn )
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for all f1 , . . . , fn ∈ C ∞ (Rn ). This is just the example which Nambu gave in his paper [6]. The corresponding Nambu–Poisson tensor η is given by η=
∂ ∂ ∧··· ∧ . ∂x1 ∂xn
We shall call this η the standard Nambu–Poisson tensor . And the pair (Rn , η) is said to be the standard Nambu–Poisson manifold . Moreover for a Nambu–Poisson manifold (M, η), it is said to be trivial if η = 0 on M . For classical Poisson manifolds, some fundamental notions were already defined [9]: Poisson brackets, Casimir functions, Hamiltonian vector fields, Poisson vector fields, etc. In the case of Nambu–Poisson manifolds, we also define the same notions as classical Poisson manifolds. Let (M, η) be a Nambu–Poisson manifold, where η is a Vn−1 P F, Nambu–Poisson tensor of order n. For an element A = fi1 ∧· · ·∧fin−1 ∈ we define a vector field XA by XA (h) =
X
{fi1 , . . . , fin−1 , h}
for all h ∈ F. Then XA is called a Hamiltonian vector field corresponding to Vn−1 Vn−1 F . If XA (h) = 0 for all h ∈ F, A ∈ F is called a Casimir function. A∈ We denote by C the space of Casimir functions. We denote by L the Lie algebra of infinitesimal automorphisms of (M, η). That is, L = {X ∈ χ(M )|L(X)η = 0} , where L(X) denotes the Lie derivative along X. Then we can easily prove the following: Proposition 2.3. A vector field X is contained in L if and only if it satisfies X · {f1 , . . . , fn } = {X · f1 , f2 , . . . , fn } + {f1 , X · f2 , . . . , fn } + · · · + {f1 , f2 , . . . , X · fn } for all f1 , . . . , fn ∈ F. Moreover we denote by H the Lie algebra of Hamiltonian vector fields. Then we shall prove in the next section that H is an ideal of L. 3. Structure of Poisson Brackets P Vn−1 F . Let A = fi1 ∧ · · · ∧ fin−1 First we define Poisson bracket [ , ] on P Vn−1 and B = gj1 ∧ · · · ∧ gjn−1 be any elements of F . Then Poisson bracket of A and B is defined by
502
N. NAKANISHI
[A, B] =
X
{fi1 , . . . , fin−1 , gj1 } ∧ gj2 ∧ · · · ∧ gjn−1
+ gj1 ∧ {fi1 , . . . , fin−1 , gj2 } ∧ gj3 ∧ · · · ∧ gjn−1
+ · · · + gj1 ∧ · · · ∧ gjn−2 ∧ {fi1 , . . . , fin−1 , gjn−1 } . Vn−1 F is obtained by exAs is easily seen, this definition of Poisson bracket on Vn−1 F . We may also write this bracket as L(XA )B. tending the action of XA to If n = 2, our definition of Poisson bracket agrees with the usual Poisson bracket. Hence it follows that [A, B] = −[B, A]. If n ≥ 3, the situation is quite different. In fact, in this case the bracket operation is not generally skew-symmetric. But we can prove Vn−1 F , [A, B] + [B, A] is a Casimir Lemma 3.1. Let n ≥ 3. For all A, B ∈ function. In particular, if C is a Casimir function, then [C, A] = 0 and [A, C] ∈ C. Proof. By using the Fundamental Identity, we have for all h ∈ F: X[A,B] (h) =
X {{fi1 , . . . , fin−1 , gj1 }, gj2 , . . . , gjn−1 , h} + {gj1 , {fi1 , . . . , fin−1 , gj2 }, gj3 , . . . , gjn−1 , h} + · · · + {gj1 , . . . , gjn−2 , {fi1 , . . . , fin−1 , gjn−1 }, h}
=
X {fi1 , . . . , fin−1 , {gj1 , . . . , gjn−1 , h}} − {gj1 , . . . , gjn−1 , {fi1 , . . . , fin−1 , h}}
=
X {fi1 , . . . , fin−1 , {gj1 , . . . , gjn−1 , h}} − {{gj1 , . . . , gjn−1 , fi1 }, fi2 , . . . , fin−1 , h} − · · · − {fi1 , . . . , fin−2 , {gj1 , . . . , gjn−1 , fin−1 }, h} − {fi1 , . . . , fin−1 , {gj1 , . . . , gjn−1 , h}}
= −X[B,A] (h) . This implies that [A, B] + [B, A] ∈ C. By the definition of Poisson bracket, it is obvious that if C ∈ C, then [C, A] = 0, and hence we have [A, C] ∈ C. Remark. Even in the case of n = 2, since C contains zero, we can say that [A, B] + [B, A] ∈ C. Thus for every n ≥ 2, Lemma 3.1 is valid. Lemma 3.2. [XA , XB ] = X[A,B] , for all A, B ∈
Vn−1
F.
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ON NAMBU–POISSON MANIFOLDS
Proof. For all h ∈ F, we have [XA , XB ](h) = XA (XB (h)) − XB (XA (h)) X {fi1 , . . . , fin−1 , {gj1 , . . . , gjn−1 , h}} = − {gj1 , . . . , gjn−1 , {fi1 , . . . , fin−1 , h}} =
X {{fi1 , . . . , fin−1 , gj1 }, gj2 , . . . , gjn−1 , h} + · · · + {gj1 , . . . , gjn−2 , {fi1 , . . . , fin−1 , gjn−1 }, h}
= X[A,B] (h) . This completes the proof.
V V Lemma 3.3. Let π : n−1 F → ( n−1 F )/C be the natural projection. Put ¯ B] ¯ = ¯ Then the bracket operation on (Vn−1 F )/C can be defined by [A, π(A) = A. Vn−1 [A, B] for all A, B ∈ F. Proof. By Lemma 3.1, we know that [A + C1 , B + C2 ] = [A, B] + [A, C2 ] for all Vn−1 F )/C C1 , C2 ∈ C. This implies that we can define the bracket operation on ( ¯ ¯ by [A, B] = [A, B]. Vn−1 F )/C Poisson bracket , and denote it by the We also call this bracket on ( Vn−1 F with Poisson bracket [ , ] does not same symbol. If n ≥ 3, recall that admit Lie algebra structure, because of lack of skew-symmetry. Hence we should V move to ( n−1 F )/C to make use of the theory of Lie algebras. Vn−1 F )/C has a structure of a Lie algebra, and Proposition 3.4. The space ( it is isomorphic to H as Lie algebras. Proof. By Lemma 3.1, it is clear that ¯ B] ¯ + [B, ¯ A] ¯ = [A, B] + [B, A] = 0 . [A, ¯ A] ¯ = −[A, ¯ B]. ¯ Next we prove Jacobi identity. Since X[F,G] (H) = [XF , XG ] Thus [B, Vn−1 F by Lemma 3.2, we have (H) for all F , G, H ∈ [[F, G], H] = [F, [G, H]] − [G, [F, H]] . Hence ¯ H] ¯ = [F¯ , [G, ¯ H]] ¯ − [G, ¯ [F¯ , H]] ¯ . [[F¯ , G], Vn−1 F )/C is skew-symmetric, we obtain Since Poisson bracket on ( ¯ H] ¯ + [[G, ¯ H], ¯ F¯ ] + [[H, ¯ F¯ ], G] ¯ = 0. [[F¯ , G],
504
N. NAKANISHI
Vn−1 A linear mapping F → H, (A 7→ XA ) is surjective and its kernel is C. ComVn−1 F )/C ∼ bining this with Lemma 3.2, we obtain that ( = H as Lie algebras. For any A =
P
fi1 ∧ · · · ∧ fin−1 ∈
Vn−1
F and g1 , . . . , gn ∈ F, it is clear that
XA · {g1 , . . . , gn } = {XA · g1 , g2 , . . . , gn } + · · · + {g1 , . . . , gn−1 , XA · gn } . (This is just the Fundamental Identity.) This implies H ⊂ L. Moreover for any Y ∈ L and h ∈ F, we have X X {fi1 , . . . , fin−1 , h} − {fi1 , . . . , fin−1 , Y · h} [Y, XA ](h) = Y · =
X
{Y · fi1 , . . . , fin−1 , h} + · · · + {fi1 , . . . , Y · fin−1 , h}
= XB (h) , P
where B = L(Y )(fi1 ∧ · · · ∧ fin−1 ) ∈ proved the following:
Vn−1
F . Hence [Y, XA ] ∈ H. Thus we have
Proposition 3.5. H is an ideal of L. 4. The Spaces L/H of Nambu Poisson Manifolds In the theory of Poisson manifolds, it is well known that the notion of Poisson cohomologies is a matter of great importance. But unfortunately it is difficult to calculate them, when, in particular, Poisson tensors have singularities. (See [5, 8].) For a Nambu–Poisson manifold (M, η), any suitable generalizations of the usual Poisson cohomology have not been found yet. In this section, the spaces L/H are calculated for some Nambu–Poisson manifolds. In the case of Poisson manifolds, these spaces L/H are nothing but the spaces of the first Poisson cohomology groups. As the first step for the cohomology theory, it may be interesting to determine the space L/H even if some conditions are imposed. In the first place, we assume that a manifold M is paracompact and that the dimension of M is equal to the order of η. Recall that any n-vector field η on an n-dimensional manifold becomes a Nambu–Poisson tensor by the result of P. Gautheron [3]. Moreover η is assumed to be nowhere vanishing on M . Thus in our case, η is non-vanishing Nambu–Poisson tensor of order n. The volume form ω corresponding to η, can be defined as follows [3]: X1 ∧ · · · ∧ Xn = (−1)n−1 ωx (X1 , . . . , Xn )ηx , for all X1 , . . . , Xn ∈ Tx (M ). Using this volume form ω, the following facts are easily proved: Lemma 4.1. For a vector field X, it satisfies L(X)η = 0 if and only if L(X) ω = 0.
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ON NAMBU–POISSON MANIFOLDS
Lemma 4.2. Let A = ∧ · · · ∧ dfin−1 .
P
fi1 ∧ · · · ∧ fin−1 ∈
Vn−1
F . Then i(XA )ω =
P
dfi1
Since ω is non-degenerate, the following mapping n−1 (M ) , τ : L → HdR
(X 7→ [i(X)ω])
is surjective, where [i(X)ω] denotes the cohomology class of a closed (n − 1)-form i(X)ω. By Lemma 4.2, it holds that [i(XA )ω] = 0. Thus we have H ⊂ ker τ . Conversely let X be an element of ker τ . First note that any (n − 2)-form β P fi1 dfi2 ∧ · · · ∧ dfin−1 . on a paracompact C ∞ -manifold M can be written as β = Since i(X)ω is an exact (n − 1)-form, there exists an (n − 2)-form β such that X i(X)ω = dβ = dfi1 ∧ · · · ∧ dfin−1 . Then by Lemma 4.2, we have X = XA
for A =
X
fi1 ∧ · · · ∧ fin−1 ∈
^
n−1
F.
This means that ker τ ⊂ H. Thus we have proved: Theorem 4.3. Let (M, η) be an n-dimensional Nambu–Poisson manifold with n−1 (M ). non-vanishing η of order n. Then L/H is isomorphic to HdR Secondly let us consider the case that η is a tensor of order n on Rn+1 which ∂ ∧ · · · ∧ ∂x∂n+1 on Rn+1 . is defined by the standard Nambu–Poisson tensor η0 = ∂x 1 More precisely, η is defined by {f1 , . . . , fn }η = {f1 , . . . , fn , f }η0 , where f1 , . . . , fn are arbitrary C ∞ -functions on Rn+1 , and f ∈ C ∞ (Rn+1 ) is a fixed function. Then it is easy to see that η actually becomes a Nambu–Poisson tensor. We denote by Ω the standard volume form on Rn+1 . Under these notations, we prove: Lemma 4.4. X ∈ L if and only if d(Xf ) = (divΩ X) · (df ). Proof. For any f1 , . . . , fn ∈ C ∞ (Rn+1 ) and any vector field X, we have (L(X)η)(df1 , . . . , dfn ) = X · ({f1 , . . . , fn }η ) − {Xf1, f2 , . . . , fn }η − · · · − {f1 , . . . , fn−1 , Xfn }η = X · ({f1 , . . . , fn , f }η0 ) − {Xf1 , f2 , . . . , fn , f }η0 − · · · − {f1 , . . . , fn−1 , Xfn , f }η0 − {f1 , . . . , fn , Xf }η0 + {f1 , . . . , fn , Xf }η0 = (L(X)η0 )(df1 , . . . , dfn , df ) + η0 (df1 , . . . , dfn , d(Xf )) = −(divΩ X) · η0 (df1 , . . . , dfn , df ) + η0 (df1 , . . . , dfn , d(Xf )) .
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N. NAKANISHI
Thus if X ∈ L, then η0 (df1 , . . . , dfn , d(Xf )) = (divΩ X) · η0 (df1 , . . . , dfn , df ). This implies that d(Xf ) = (divΩ X) · (df ). The converse is clear. Since η is defined by the standard Nambu–Poisson tensor η0 , it is quite easy to show the following lemma. Lemma 4.5. For XA ∈ H, where A ∈ divΩ XA = 0.
Vn−1
C ∞ (Rn+1 ), we have XA f = 0 and
If we take f = 12 (x21 + x22 − x23 − · · · − x2n+1 ), a linear Nambu–Poisson tensor η on Rn+1 is defined by {f1 , . . . , fn }η = {f1 , . . . , fn , f }η0 . Let m(t) be a C ∞ -function of one variable which is flat at the origin and is zero if t ≤ 0. Using these functions, let us define a vector field X by x2 ∂ ∂ x1 + 2 . X = m(f ) x21 + x22 ∂x1 x1 + x22 ∂x2 Then X satisfies d(Xf ) = (divΩ X)(df ), hence X ∈ L by Lemma 4.4. On the contrary, since Xf = m(f ) 6= 0, X is not contained in H by Lemma 4.5. Thus we can easily conclude as follows: Proposition 4.6. If η is a linear Nambu–Poisson tensor on Rn+1 induced from f = 12 (x21 + x22 − x23 − · · · − x2n+1 ), then the space L/H is infinite dimensional. Next if we take f = 12 (x21 + x22 + · · · + x2n+1 ), we have another linear Nambu– Pn+1 ∂ , n ≥ 3, be an element of L. Poisson tensor π of order n. Let X = i=1 fi ∂x i Pn+1 Then by using the same method as [4], we can obtain that i=1 xi fi = 0, and divΩ X = 0. By Lemma 4.5, these are necessary conditions for X to be contained in H. If n = 2, π is just a linear Poisson tensor on so(3, R)∗ , and it is well known that L = H. So it may be natural that we conjecture as follows: Conjecture. For a linear Nambu–Poisson tensor π of order n, (n ≥ 3), it also holds that L = H. 5. Canonical Local Coordinates of Nambu Poisson Manifolds Let (M, η) be a Nambu–Poisson manifold of order n. A point x0 ∈ M is called regular if η(x0 ) 6= 0. Then the main theorem of this section states that around a regular point every Nambu–Poisson manifold with a Nambu–Poisson tensor η of order n ≥ 3 is locally written as the product of a standard Nambu–Poisson manifold and a trivial Nambu–Poisson manifold. Note that the product of two Nambu–Poisson manifolds is, in general, no longer a Nambu–Poisson manifold. But one of them is, in our case, a “trivial” Nambu– Poisson manifold. So the product manifold is also a Nambu–Poisson manifold in a natural manner.
ON NAMBU–POISSON MANIFOLDS
507
Suppose that η 6= 0 at x0 . Then there are n-local functions x1 , . . . , xn−1 , x0n such that {x1 , . . . , xn−1 , x0n }(x0 ) 6= 0. Since a Hamiltonian vector field Xx1 ∧···∧xn−1 (x0 ) 6= 0, there exists a system of local coordinates (z1 , . . . , zm ) around x0 such that Xx1 ∧···∧xn−1 = ∂z∂ 1 . Rewriting z1 = xn , we have {x1 , . . . , xn−1 , xn } = 1. We shall define n-Hamiltonian vector fields {Yi }1≤i≤n by Yi = (−1)n−i Xx1 ∧···∧ˆxi ∧···∧xn , where the symbol x ˆi denotes the absence of the corresponding factor. Lemma 5.1. Y1 , . . . , Yn are n-linearly independent vector fields which commute each other around x0 . Proof. Since Yi (xj ) = δij , it is clear that Y1 , . . . , Yn are linearly independent. For proving the commutativity of {Yi }, it suffices to show the case i > j. If i > j, one has [x1 ∧ · · · ∧ xˆi ∧ · · · ∧ xn , x1 ∧ · · · ∧ xˆj ∧ · · · ∧ xn ] ˆj ∧ · · · ∧ xi−1 ∧ 1 ∧ xi+1 ∧ · · · ∧ xn . = (−1)n−i x1 ∧ · · · ∧ x Hence [Yi , Yj ] = (−1)2n−i−j [Xx1 ∧···∧ˆxi ∧···∧xn , Xx1 ∧···∧ˆxj ∧···∧xn ] = (−1)n−j Xx1 ∧···∧ˆxj ∧···∧xi−1 ∧1∧xi+1 ∧···∧xn = 0,
and this proves lemma.
By virtue of Lemma 5.1 and the theorem of Frobenius, one can find local coordinates (a1 , . . . , an , b1 , . . . , bs ), (n + s = m) with Yi =
∂ , ∂ai
(i = 1, 2, . . . , n) .
Each Yi clearly satisfies Yi (bj ) = 0 ,
(1 ≤ i ≤ n, 1 ≤ j ≤ s) .
Lemma 5.2. (x1 , . . . , xn , b1 , . . . , bs ) is a system of local coordinates around x0 . Proof. Since
∂xi ∂aj
= Yj (xi ) = δij , we have ∂(x1 , . . . , xn , b1 , . . . , bs ) 6= 0 . ∂(a1 , . . . , an , b1 , . . . , bs )
Lemma 5.3. With respect to the new local coordinates (x1 , . . . , xn , b1 , . . . , bs ), it holds
508
N. NAKANISHI
∂ (a) Yi = ∂x , (1 ≤ i ≤ n), i (b) {xi1 , . . . , xin−1 , bj } = 0, (c) {xi1 , . . . , xik , bj1 , . . . , bjl } = 0, (k + l = n, k ≥ 1) .
Proof. Note that Yi (xj ) = δij and Yi (bj ) = 0. Then (a) is clear. For the proof of (b), it suffices to show that {x1 , . . . , xn−1 , bj } = 0. (Other cases can be shown in the same manner.) In fact we have {x1 , . . . , xn−1 , bj } = Yn (bj ) = 0. It will be enough to prove (c) in the case where {x1 , . . . , xk , bj1 , . . . , bjl } = 0 only. 1 k−1 2 (−1) x1 , x2 , . . . , xn−1 , {x2 , . . . , xk , xn , bj1 , . . . , bjl } 2 using the Fundamental Identity 2 1 k−1 x1 , x2 , . . . , xn−1 , xn , bj1 , . . . , bjl = x2 , . . . , xk , (−1) 2 = x2 , . . . , xk , (−1)k−1 x1 , bj1 , . . . , bjl = x1 , . . . , xk , bj1 , . . . , bjl on the other hand, by the Leibniz rule = (−1)k−1 x1 {x1 , x2 , . . . , xn−1 , {x2 , . . . , xk , xn , bj1 , . . . , bjl }} = (−1)k−1 x1 {x2 , . . . , xk , 1, bj1 , . . . , bjl } = 0. Hence we have {x1 , . . . , xk , bj1 , . . . , bjl } = 0.
Lemma 5.4. Nambu–Poisson brackets {bj1 , . . . , bjn } are functions of b1 , . . . , bs only. Proof. By Lemma 5.3 (a), we have for any xi (1 ≤ i ≤ n) ∂ {bj , . . . , bjn } = Yi {bj1 , . . . , bjn } ∂xi 1 ˆi , . . . , xn , {bj1 , . . . , bjn }} = (−1)n−i {x1 , . . . , x using Lemma 5.3 (b) and the Fundamental Identity = 0.
This completes the proof.
Assume that η 6= 0 at x0 ∈ M . Then by Lemma 5.3 (b) and (c), we can find a system of local coordinates (x1 , . . . , xn , b1 , . . . , bs ) such that η=
X ∂ ∂ ∂ ∂ ∧ ···∧ + Pj1 ...jn ∧ ···∧ . ∂x1 ∂xn j π(k + 1) and let π 0 := tk π. If tk1 . . . tkl is a decomposition of π 0 into a minimal number of transpositions then tk tk1 . . . tkl = tk π 0 is a decomposition of π into a minimal number of transpositions and l(π) = l(π 0 ) + 1. Therefore for all k Sn =
X
π∈Πn π(k)>π(k+1)
π∈Πn π(k) 0, p0 ∈ πT (2Z + 1). For T = 0, p0 ∈ R. The function h is a C ∞ cutoff function that is zero for x ≤ 1/4 and one for x ≥ 1. Without the infrared cutoff ε, G(χ, χ) ¯ would not be defined because of the infrared divergences mentioned above. The cutoff removes these divergences because it cuts off the singularity of the propagator C. When ε → 0, the coefficients in the perturbation expansion of Gε in powers of λ diverge because they are not yet renormalized. Renormalization will remove these divergences and we can then take the limit ε → 0. In the following, I set T = 0 and Λ = Zd to discuss the singularities that give rise to the infrared divergences. If λ = 0, the electrons are independent and one can calculate the correlation functions simply by doing a Fourier transform. For the lattice systems discussed above, Fourier space is given by B = Rd /Γ# where
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
557
Γ# is the lattice dual to the position space lattice, e.g. Γ# = 2πZd for Γ = Zd . The Fourier transform of the hopping term gives the band structure (or dispersion relation) X tx e−ip·x − µ (2.11) e(p) = − x
Pd which, for the Hubbard case, reduces to e(p) = −2t i=1 cos pi − µ. At zero temperature, all states with e(p) < 0 are filled. The boundary of the occupied region in k-space is the Fermi surface S = {p ∈ B : e(p) = 0} .
(2.12)
1 N is given by the volume enclosed by S (in two dimensions The density ρ = |Λ| by the area inside S). In the example of the Hubbard model, the function e has its minimum at p = 0, and it is strictly convex (and analytic) near this minimum. Consequently, the Fermi surface is strictly convex for µ slightly larger than −2td. This is the generic behaviour of systems in solid state models: the band function is strictly convex around a minimum, so a small (but macroscopic) occupation of electrons in that conduction band gives rise to a strictly convex, curved Fermi surface. As the filling increases (which happens if µ is increased), the shape of the Fermi surface changes and it even becomes diamond-shaped at µ = 0 (half-filling) for the H0 of the Hubbard model when d = 2. Before discussing renormalization, I state our hypotheses more precisely. We assume that the Fourier transform vˆ of the two-body potential v is vˆ ∈ C 2 (R×B, C), that (2.13) vˆ(−p0 , p) = vˆ(p0 , p) ,
and that all derivatives of vˆ up to second order are bounded functions on R × B. Since λ and vˆ appear only in the combination λˆ v , we may assume that |ˆ v2 | ≤ 1, P where |f |2 = |α|≤2 kDα f k∞ . Note that the interaction potential may depend on p0 as well. To show convergence at large p0 we need only that vˆ approaches a finite limit as p0 → ±∞ (see [8, Hypothesis (H1)] for details). These assumptions about the behaviour at large |p0 | are satisfied in all the models discussed above, they are in fact much weaker than the usual analyticity assumptions (if the interaction is instantaneous, as in the Hamiltonian used in the above motivation, its Fourier transform vˆ is even independent of p0 ). The physically relevant assumption is the regularity of vˆ because it requires sufficient decay in position space. We assume that e ∈ C 2 (B, R) and that for all p ∈ S, ∇e(p) 6= 0. This implies that the Fermi surface is a C 2 -submanifold of B. Moreover, we assume one of the following: (A) S has no identically flat sides (for a precise definition see [7, Assumption A3]) (B) S is strictly convex with strictly positive curvature (for a precise definition, see [8, Hypotheses (H3)–(H5)]) These assumptions exclude half-filling (µ = 0) because there the Fermi surface has flat sides and because the gradient vanishes at the corners of the diamond. I will
558
M. SALMHOFER
not discuss this case further here; the renormalization problem is actually simpler in that case because the Fermi surface stays fixed by the particle–hole symmetry. Obviously, assumption (B) implies assumption (A). The connection between the singularity of C and the infrared divergences in the model is easy to see: the functional Gε has an expansion in the coupling and the fields, ¯ = Gε (ψ, ψ)
X r≥1
r
λ
2(r+1) Z 2m X Y m=0
dpi δ(p1 + · · · + pm − pm+1 − · · · − p2m )
i=1
×Gε,mr (p1 , . . . , p2m )
m Y
¯ i )ψ(pm+i ) ψ(p
(2.14)
i=1
with the kernels Gε,mr given by a sum over values of Feynman diagrams. Every such contribution is a finite-dimensional integral. The integrand consists of various combinations and powers of C given by the Feynman rules. However, powers of C are in general not locally integrable: introducing variables ρ transversal and ω tangential to S, the integral Z |ip0 −e(p)| 0 if d = 2 and σ ∈ C 2 for d ≥ 3. We now proceed to do renormalization using counterterms instead of putting the σ into the denominator. For the purposes of the following discussion, putting counterterms in the action makes the concepts clearer in this problem. Another reason to do that is that we can show more regularity of the counterterm function K which essentially restricts σ to the Fermi surface than of the self-energy σ itself: we have shown that K is C 2 for all d ≥ 2. If one prefers to change the propagator, one should put K in there instead of σ to use our bounds. One way to motivate putting counterterms is as follows: since turning on λ makes the Fermi surface move, and since this movement causes all the trouble with the expansion, one can try to add a function K(λ, p) to the bilinear part of the action such that the Fermi surface S stays fixed. In other words, K compensates all self-energy corrections that would move the Fermi surface under the interaction. Theorem 2.1. Assume (A). There is a formal power series K (ε) (λ, p) =
∞ X
λr Kr(ε) (p)
(2.19)
r=1
such that the model defined as Z R ¯ ¯ G ren (ψ,ψ) ¯ −λV (χ,χ)− = dµCε (χ − ψ, χ ¯ − ψ)e e
(ε) dpχ(p)K ¯ (λ,p)χ(p)
(2.20)
ren has Fermi surface fixed to S = {p : e(p) = 0}. Moreover, the kernels Gren m,r of G (ε) all have finite limits as ε → 0, and K (λ, p) has a finite limit K(λ, p). The Borel transform in λ of G has a positive radius of convergence uniformly in ε.
560
M. SALMHOFER
Thus, fixing the Fermi surface indeed removes all infrared divergences. It is interesting to note that the counterterms are finite. This theorem [7, Theorem 1.2] is a nontrivial extension of the statements proven in [5] because the counterterms are momentum dependent. The dependence of K on p is really there in absence of rotational symmetry, and it leads to substantial technical complications. K is also a functional of e, so K = K(λ, e, p). Using (2.13), one can show that the self-energy σ satisfies σ(−p0 , p) = σ(p0 , p). This implies that K(λ, p) ∈ R for all p because K is constructed from the self-energy by evaluating at p0 = 0 and p ∈ S. More technically speaking, the graphs contributing to K are the two-legged one-particle irreducible graphs that also contribute to σ, but they are evaluated at p0 = 0 and p ∈ S (see [7, Sec. 2.3]). Although we have now removed the infrared divergences, we have done so at the price of changing the model. Because of the counterterm function K, the quadratic part of the action is now Z ¯ (2.21) A0 = dpψ(p)(ip 0 − e(p) − K(λ, p))ψ(p) and corresponds to a free Hamiltonian with dispersion relation e + K, which is λ-dependent, instead of e. Thus, if e is the free dispersion relation, Theorem 2.1 makes a statement not about the original model but about a changed model. To do the renormalized expansion for a prescribed free model with band structure E and interaction V , one has to solve the equation E(p) = e(p) + K(λ, e, p)
(2.22)
for e. Equation (2.22) is the central equation of the problem. I first explain how we solve it and then show how to renormalize without changing the model by using the solution e = R(E, λ) of (2.22) (e also depends on the two-body potential vˆ, but all bounds are uniform in vˆ for the set of vˆ specified above, so I suppress that dependence in the notation). Let K (R) (λ, p) =
R X
λr Kr (p)
(2.23)
r=1
be the function K up to order R in perturbation theory. Crudely speaking, the right-hand side of (2.22) is the identity plus a small term, because K (R) is of order λ, so an iteration is the natural strategy to get a solution. However, because of the various dependences of K (R) on e and p one has to be very careful what one means by small (no matter how small λ is chosen, the properties of the sum f + λg will differ very much from those of f if g is more singular than f ). Since e ∈ C 2 is the basic condition for all our bounds, we need at least K (R) ∈ C 2 , because in every step of the iteration, e gets replaced by e + K. Also, to use a fixed (R) point theorem, one needs control over δKδe . But since 1 1 δ ∼ , δe ip0 − e (ip0 − e)2
(2.24)
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taking such derivatives seems to lead to new divergences. Nonetheless we have the following theorem [7, Theorems 1.2 and 1.6]. Theorem 2.2. Assume (A). Then for all R ∈ N, K (R) (λ, ·) ∈ C 1 (B, R), and is also C 1 in e. Denote the Fr´echet derivative of K with respect to e by K (R) δK ∈ L(C 1 , C 0 ) and the sup norm on C 0 by |·|0 . Then for all h ∈ C 1 (B, R) δe δK (R) (2.25) δe (h) ≤ const |λ| |h|0 . 0 (R)
(R)
Because of (2.25), δKδe extends uniquely to a bounded linear operator from C 0 to C 0 . The set of e satisfying (A) is open, so if e1 and e2 are close enough, (A) holds for all e on the line connecting e1 and e2 . Then e1 + K (R) (e1 ) = e2 + K (R) (e2 ) implies R1 (R) by Taylor expansion that (1+L)(e2 −e1 ) = 0, where L = 0 dt δKδe ((1−t)e1 +te2 ). (R)
Since R is fixed and δKδe is a bounded operator for all t, 1 + L is invertible for λ small enough. Thus, we have [7, Theorem 1.7]: Theorem 2.3 Assume (A). For all R ∈ N, there is λR > 0 such that for all λ ∈ (−λR , λR ), the map e 7→ e + K (R) is locally injective. This implies uniqueness of the solution under the quite general conditions (A). The existence proof requires the stronger assumptions (B) because for that, we need to show that K is even in C 2 . It is a priori not clear that K must have the same differentiability properties as e. One might be in the situation that one always loses some regularity, i.e. that e ∈ C k only implies K ∈ C k−1 , or that even e ∈ C ∞ leads only to K ∈ C k0 for some fixed k0 . It took us some time and optimal bounds to prove that for k = 2, there is no loss of regularity. Theorem 2.4. There is an open set E ⊂ C 2 of dispersion relations e fulfilling (B) and e(−p) = e(p) such that K (R) ∈ C 2 for all e ∈ E. There is an open subset E 0 ⊂ E and for all R ∈ N, there is λR > 0 and a map R : (−λR , λR ) × E 0 → E such that for all (λ, E) ∈ (−λR , λR ) × E 0 , e = R(λ, E) solves (2.22), with K replaced by its truncation to order R, K (R) . The regularity statement is [9, Theorem 1.1]. The inversion statement is proven in [10]. To state a similar theorem for the nonsymmetrical case requires introducing older continuity of the second spaces of C 2+ -functions (the extra is meant as -H¨ derivative), because in the nonsymmetric case, one needs to prove more regularity to bound the particle–particle ladders. This was done in [8], and the regularity statements of the above theorems in the C 2+ class of functions are proven in [8, 9]. For conciseness, I do not state all the details here; they are provided in [8]. Using Theorem 2.4, we can now do renormalization without changing the model, as follows. Let the model be given by a potential V and by a dispersion relation E for the independent electrons, with E ∈ E 0 . Let R ∈ N and λ ∈ (−λR , λR ). Use Theorem 2.4 to determine e = R(E, λ), and set κ(E, λ, p) = K(λ, R(E, λ), p).
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M. SALMHOFER
Then E = e + κ = e + K(e). Denoting the propagator with E by C(E) and the one with e by C(e), we have, by standard shift formulas for Gaussian measures, the identity Z ¯ ¯ −λV (χ,χ) dµC(E) (ψ − χ, ψ¯ − χ)e =
Ze ZE
Z
¯
¯ −(ψ−χ,K(ψ−χ)) ¯ dµC(e) (ψ − χ, ψ¯ − χ)e ¯ −λV (χ,χ) e
Ze −(ψ,Kψ) ¯ = e ZE
Z
(2.26)
¯ ¯ χ,Kχ) ¯ ¯ dµC(e) (ψ − χ, ψ¯ − χ)e ¯ −λV (χ,χ)−( e(ψ,Kχ)+(χ,Kψ) .
This is an identical rewriting of the generating functional for the model given by E and V in terms of the quantities e and K that appear in the renormalized expansion, obtained by moving the K from the propagator to the interaction. Since E = e+K, this leaves e in the propagator. The change in normalization factor is irrelevant for any correlation function, and the extra source terms in the integrand just modify the external legs in a trivial way. This identity also holds if a cutoff ε > 0 is in place. In that case, all bounds are uniform in ε and Theorem 2.1 implies that the (R) kernels Gm,ε converge as ε → 0 (here the superscript R indicates G up to order R in λ, similarly to the definition of K (R) ). Physically, this procedure means the following. Applying the map R, i.e. going from E to e, shifts the Fermi surface from the free surface S(E) to the interacting Fermi surface S(e). Thus, in this step, the deformation of the surface caused by the interaction is taken into account. The renormalized expansion is then done at fixed interacting Fermi surface S(e), and it can be used to calculate other self-energy effects, and the other correlation functions. As mentioned in the statement of Theorem 2.1, the bounds that we prove for the kernels are not sufficient (and shouldn’t be sufficient) to show convergence of the perturbation series in λ. This is the reason for the explicit restriction to a finite order in perturbation theory in the other theorems. The general bound we obtain is the standard de Calan–Rivasseau bound, e.g. for the two-point function G2 (p) =
∞ X
G2,r (p)λr
(2.27)
r=0
it reads |G2,r (p) ≤ r! Qr ,
(2.28)
where Q is some constant. If this bound is saturated, the perturbation series has convergence radius zero, and more precisely, it means that the λR of the above theorems behaves as 1 (2.29) λR ∝ . R Even if the perturbation series diverges, it is conceivable that the map from the interacting to the free dispersion relation is invertible for some range of coupling
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
563
constants of the form (0, λ0 ), although analyticity does not hold. The investigation of the reasons for such a nonanalyticity is, of course, very important. The renormalization method yields statements about how and why these factorials can appear: (1) In order r, there are so many graphs that bounding the sum of graphs by the sum of their absolute values already gives an r factorial. This bound is not sharp because in it, the Pauli principle is ignored. For fermions, one may expect sign cancellations, such as in determinant bounds, to be useful. Determinant bounds do not work uniformly in the cutoff in these models, however, and it is a hard problem to implement the Pauli principle to show that the number of graphs does not produce a factorial. This was done by Feldman, Magnen, Rivasseau, and Trubowitz [4] for d = 2. A similar result is expected to hold for fermions in any dimension. (2) Singularities in values of individual four-legged diagrams can produce r factorials as well. The best-known example of this are the BCS ladders which produce symmetry breaking [6]. For item (2), the improved power counting method provides the following theorem. Theorem 2.5. Assume (A). The only graphs that can produce r factorials are generalized ladder graphs. For details and the proof, see [8, Theorem 2.46] and the next section. This theorem holds for any d ≥ 2 and for the very general class of Fermi surfaces satisfying only the condition (A) of non-flatness. No resummation, and hence no condition on the sign of the coupling is required. For the special case of spherical Fermi surfaces, a similar theorem was stated in [11]. The meaning of the term “generalized ladder” is explained in detail in [7, Sec. 2.4], and also below. The generalized ladders (called dressed bubble chains in [7, Definition 2.24]) are non-overlapping four-legged diagrams. By [7, Lemma 2.26], any non-overlapping four-legged graph is a generalized ladder. It is constructed from the usual ladder graphs by replacing the bare vertices by effective vertices of a higher scale. The non-overlapping four-legged graphs emerge in a natural way in the renormalization flow because their scale behaviour is marginal, which produces the factorials. All contributions to the four-point function from overlapping graphs are bounded, or, in renormalization group language, irrelevant. The importance of this is that convergence of perturbation theory can now be checked by looking at the ladders only: if they have singularities, then perturbation theory does not converge. If they don’t, the expansion in λ converges. The structure of the ladders is so simple that their properties essentially only depend on the fermion propagator, in particular, the Fermi surface. In absence of nesting (such as takes place at half-filling in the Hubbard model), the particle–hole ladders have no singularities. The particle–particle ladders always have a singularity at zero transfer momentum if the Fermi surface is symmetric, i.e. if e(−p) = e(p). The
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M. SALMHOFER
Fermi liquid of [3] has a Fermi surface for which the particle–particle ladders are uniformly bounded as well because there is no symmetry of e under p → −p. 3. Improved Power Counting In this section, I discuss the reasons behind the improved power counting bounds. I have written the present section so that it can be read as an easy introduction to the technical parts (Chaps. 2 and 3) of [7] and to the regularity analysis done in [8]. I shall discuss two examples of graphs to bring out the main point. After that, it should be obvious to generalize it to all graphs, given the graph classification of [7, Sec. 2.4]. To do estimates we need some definitions from scale analysis. As in all modern treatments of renormalization, we decompose (“slice”) the propagator around its singularity. There is a lot of freedom in doing this, but the decomposition is chosen such that the propagator has very simple behaviour on each slice. In the previous section, we introduced an infrared cutoff ε. Since all quantities scale like powers of ε and logε, we take ε of the form ε = MI ,
with M > 1 fixed and I a negative integer.
(3.1)
Removing the cutoff thus means taking the limit I → −∞. Moreover, we will now trace back the behaviour when the energy scale varies by looking at the contributions from energy shells M j−2 ≤ |e(p)| ≤ M j , for I ≤ j < 0. This decomposition is natural because it is adapted to the singularity. For definiteness, here are the details (readers interested only in the main features of the decomposition can skip this paragraph). Let r0 > 0 be chosen such that in an r0 -neighbourhood of the Fermi surface the coordinates ρ and ω of (2.15) can be used. Let M ≥ max{43 , r10 } (then |e(p)| < M −1 implies |ρ| < r0 ), and let −4 a ∈ C ∞ (R+ , a(x) = 1 for x ≥ M −2 , 0 , [0, 1]) such that a(x) = 0 for 0 ≤ x ≤ M and a0 (x) > 0 for all x ∈ (M −4 , M −2 ). Set 0 a(x)
if x ≤ M −4
if M −4 ≤ x ≤ M −2 x x = f (x) = a(x) − a M2 if M −2 ≤ x ≤ 1 1−a 2 M 0 if x ≥ 1,
(3.2)
so that, for all x > 0, f (x) ≥ 0 and 1 − a(x) =
−1 X
f (M −2j x) .
(3.3)
j=−∞
Calling fj (x) = f (M −2j x), supp fj = [M 2j−4 , M 2j ] .
(3.4)
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
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The decomposition of C is + X eip0 0 a(p20 + e(p)2 ) eip0 0 = + eip0 0 Cj (p) , ip0 − e(p) ip0 − e(p) j 0 and e ∈ R, if these solutions are superconducting solutions and if there exists 0 > 0 such that the solutions (f (., ), A(., ); h()) exist for || ≤ 0 and if the map → (f (., ), A(., ); h()) is continuous from [−0 , 0 ] to H 2 (] − d/2, d/2[) × H 2 (] − d/2, d/2[) × R and satisfies (f (., 0), A(., 0); h(0)) = (0, h0 (x + e); h0 ) . When this map is C 1 , we speak about a C 1 -bifurcation. ˆ of (GL)d is a locally stable solution for ˆ h) We also say that a solution (fˆ, A; ˆ (GL)d if it gives, at fixed h, a local minimum of the GL functional with respect to (f, A). Otherwise, it will be called an unstable solution. We recall (see [4, Proposition 0.1]) that, for any κ > 0, d > 0 and e ∈ R, ¯ = h(κ, ¯ d, e) such that the normal solutions (0, h(x + e); h) there exists a unique h ¯ d, e) and unstable when 0 < h < h(κ, ¯ d, e). The are locally stable when h > h(κ, ¯ d, e) of h will be characterized in (2.4). critical value h(κ, We adopt the following definitions.
STABILITY OF BIFURCATING SOLUTIONS
...
581
Definition 1.1.a Let κ > 0 and d > 0. 1. A bifurcating curve of solutions for (GL)d of the form (2.6) starting from a normal solution (0, h0 (x+e)) is called subcritical at (0, h0 (x+e)) if there exists 0 (κ, d) such that the bifurcating solutions are unstable for 0 < || ≤ 0 (κ, d). 2. A bifurcating curve (2.6) is called supercritical at (0, h0 (x + e)), if there exists 0 (κ, d) s.t. the bifurcating solutions are locally stable solutions for 0 < || ≤ 0 (κ, d). 3. When the bifurcating solutions are locally stable for 0 < ≤ 0 (κ, d) and unstable for ¯0 (κ, d) ≤ < 0 (or unstable for > 0 and locally stable for < 0), the bifurcation is called transcritical at (0, h0 (x + e)). In previous papers, mainly in [3] and [4], we have studied the bifurcating solutions starting from normal solutions by considering a new scaling which will be recalled later. We have then deduced existence and uniqueness results for any fixed value of the parameter κ and when d tends to 0 or to +∞. We prove, in this paper, that the domain of validity of our results is improved by considering as main parameter the product κd, in the limit κd → 0 as well as in the limit κd → +∞. In these asymptotic regimes, we study the structure of the superconducting solutions starting from normal solutions. One part of this work concerns the sign 1 of ∂h ∂ (0) when h is C. Another part concerns the local stability of the bifurcating solutions. The plan of this study is the following. In Sec. 2 we recall results on a spectral problem attached to a linearization of the GL equations at a normal solution. In Sec. 3, we extend, in new theorems, some previous existence and uniqueness results on bifurcating solutions to regims when κd tends to 0 or to +∞. Section 4 presents the main results on the structure of the bifurcating solutions. In Sec. 5, we study the sign of ∂h ∂ (0) in the symmetric case (e = 0), first in Subsec. 5.1, when κd tends to +∞, then in Subsec. 5.2, when κd tends to 0. In Sec. 6, we study the sign of ∂h ∂ (0) in the asymmetric case (e 6= 0) when κd tends to +∞. Section 7 is devoted to the stability results. We determine, in Subsec. 7.1, the two first eigenvalues of the spectral problem attached to a linearization of the GL equations at a bifurcating solution. In Subsec. 7.2, we study the stability or instability of the symmetric bifurcating solutions, first when κd tends to +∞, then when κd tends to 0, and finally in the case when one restricts the problem to the symmetric solutions. The stability or instability of the asymmetric bifurcating solutions is treated in Subsec. 7.3 when κd tends to +∞. 2. General Results on Bifurcating Solutions The starting point of this study is the observation that the rigorous results established in our previous articles can also be used to determine the structure and the stability of bifurcating solutions starting from normal solutions. Before going a Some authors adopt other definitions.
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C. BOLLEY and B. HELFFER
further, we think that there is a need for a short summary of these results. This leads us, in many cases, to actually improve their domain of validity. 2.1. Necessary and sufficient conditions for the existence of bifurcating solutions In [2] and [7], we have given necessary conditions and sufficient conditions on the parameters e and h = h0 for the existence of bifurcating solutions starting from a normal solution (0, h(x + e); h). One necessary condition is the existence of a double eigenvalue λ equal to zero for the spectral problem attached to a linearization of the GL equations at the normal solution (0, h(x + e); h). For all κ > 0, d > 0, h > 0 and e ∈ R, this spectral problem is the following: ( −κ−2 φ00 + h2 (x + e)2 φ − φ = λφ in ] − d/2, d/2[ (a) φ0 (±d/2) = 0 , ( (2.1) −v 00 = λv in ] − d/2, d/2[ (b) v 0 (±d/2) = 0 with (φ, v) ∈ (H 2 (] − d/2, d/2[))2 and λ ∈ R. This is a diagonal system which always admits φv = 0c , with c ∈ R, as an eigenvector associated to the eigenvalue λ = 0. We consequently get that a necessary condition for the existence of a bifurcation is that λ = 0 is also an eigenvalue for the problem (2.1)(a). This leads us to consider, with τ = 1 + λ, the spectral problem −2 00 2 2 −κ φ + h (x + e) φ = τ φ in ] − d/2, d/2[ (2.2) φ0 (±d/2) = 0 , 2 φ ∈ H (] − d/2, d/2[) , with the normalization condition kφkL2 (]−d/2,d/2[) = 1 .
(2.3)
The previous necessary condition is now that 1 is an eigenvalue of the spectral problem (2.2). If we add the conditionb that φ > 0, and if we denote by τ = τ (κ, d, e, h) the principal eigenvalue of the Neumann problem, the necessary condition becomes that τ (κ, d, e, h) = 1 . We have proved in [4, Proposition 0.1] that, for all κ > 0, d > 0 and e ∈ R, there ¯ d, e) such that exists a unique h = h(κ, ¯ d, e)) = 1 . τ (κ, d, e, h(κ,
(2.4)
b This condition is not justified mathematically but seems to us natural if one want to analyze the most stable solutions.
STABILITY OF BIFURCATING SOLUTIONS
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583
The condition (2.4) is not sufficient for bifurcations to appear and we gave in [7] a more precise study. We have, indeed, proved the following existence theorem of bifurcating solutions. Theorem 2.1 [Theorem 2.1 in [7] (see also [2])]. Let (κ, d) ∈]0, +∞[2 and let (e, h0 ) ∈ R×]0, +∞[ satisfying ¯ d, e) , (a) h0 = h(κ, ∂τ (κ, d, e, h0 ) = 0 , (2.5) ∂e ∂2τ (c) (κ, d, e, h0 ) 6= 0 . ∂e2 Then, there exists a constant ˜0 = ˜0 (κ, d) > 0 and a C ∞ curve, in a neighborhood of 0, → (f (., ), A(., ); h()) of superconducting solutions such that, for 0 < || ≤ ˜0 , (b)
f (x, ) = f0 (x) + 3 f1 (x) + o(3 )
in H 2 (] − d/2, d/2[) ,
A(x, ) = A0 (x) + 2 A1 (x) + o(2 )
in H 2 (] − d/2, d/2[) ,
(2.6)
h() = h0 + 2 h1 + o(2 ) , where f0 is the principal normalized positive eigenfunction φ defined by (2.2), h0 = ¯ d; e) and A0 = h0 (x + e). h(κ, Moreover, there exist constants 0 = 0 (κ, d) > 0 and γ0 = γ0 (κ, d) > 0 such that, for 0 < || ≤ 0 , the solution (f (., ), A(., ); h()) is the unique solution of (GL)d such that (i)
kf (., )kH 2 (]−d/2,d/2[) ≤ γ0 ,
(ii)
kA(., ) − A0 kH 2 (]−d/2,d/2[) ≤ γ0 ,
(iii)
|h() − h0 | ≤ γ0 ,
(iv)
(f (., ), f0 )L2 (]−d/2,d/2[) = .
Condition (2.5)(b) means that 1 is a critical value for τ , and (2.5)(c) is a condition of non degeneracy. The two conditions (2.5)(a)–(b) are necessary conditions for the existence of bifurcating solutions starting from (0, h0 (x + e); h0 ) (see [2] and [7]). ¯ d, 0) verifying (2.4) satisfies automatically When e = 0, a value of h = h(κ, (2.5)(b). We know that condition (c) is at least satisfied when d is small enough or large enough and that there is at least one point for which condition (c) is not satisfied (see [3]). We also know the existence of values of e, different from zero giving bifurcating solutions. These results are extended in Sec. 3. 2.2. The spectral problem 2.2.1. Scalings We have used, in preceding papers, two different scalings for studying (2.2) and we shall refer to them often.
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C. BOLLEY and B. HELFFER
(i) The first one, is given by y=
√ κh(x + e) ,
d√ κh , a= 2
¯ = (κh)−1/4 φ(x) , f(y)
√ c = e κh ,
(2.7)
κτ . µ= h
It will be used in this paper when κd is large. The spectral problem (2.2)–(2.3) becomes, for a > 0 and c ∈ R, ¯ ¯ P f = µf in ] − a + c, a + c[ , 0 ¯ f (±a + c) = 0 , ¯ f ∈ H 2 (] − a + c, a + c[) ,
(2.8)
kf¯kL2 (]−a+c,a+c[) = 1 ,
(2.9)
with f¯ = f¯(y; a, c) and
where P is the harmonic oscillator P ≡−
d2 + y2 . dy 2
(2.10)
(ii) The second scaling gives an interval independent of d. It is defined by x ¯ , φ(u) = d1/2 φ(x) , d e d√ ¯ = κ2 d2 τ . κh , c˜ = , λ a= 2 d
u=
(2.11)
It will be used when κd is small. The spectral problem becomes 4 2¯ ¯¯ ¯00 −φ + 16a (u + c˜) φ = λφ in ] − 1/2, 1/2[ , 0 ¯ φ (±1/2) = 0 , ¯ φ ∈ H 2 (] − 1/2, 1/2]) ,
(2.12)
¯ a, c˜) and with φ¯ = φ(u; kφ¯ kL2 (]−1/2,1/2[) = 1 .
(2.13)
(iii) We remark that (2.8) and (2.12) are linked by the change of variables and parameters ¯ λ c. (2.14) y = 2au + c , µ = 2 , c = 2a˜ 4a
STABILITY OF BIFURCATING SOLUTIONS
...
585
2.2.2. The principal eigenvalue ¯ and φ¯ depending only on Considering (2.12) with c˜ = 0 gives eigen-elements λ a and verifying ( ¯ φ¯ in ] − 1/2, 1/2[ , −φ¯00 + 16a4 u2 φ¯ = λ(a) (2.15) φ¯0 (±1/2) = 0 , √ where, as in (2.7), a = d2 κh. ¯ are simple and are C ∞ This is a Sturm–Liouville Problem whose eigenvalues λ functions on ]0, +∞[ with respect to the parameter a (see [20]). By differentiating this equation with respect to a, we get ¯ ∂λ (a) = 64a3 ∂a
Z
1 2
u2 φ¯2 (u)du ,
(2.16)
− 12
so that ¯ a → λ(a) is a strictly increasing function of a on ]0, +∞[ .
(2.17)
Moreover, it satisfies ¯ ¯ λ(0) = 0 and λ(+∞) = +∞ ,
(2.18)
with, when a → 0, (a)
4 ¯ λ(a) = a4 + O(a8 ) , 3
(b)
¯ a) = 1 + O(a4 ) , φ(u,
(2.19)
and, when a → +∞, 5 ¯ . λ(a) = 4a2 + O exp − a2 8
(2.20)
(other asymptotic results, when a → +∞, are recalled in Subsec. 5.1). Using now (2.16) and (2.19(b), we get ¯ 16 3 ∂λ (a) = a + O(a7 ) as a → 0 . ∂a 3
(2.21)
Let us now consider the scaling (2.7) with c = 0. We get, using (2.14), (2.19) and (2.21), for a > 0 small enough, a2 + O(a6 ) , 3
(a)
µ(a) =
(b)
2 ∂µ (a) = a + O(a5 ) . ∂a 3
(2.22)
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C. BOLLEY and B. HELFFER
¯ d, e) 2.2.3. Universal lower bounds for h(κ, ¯ d, e). Let us now give lower bounds for h(κ, Lemma 2.2. (a) Let e = 0 and (κ, d) ∈]0, +∞[2 , then ¯h(κ, d, 0) > κ .
(2.23)
(b) Let (κ, d) ∈]0, +∞[2 and e ∈ R. If ¯h(κ, d, e) satisfies (2.5)(b) then, e ∈ [−d/2, d/2] . (c) Let (κ, d) ∈]0, +∞[2 , then, (i)
when e ∈ [−d/2, d/2] ,
(ii)
when e = 0 ,
¯ d, e) ≥ dh(κ,
√ 3.
√ ¯ d, 0) ≥ 2 3 . dh(κ,
(2.24)
Part (a) is proved in [3, Proposition 2.5)] Part (b) in [4] (Formula (2.7)). Part (c)–(i) results immediatly from [4, relation 2.14]. Part (c)–(ii) results from Corollary 2.2 and Formula (2.15) in [4]. 2.2.4. Heilman–Feynman relations We shall also use the following relations between the derivatives of τ and those ¯ of h, Lemma 2.3. (a) Let (κ, d) ∈]0, +∞[2 and e ∈ R, then, (i)
(ii)
Z d2 ¯ ∂τ ¯ d, e) · ∂ h (κ, d, e) (κ, d, e, ¯ h(κ, d, e)) = −2h(κ, (x + e)2 φ(x)2 dx , d ∂e ∂e −2 Z d2 ∂τ ¯2 (κ, d, e, ¯ h(κ, d, e)) = 2h (x + e)φ(x)2 dx . ∂e −d 2
¯ d, e)) satisfying (2.5)(a, b), then, (b) Let (κ, d) ∈]0, +∞[2 and (e, h0 = h(κ, Z d2 2¯ ∂2τ ¯ ¯ d, e)· ∂ h (κ, d, e) (i) (κ, d, e, h(κ, d, e)) = −2 h(κ, (x+e)2 φ(x)2 dx. d ∂e2 ∂e2 −2 # " d Z 2 ∂φ ∂2τ 2 ¯ (κ, d, e) 1 + 2 (κ, d, e, ¯ h(κ, d, e)) = 2h (x + e) (x)φ(x)dx . (ii) ∂e2 ∂e −d 2 The lemma is proved by differentiating (2.4) with respect to e, and (2.2), when τ = 1, with respect to h. 2.3. Formulas for h1 The computation of the Ginzburg–Landau functional ∆G(f (., ), A(., ); h()) on a bifurcating solution gives:
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Proposition 2.4 [see Formula (2.20) in [7]]. Let κ > 0 and d > 0. Let (e, h0 ) satisfying (2.5). Then, there exists 0 > 0 such that for || ≤ 0 , !# " Z d 2
∆G(f (., ), A(., ); h()) = h1 /
−d 2
4 + O(6 ) .
h20 (x + e)2 f02 dx
(2.25)
This formula shows that the energy ∆G, when calculated at a bifurcating solution, has, for small enough, the same sign as the constant h1 (when h1 6= 0). Now, with the notations of [7], we split A1 defined in (2.6) as A1 = a1 + A1,0 + h1 x ,
with a1 ∈ R ,
where A1,0 is the unique solution in H 2 (] − d/2, d/2[) of the following problem 00 2 −A1,0 + f0 A0 = 0 in ] − d/2, d/2[ , 0 A1,0 (±d/2) = 0 , A1,0 (0) = 0 ,
(2.26)
and get the following formula (see [7, Sec. 2] Sec. 2 or [21]): 2
h1 h0
Z
d 2
−d 2
Z (A0 (x))2 (f0 (x))2 dx = −
d 2
−d 2
Z (f0 (x))4 dx + 2
d 2
−d 2
(A01,0 (x))2 dx ,
(2.27)
which will be useful for the determination of the sign of h1 . An analogous formula is studied in S. J. Chapman [9] and in C. Bolley-B. Helffer [7] in two limiting problems associated with d = ∞: the symmetric case (e = 0) and the asymmetric one (e 6= 0). The first case gives the condition h1 (κ − 2−1/2 ) > 0 and exhibits consequently the well-known critical value κ = 2−1/2 (between type 1 and type 2 superconductors). In [9], the author also gives a formal study of the stability of bifurcating solutions for this limiting problem, but the splitting of the first eigenvalue of (2.1), which has a multiplicity two at the bifurcation, is not considered (see our study in Sec. 7). The sign of ∆G is also studied in the paperc by S. P. Hastings and W. C. Troy [17], when κ is large, for particular asymmetric solutions, by using the limiting problem d = +∞. They also show the existence of stable asymmetric superconducting solutions for κ large and suitable h’s, but they don’t exhibit them. 2.4. The critical value Σ0 The analysis of the second case (the asymmetric case) leads to the introduction of another limiting critical value for κ, denoted by Σ0 , which can be defined as follows. c In a first version of their paper, these authors assert that our study of bifurcating solutions starting from normal solutions, given in previous papers [2, 3, 5, 4], concern only the case κ → 0. We emphasize that these results are true for any κ.
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We consider the problem P φ = µφ
in R ,
(2.28)
2
d 2 where P is, as in (2.10), defined by P ≡ − dy 2 + y . For every parameter µ, we can choose a basis {φ1 (., µ), φ2 (., µ)} for the set of the solutions of (2.28), such that, when y → +∞: 2 y · y (µ−1)/2 · (1 + O(y −2 )) , (2.29) φ1 (y, µ) = exp − 2
φ2 (y, µ) = exp
y2 2
· y −(µ+1)/2 · (1 + O(y −2 )) ,
(2.30)
and where the O are locally uniform with respect to µ (see Y. Sibuya [24]). The function φ1 (., µ) is the solution of P − µ on R whose behavior, when y → +∞, is such that lim φ1 (y, µ) = 0 . y→+∞
Let α > 0, and let µ1 (α) be the first eigenvalue of the harmonic oscillator P in ] − α, +∞[ with the Neumann condition at −α. Then, the first eigenfunction is necessary given by γφ1 (., µ1 (α)), where γ is a constant. It is proved in [12] that the function [0, +∞[3 α → µ1 (α) has a minimum µ01 which is reached at a unique α0 > 0. Moreover, µ01 ≤ µ1 (α) ≤ 1 ,
with µ01 = µ1 (α0 ) .
(2.31)
Computations in [4], Appendix 2, give µ1 (α0 ) ≈ 0, 59 and α0 ≈ 0, 73. Let us now introduce, for α > 0, the two functions, Z ∞ φ1 (y, µ1 (α))4 dy ρ(α) =
(2.32)
for any α > 0 ,
−α
and
Z Σ0 (α) = 2
∞
−α
Z
−α
Then, we define Σ0 by
Σ0 =
2
y
t · φ1 (t, µ1 (α))2 dt
σ(α0 ) ρ(α0 )
dy .
(2.33)
12 .
(2.34)
The constant Σ0 is computed in [5, Subsec. 9.4.2]. We got, Σ0 ≈ 0, 4 . We shall meet, in Theorems 4.2 and 6.2, the two critical values 2−1/2 and Σ0 of κ, when the parameter κd tends to +∞. 3. Existence and Uniqueness Theorems for Bifurcating Solutions Existence theorems are given in earlier papers, but we give, in this subsection, more general and new results by considering the natural parameter κd.
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The following uniqueness theorem extends Theorem 0.7 in [4]. Theorem 3.1. (a) There exists a constant a0 > 0 such that, ∂τ ¯ (κ, d, e); (κ, d, e, h(κ, d, e)) = 0 and 0 < κd ≤ a0 ∂e = {(κ, d, 0); 0 < κd ≤ a0 } . (b) Let us suppose that e = 0. There exists a0 > 0 such that, (i) (ii)
∂2τ ¯ d, 0)) > 0 , (κ, d, 0, h(κ, ∂e2 ∂2τ ¯ d, 0)) < 0 . for κd ≥ a−1 (κ, d, 0, h(κ, 0 , ∂e2 for κd ≤ a0 ,
The first part was proved in [4, Theorem 0.7], in the case when κ is fixed and d small, and the second part was given in [5], for also fixed κ, (see (1.23)2,3 ) using estimates of [3]. Proof of Theorem 3.1. In [4, Theorem 0.7], we have established, using ¯ 1/2 small enough, the relation the scaling (2.7) with e = 0, that for a = d2 (κh) ∂τ ¯ ∂e (κ, d, e, h(κ, d, e)) = 0 implies e = 0. So, for proving (a), we only need a control ¯ d, 0) as κd tends to 0. This is given by the following lemma: of dh(κ, Lemma 3.2. There exists a constant a0 > 0 such that, for (κ, d) satisfying 0 < κd ≤ a0 , then √ ¯ d, 0) = 2 3(1 + O(κ2 d2 )) , (3.1) dh(κ, This lemma completes (2.24)(ii). Proof of the lemma. We take back the proof of Corollary 7.6 in [4], and deduce a more accurate asymptotic formula than thatpgiven in this corollary. ¯ d, 0) is small enough, It is proved in [4] that for (κ, d) such that a = d2 κh(κ, ∂τ ¯ the unique h solution of ∂e = 0 is given by, ¯ = 4u , h κd2
(3.2)
√ κ2 d2 , Ψ(u) ≡ uµ( u) = 4
(3.3)
where u is the solution of
2
d 2 and were µ(a) is the first eigenvalue of the harmonic oscillator P ≡ − dy 2 + y , for the Neumann problem on the interval ] − a, a[ (or on ]0, a[). The function ]0, +∞[3 a → µ(a) can be extended as a positive C ∞ even function on R verifying (2.22) when a ∈] − a0 , a0 [ with a0 small enough. Consequently, Ψ
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can be extended as a C ∞ positive eigenfunction and can be seen as the square of a C ∞ function χ which, using (2.22) satisfies u χ(u) = √ + O(u3 ) as u → 0 . 3
(3.4)
Equation (3.3) can then be written, χ(u) =
κd , 2
with χ invertible in a neighborhood of u = 0. The implicit function Theorem gives, for κd small, the existence of a unique u = u(κd), with κd u √ = + O((κd)3 ) . 2 3 Using (3.2), (3.1) follows for a and κd small enough. 1/2 ¯ , we deduce that Now, from (2.17), (2.19)(a) and the relation κd = (λ(a))
a → 0 as κd → 0 . Consequently, Lemma 3.2 is proved and implies, for κd small enough, a=
1 1 1 1 d ¯ (κh(κ, d, e)) 2 = 3 4 2− 2 (κd) 2 (1 + O(κ2 d2 )) , 2
(3.5)
Proof of (a) in Theorem 3.1. This results from (3.5) and from Theorem 3.4 in [4]. Proof of (b)(i) in Theorem 3.1. According to Lemma 3.2 and (3.5), the proof of (b)(i) is the same as in [3, Proof of Proposition 2.18].d We reproduce it after rescaling for completeness. 2
We shall use the expression of ∂∂eτ2 given by Lemma 2.3 (b)(ii) with e = 0 in the scaling (2.11). We get, for all κ > 0, d > 0 and for c = 0 (or equivalently e = 0) ¯ d, 0), and h = h(κ, # " Z 12 ∂ φ¯ ¯ ∂ 2τ 2 ¯ · φ(u)du . (3.6) (κ, d, 0, h) = 2h 1 + 2 u· ∂e2 ∂˜ c − 12 We are going to prove that the integral term tends to 0 as κd → 0. ¯
Let us first prove that ∂∂φc˜ → 0 in H2 (] − 1/2, 1/2[), as κd → 0. ¯ We denote ∂∂˜φc by ψ. The equations satisfied by ψ are given by differentiating ¯ (2.12) and (2.13). Using that ∂∂˜λc (κ, d, 0, ¯h) = 0 (see Lemma 2.3 (a)(ii)), we get with p ¯ d, 0), when e = 0 and h = h(κ, ¯ d, 0), a = d2 κh(κ, d In the right-hand side of the last relation in [3, p. 268] one has to read 1 a4 h2 κ2 and not 2 (λ1 −a2 κ2 ) a4 h2 1 . 2 (λ1 −a2 κ2 )
In this proof, a = d and x is changed in
x . a
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...
−ψ 00 + 16a4 u2 ψ − κ2 d2 ψ = −32a4 uφ¯ in ] − 1/2, 1/2[ ψ 0 (±1/2) = 0 , ¯ L2 (]−1/2,1/2[) = 0 , (ψ, φ) ψ ∈ H 2 (] − 1/2, 1/2[) .
591
(3.7)
We define the operator T from H 2 (] − 1/2, 1/2[) to L2 (] − 1/2, 1/2[) (see Eq. (2.12)) by T : ξ → T ξ ≡ −ξ 00 + 16a4 u2 ξ . The spectral problem attached to this operator, with the Neumann conditions at ¯ d, 0), the first ±1/2, is a Sturm–Liouville Problem. With the choice h = h(κ, eigenvalue for this operator, with the Neumann condition at ±1/2, is ¯ = κ2 d2 , λ0 = λ and the second eigenvalue is given by λ1 =
inf
ξ∈H 1 (]−1/2,1/2[) ¯ (ξ,φ) =0 L2 (]−1/2,1/2[)
(T ξ, ξ)L2 (]−1/2,1/2[) . kξk2L2 (]−1/2,1/2[)
¯ we get Therefore, with the choice ξ = ψ (ψ is orthogonal to φ), (T ψ, ψ)L2 (]−1/2,1/2[) ≥ λ1 kψk2L2 (]−1/2,1/2[) .
(3.8)
Now, (3.7) implies ¯ ψ)L2 . (T ψ, ψ)L2 − κ2 d2 kψk2L2 = 16a4 (uφ, So (3.8) and the Cauchy–Schwarz inequality give ¯ L2 · kψkL2 . (λ1 − κ2 d2 )kψk2L2 ≤ 16a4 kuφk Using now (2.13) and the strict inequality λ1 > λ0 = κ2 d2 , we get kψkL2 ≤
16a4 1 · . 2 (λ1 − κ2 d2 )
(3.9)
Using the continuity of the eigenvalues with respect to the coefficients, we get that λ1 tends, when a → 0, to the second eigenvalue of the Neumann problem, 00 −ψ = µψ in ] − 1/2, 1/2[ , ψ 0 (±1/2) = 0 , ψ ∈ H 2 (] − 1/2, 1/2[) , which is equal to 4π 2 . Consequently, lim ψ = 0
a→0
in L2 (] − 1/2, 1/2[) .
(3.10)
A bootstrap argument implies the convergence in H 2 (] − 1/2, 1/2[). It results from (3.5) that ψ tends to 0 in H 2 (] − 1/2, 1/2[) as κd tends to 0.
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C. BOLLEY and B. HELFFER
2 Let us prove that ∂∂e2τ (κ, d, 0, ¯ h(κ, d, 0)) > 0 for κd small. We apply the Cauchy–Schwarz inequality to the integral appearing in (3.6). We get, using (2.13), Z 1
1 2
∂ φ¯ ∂ φ¯ ¯
(u) · φ(u)du ≤ u· . 2 ∂˜ −1 ∂˜ c c L2 (]−1/2,1/2[) 2 2
Therefore, using now (3.10), ∂∂eτ2 (κ, d, 0, h) is positive for κd small enough, and the relation (b)(i) of Theorem 3.1 follows. Proof of (ii). It is a consequence of the proof of [3, Proposition 2.21] where, √ ¯ using the scaling (2.7) with e = 0 (or c = 0), we have established that, for a = d2 κh large enough, then ∂2τ (κ, d, 0, ¯h(κ, d, 0)) < 0 . ∂e2 For getting (ii), we use√once again (2.17)–(2.18) or we simply remark, using ¯ > κd when e = 0. Therefore, a tends to +∞ as Lemma 2.2 (a), that, d2 κh 2 κd tends to +∞ and the assertion (ii) follows. We have studied in [3, 4] the existence of pairs (e, h0 ) with e = e(κ, d) 6= 0 ¯ d, e(κ, d)) such that (2.5) is satisfied. The results can be extended and h0 = h(κ, as follows. Theorem 3.3. There exists a constant a1 > 0 and a function (κ, d) → e¯(κ, d) defined for (κ, d) satisfying κd > a1 , such that ∂τ (κ, d, e, ¯h(κ, d, e)) = 0 and κd ≥ a1 (κ, d, e); ∂e [ = {(κ, d, 0); κd ≥ a1 } {(κ, d, e¯(κ, d)); κd ≥ a1 } [
{(κ, d, −¯ e(κ, d)); κd ≥ a1 } .
Proof. When κ > 0 is fixed and d is large, the result is proved in [4, Theorem 0.5] using the existence of e¯(κ, d) > 0 established in [3, Proposition 2.25]. Let us prove Theorem 3.3. We proceed as in [3], but for large κd instead of d large. Theorem 3.1 (ii) gives 2 that for κd ≥ a1 , with a1 large enough, ∂∂eτ2 (κ, d, 0) is strictly negative. Then, from 2¯ Lemma 2.3 (b)(ii), ∂∂eh2 (κ, d, 0) is strictly positive. Moreover, in [3, Proposition 2.24] we have established that for κ > 0 and d > 0 fixed, lim ¯h(κ, d, e) = 0 , e→+∞
Therefore, by continuity and differentiability of e → ¯h(κ, d, e), the existence for κd ¯ large enough of some e¯(κ, d) > 0 such that ∂∂eh (κ, d, e¯(κ, d)) = 0 follows from the Rolle Theorem. Lemma 2.3 (a)(i), gives then the equivalence with the condition
STABILITY OF BIFURCATING SOLUTIONS
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593
∂τ ¯ d, e¯(κ, d))) = 0 . (κ, d, e¯(κ, d), h(κ, ∂e Let us prove the uniqueness of e¯(κ, d) > 0 for large κd. In the scaling (2.7), we have proved in [4, Theorem 3.3] the uniqueness, for large a, of a value c¯ of c such that ∂µ ∂c (a, c) = 0. Using that, from (2.7), h3/2 ∂µ ∂τ (κ, d, e, h) = 1/2 (a, c) , ∂e ∂c κ ¯ d, e) the only point is to verify that κd large implies a large for any e such that h(κ, satisfies (2.5)(b). This is given by Lemma 2.2 (b) and (c)(i) which imply that d ¯ 1/2 ≥ 31/4 (κd)1/2 . We then get the uniqueness of e¯(κ, d) for large κd and 2 (κh) 2 eventually Theorem 3.3. In the following, we will say that a bifurcating solution is symmetric when f (., ) is even and A(., ) is odd and that it is asymmetric otherwise. When e = 0, an eigenfunction f0 = φ of (2.2) is then an even function and A0 is odd. We observe also that, if (f (x, ), A(x, ), h()) is a solution of the GL equations, then (f (−x, ), −A(−x, ), h()) and (−f (x, −), A(x, −), h(−)) are also solutions and that these solutions are equal when = 0. By uniqueness of the curve of bifurcation, we conclude, in the symmetric case, that, ( f (x, ) = f (−x, ) = −f (x, −) , (3.11) A(x, ) = −A(−x, ) = A(x, −) , so that f is even and A is odd in a neighborhood of = 0. We summarize the existence results of bifurcating solutions by the following theorem. Theorem 3.4. (i) There exists a constant a0 > 0 such that, for (κ, d) satisfying κd ≤ a0 , there exists a unique C ∞ curve → (f (., ), A(., ); h()) of bifurcating solutions starting from all the normal solutions (0, h0 (x + e); h0 ). These solutions ¯ d, 0) are starting from the particular normal solutions (0, h0 x; h0 ), where h0 = h(κ, satisfies (2.5), and are symmetric solutions. (ii) There exists a constant a1 > 0 such that for (κ, d) verifying κd ≥ a1 , there exist exactly three C ∞ curves → (f (., ), A(., ); h()) of bifurcating solutions starting from normal solutions (0, h0 (x + e); h0 ). One curve of solutions, starting from ¯ d, 0), is composed with symmetric solutions, the other (0, h0 x; h0 ) with h0 = h(κ, two ones are composed with asymmetric solutions corresponding to e = e¯(κ, d) and e = −¯ e(κ, d) and deduced from each other by symmetry. Proof. Theorem 3.1 (a) gives the uniqueness of a value of e (which is e = 0) leading to a curve of bifurcating solutions when κd is small enough. Theorem 2.1, combined with Theorem 3.1 (b), gives, for κd small or large enough, the uniqueness of the curve of bifurcating solutions when e = 0. Their symmetry follows from this uniqueness. We get (i).
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Theorem 3.3 implies the existence of exactly three values of e (e = 0, e = e¯(κ, d) and e = −¯ e(κ, d)) such that the conditions (2.5)(a, b) are satisfied when κd is large enough. The value e = 0 leads as in (i) to symmetric solutions. The condition (2.5)(c) was not proved to be satisfied in earlier papers when e = e¯(κ, d), but will result from the relations (7.16) and (7.8) established in Sec. 7. The second equation in (2.6) shows that, in that case, the first term for A is not symmetric so that the bifurcating solutions are asymmetric solutions. We get (ii). Remark 3.5. The existence of asymmetric solutions for κ fixed and d large enough was given by [2, Theorem 3.1] (which proves that the two conditions (2.5)(a, b) are sufficient for getting the existence of bifurcating solutions satisfying (2.6)) combined with [3, Proposition 2.25 (i)] (which proves that (2.5)(a, b) is satisfied for d large enough). 4. Main Results on the Structure of the Bifurcating Solutions Let us now give the main results proved in this paper concerning the bifurcating solutions. It is generally admitted in the literature (see for example [11] or [1]) that the bifurcating solutions starting from a normal solution are supercritical for all d when κ is large, and that they are supercritical for small d and subcritical for large d when κ is small. But it seems that no proof of any part of these results is available, except in the case of a bounded interval with d small which is studied in [7, Sec. 2]. The situation is in fact more complicated as we will see later. The a main purpose of this paper is to study two properties of the bifurcating solutions constructed in Theorem 2.1, which were not analyzed in our previous papers. The first one is to calculate the sign of h1 as function of the parameters, near the bifurcating points. The second one is to analyze the stability of the bifurcating solutions. The two studies are performed as function of κd and κ, or of κd and d according to the asymptotic regime in study. When considering the sign of h1 , we distinguish between the symmetric case (associated to e = 0) and the asymmetric case (associated to e = e¯(κ, d)). We first analyze the symmetric case. We prove in particular the two following theorems. Theorem 4.1 [κd large]. Let e = 0. For any η > 0, there exists a constant a1 > 0 such that, for (κ, d) satisfying κd ≥ a1 and |κ − 2−1/2 | > η, and for h0 satisfying (2.5), there exists a constant 1 = 1 (κ, d) > 0 s.t. the curve of superconducting solutions (f (., ), A(., ); h()) starting from the normal solution (0, h0 x; h0 ) satisfies for 0 < || ≤ 1 , (a) (b)
(2−1/2 − κ) · [h() − h0 ] > 0, (2−1/2 − κ) · ∆G(f (., ), A(., ); h()) > 0.
When d is fixed and κ is large, we have a more precise description for the asymptotic result.
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Theorem 4.2 [d fixed, κ large]. Let d > 0 and e = 0. There exists a constant κ0 = κ0 (d) > 2−1/2 such that, for κ ≥ κ0 and for h0 satisfying (2.5), there exists 1 = 1 (κ, d) > 0 s.t. the curve of bifurcating solutions (f (., ), A(., ); h()), starting from the normal solution (0, h0 x; h0 ) verifies, for 0 < || ≤ 1 , (a) (b)
h() < h0 , ∆G(f (., ), A(., ); h()) < 0.
There exists a constant C > 0 such that κ0 can be chosen s.t. 1 1 √ < κ0 (d) = √ + O(exp(−Cd2 )) 2 2
when d → +∞ .
(4.1)
The relation (4.1), together with Theorem 4.1, gives that κ = √12 is the limiting critical value for κ, when the thickness d of the film tends to +∞, deliminating different behaviors for the solutions. This asymptotic value is often used (as an approximation) for distinguishing a superconductor of type 1 and a superconductor of type 2. This critical value has also been discussed in [7]. We then study the asymptotics when κd √ tends to 0 and verify that, as in the limiting f − constant model, the value d = 5 is the theoretical value determining the locally stable solutions and the unstable ones. In the asymmetric case, we get similar results to Theorems 4.1 and 4.2 when κd tends to +∞, and we recover the critical value Σ0 , for κ, defined in (2.34). The stability of the bifurcating solutions is then studied by first considering the behavior of the spectral problem attached to a linearization of the GL equations at a bifurcating solution when is small. For = 0 (problem (2.1)), the lowest eigenvalue is of multiplicity two. When is small, this double eigenvalue splits in general into two distinct eigenvalues that we compute in Subsec. 7.1. The sign of one of them is opposite to the sign of h1 and the sign of the other is given by the 2 sign of ∂∂eτ2 (κ, d, e, h0 ); this value is linked to the existence or not of asymmetric solutions and can be positive or negative in function of (κ, d, e) (but independently of h1 ). We then study the sign of these two eigenvalues in the different asymptotic cases considered before. We prove, in particular: Theorem 4.3 [Symmetric bifurcating solutions, κd small]. √ Let e = 0. For any η > 0, there exists a0 > 0 such that for κd ≤ a0 and |d − 5| ≥ η, and for h0 satisfying (2.5), there exists 1 > 0 such that, for 0 < || ≤ 1 , the following properties are satisfied. √ (i) When d < 5, the bifurcating solutions (f (., ), A(., ); h()) starting from the normal solution (0, h0 x; h0 ) are locally stable and consequently supercritical. √ (ii) When d > 5, the bifurcating solutions starting from the normal solution (0, h0 x; h0 ) are unstable and consequently subcritical. Remark 4.4. We emphasize that the assumptions of Theorem √ 4.3 allow us, in particular, to treat the case κ > 0 and d → 0, and the case d 6= 5 and κ → 0.
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We complete the study of the symmetric case by an asymptotic instability result for large κd. Theorem 4.5 [Symmetric bifurcating solutions, κd large]. There exists a1 > 0 such that for (κ, d) satisfying κd ≥ a1 and for h0 satisfying (2.5), there exists 1 > 0 such that, for 0 < || ≤ 1 , the bifurcating solutions starting from the normal solution (0, h0 x; h0 ) are unstable and consequently subcritical. We remark that, to our knowledge, the instability property presented in this theorem in the case of a fixed large κ, does not seem to be proved nor even be mentioned in the literature, but it corresponds probably to the linear instability with respect to a two-dimensional perturbation introduced by S. J. Chapman in [9]. However, when we restrict our study to the research of symmetric solutions, as was done in another context in [6] and in previous articles [15] or [9], we consider another Ginzburg–Landau functional for which the symmetric bifurcating solutions are locally stable for κd and κ large enough (see Theorem 7.5). Concerning the asymmetric solutions, we prove Theorem 4.6 [Asymmetric bifurcating solutions]. Let a1 be the positive constant defined in Theorem 3.3 and Σ0 > 0 defined in (2.34). For any η > 0, there exists a constant a2 ≥ a1 such that for (κ, d) satisfying κd ≥ a2 and |κ−Σ0 | ≥ η and for e = e¯(κ, d) and h0 satisfying (2.5), there exists 1 > 0 such that, for 0 < || ≤ 1 , the following properties are satisfied: (i) When κ < Σ0 , the bifurcating solutions starting from the normal solution (0, h0 (x + e¯(κ, d)); h0 ) are subcritical. (ii) When κ > Σ0 , the bifurcating solutions starting from the normal solution (0, h0 (x + e¯(κ, d)); h0 ) are supercritical. Remark 4.7. Again, the assumptions of Theorem 4.6 allow us to treat, in particular, the case d > 0 and κ → 0 or +∞, or the case κ > Σ0 and d → +∞. Let us first consider h1 in the symmetric case. 5. The Sign of h1 in the Symmetric Case We have established in [3, Proposition 2.9] in the case when e = 0 and in [4, Proposition 5.3] in the case when e 6= 0, asymptotics for the eigenfunctions of the linearized problem (2.2)–(2.3), when the length d of the interval tends to +∞. For this purpose, we have used the scaling (2.7) which reduces the problem to the study of the Neumann realization of the harmonic oscillator: P ≡−
d2 + y2 dy 2
on some bounded interval ] − α, β[ .
In this section, we use these results for the study of h1 in the case when e = 0, with a symmetric interval ] − a, a[. The parameters are here κd and κ with κd large. Another part, Subsec. 5.2, concerns the study of h1 as function of κd and d, when κd is small.
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...
597
5.1. The asymptotics for large κd For studying the asymptotics of the principal normalized eigenfunction f0 when d tends to +∞, we have studied, in [3], the problem (2.8) with c = 0. In the spirit of B. Helffer–J. Sj¨ ostrand [18], we have constructed an approximation of f¯ and of µ valid when a tends to +∞. For this purpose, we have considered the problem (2.28) with µ = 1, that is Pf = f
in R ,
and the basis {φ1 (., 1), φ2 (., 1)}, defined by (2.29)–(2.30), of the solutions of this equation. In that particular case, we denote these functions more simply by φ1 (.) and φ2 (.). They satisfy 2 y for y ∈ R φ1 (y) = exp − 2 1 1 y2 · · 1+O as y → +∞ . 2 y y2 We have proved, using techniques from [18], the following result:
and
φ2 (y) = exp
(5.1)
Proposition 5.1 [see Proposition 2.9 in [3]]. There exist positive constants C1 , C2 , δ1 , δ2 (with δ1 and δ2 > 12 ) and a0 such that for a ≥ a0 , (i)
|µ(a) − 1| ≤ C1 exp(−δ1 a2 ) .
(5.2)
(ii)
supy∈[−a,a] |f¯(y) − f¯a (y)| ≤ C2 exp(−δ2 a2 ) ,
(5.3)
where f¯a is an even approximation of f¯ in the form 2 y y a ¯ + ρ(a) · φ2 (|y|) · Ξ , f (y) = β(a) exp − 2 a
(5.4)
with β(a) = π −1/4 + O(a exp(−a2 )) as a → +∞ , 1 −1/4 2 , a exp(−a ) · 1 + O ρ(a) = π a2
(5.5) (5.6)
and where Ξ is an even C ∞ (R) cutoff function s.t. Ξ(x) = 0if |x| ≤ 1/2 ;
Ξ(x) = 1 if |x| ≥ 3/4 .
Remark 5.2. The proof shows that we can choose δ1 =
11 16
and δ2 = 58 .
The idea of the proof was that the positive function φ1 could be a good approximation of a positive solution of P f¯ = µf¯ on ] − a, a[, when a is large enough, but φ1 does not satisfies the Neumann conditions at ±a. Therefore, using the cutoff function Ξ, we have added in (5.4), for large y > 0, a term ρ(a)φ2 (y)Ξ( ya ) in such a way that the boundary condition at x = a is satisfied. Then, we use the symmetry for y = −a.
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C. BOLLEY and B. HELFFER
The constant β is chosen such that kf¯a kL2 (]a,a[) = 1. ¯ d, 0) as d tends to +∞ In [3], we have deduced asymptotics for f0 and h0 = h(κ, with κ fixed. Here, we keep the scaling (2.7) for the study of h1 by rewriting the relation (2.27) in these new variables and parameters. We get, with p y = κh0 x 2
h1 h0
Z
a
−a
¯0 (y)2 f¯(y)2 dy B
Z p = κh0 · −
a
f¯(y)4 dy + 2κ−2
−a
Z
a −a
0 2 ¯ B1,0 (y) dy ,
(5.7)
¯1,0 ∈ H 2 (] − a, a[) are defined by ¯0 and B where B ( ¯0 (y) = A0 (x) p B with y = κh0 x , ¯1,0 (y) = κA1,0 (x) , B ¯1,0 satisfies In particular, B 00 ¯1,0 (y) = y f¯(y)2 B ¯ 0 (±a) = 0 , B 1,0 ¯ B1,0 (0) = 0 . If we define Λ by
Z Λ≡ −
a
for y ∈] − a, a[ , (5.8)
f¯(y)4 dy + 2κ−2
−a
Z
a
−a
0 2 ¯ B1,0 (y) dy ,
(5.9)
and the function sign by, sign(x) = +1 when x > 0 ,
sign(x) = −1 when x < 0 .
(5.10)
then, from (5.7), sign(h1 ) = sign(Λ) .
(5.11)
Let us show Lemma 5.3. Let e = 0. For any η > 0, there exists a constant a1 such that for a ≥ a1 and |κ − 2−1/2 | ≥ η, we have 1 √ −κ ·Λ > 0. 2 Proof. We compute the two integrals appearing in the right-hand side of (5.9). Using (5.3) with δ2 = 58 , we get Z a Z a 5 2 4 a 4 ¯ ¯ f (y) dy = 2 f (y) dy + O a exp − a 8 −a 0 as a tends to +∞.
STABILITY OF BIFURCATING SOLUTIONS
Let us now compute
Z
...
599
a
f¯a (y)4 dy , 0
by using (5.4) and the binomial formula. We remark that, for a large enough, the product φ1 φ2 is bounded, and that by the choice of the cutoff function and the variations of the function φ2 , we have, for some constants C > 0 and C˜ > 0, 2 y a φ2 (y) ≤ C exp , ∀y ∈ [0, a] , 0 ≤ Ξ a 2 and ∀y ∈ [0, a] ,
2 a ˜ 0≤Ξ φ1 (y) ≤ C exp − . a 4 y
Therefore, as a → +∞, Z a Z f¯a (y)4 dy = β(a)4 0
a
0
2
5 , exp(−2y 2 )dy + O a exp − a2 4
so that, using (5.5): Z a f¯a (y)4 dy = 2−3/2 π −1/2 + O(a exp(−a2 )) ,
as a tends to + ∞ .
(5.12)
0
Let us now compute the last integral in (5.7). ¯1,0 verifies, for y ∈] − a, a[, The function B Z y 0 ¯ t(f¯(t))2 dt . B1,0 (y) = −a
Using again (5.3), we get, as a tends to +∞: Z y 5 2 0 2 2 ¯ t exp(−t )dt + O a exp − a B1,0 (y) = β(a) 8 −a 5 2 −1 −1/2 2 . exp(−y ) + O a exp − a = −2 π 8 Therefore, as a tends to +∞, Z a Z 0 2 −2 −1 ¯ B1,0 (y) dy = 2 π −a
5 2 2 exp(−2y )dy + O a exp − a 8 −a 5 . = 2−5/2 π −1/2 · 1 + O a2 exp − a2 8 a
2
We then get that Λ is equal, as a tends to +∞, to 5 2 −3/2 −1/2 −2 2 2 2 Λ=2 1 − 2κ + (1 + κ )O a exp − a . π κ 8
(5.13)
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C. BOLLEY and B. HELFFER
Consequently, for any η > 0, |κ − 2−1/2 | ≥ η, and then for a ≥ a1 for a1 large enough, sign(Λ) = sign(1 − 2κ2 ) . Lemma 5.3 follows. Coming back to the initial units (see (2.7)), the estimate (5.2) gives the existence ˜0 > 0 such that we have, of positive constants C > 0, δ3 and a p κ δ1 . (5.14) for d κh0 ≥ a ˜0 , − 1 ≤ C exp(−δ3 d2 κh0 ) with δ3 = h0 4 Actually, δ1 > 12 gives δ3 > 18 . √ Using Lemma 2.2 (a), we have d κh0 ≥ κd. Therefore, from (5.14), when κd ≥ a ˜0 , h0 = κ + O(exp(−δ3 (dκ)2 )) . and Λ=2
−3/2 −1/2 −2
π
κ
2 2 d κ 2 2 . 1 − 2κ + (1 + κ )O exp − 8
(5.15)
We get Proposition 5.4 [κd large]. Let e = 0. For any η > 0, there exists a ˜0 > 0 ˜0 , and for h0 satisfying (2.5), then, such that for (κ, d) s.t. |κ − √12 | ≥ η and κd ≥ a
1 √ − κ · h1 > 0 . 2
Proof. The proposition results immediately from (5.15) and (5.7) or (5.11). Theorem 4.1 follows then using (2.6) and (2.25). We shall use (5.15) in the particular situation when d is keeped fixed, and κ tends to +∞. We get the following result: Proposition 5.5 [d fixed, κ large]. Let d > 0 and e = 0. Then, there exists a constant κ0 = κ0 (d) > 2−1/2 such that, for κ ≥ κ0 and for h0 satisfying (2.5), we have h1 < 0. At last, using (2.6) we get (a) and (b) in Theorem 4.2 and using once again (5.15), we easily get (4.1) and the limiting value 2−1/2 for κ. 5.2. The asymptotics for small κd 5.2.1. Main results for κd small When e = 0, we have proved in [7, Lemma 2.6] the following result: ¯ 2 (κ, d, 0) < 32, then h1 < 0. Lemma 5.6. Let e = 0. If (κ, d) satisfies d4 h
STABILITY OF BIFURCATING SOLUTIONS
Using that ¯ h(κ, d, 0)d tends to
√
...
601
12 as κd tends to 0 (see Lemma 3.2), we get
Proposition 5.7 [d small]. Let e = 0. There exists constants C0 and d0 > 0 such that for (κ, d) s.t. d ≤ d0 and κd ≤ C0 , and for h0 satisfying (2.5), then h1 < 0. The following result, which was suggested by the formal computations and numerical results of [15, 13, 5], improves Proposition 5.7 when κd is small. Theorem 5.8 [κd small]. Let e = 0. For any η > 0, there exists a0 such that √ for (κ, d) satisfying |d − 5| ≥ η, and 0 < κd ≤ a0 , and for h0 satisfying (2.5), then (i) √ ( 5 − d) · h1 < 0 . (ii) There exists 0 = 0 (κ, d) > 0 s.t. the curve of bifurcating solutions (f (., ), A(., ); h()) starting from (0, h0 x; h0 ) verifies for 0 < || ≤ 0 , √ ( 5 − d) · (h() − h0 ) < 0 . Theorem 5.8 is proved in Subsec. 5.2.3. In [8, Sec. 2.2] is studied (with more details given in the preliminary [5]) the behavior of the bifurcating solutions when κ and tend to 0. We got, in that paper, that, for fixed d, for κ ≤ κ1 and ≤ 0 (with 0 = 0 (d) and κ1 = κ1 (d) small), these solutions are uniquely defined, in a neighborhood of the bifurcating point (0, h0 x; h0 ) by the parameters κ and . The limiting model is the f − constant model studied in Subsec. 3.1 of [7]. In Theorem 10 of [8] is also studied the case when d and tend to 0, where the limiting problem is also the f − constant model. We first prove an analogous result when κd tends to 0. Here, we consider the behavior of the bifurcating solutions when both and κd tend to 0. 5.2.2. Bifurcations when and κd tend to 0 We consider, as in the proof of Theorem 3.1 (b)(i), the scaling (2.11), with g(u) = f (x) ;
V (u) = A(x)
for u =
x ∈ [−1/2, 1/2] , d
(5.16)
which gives a study interval independent of d. The Ginzburg–Landau equations become, for κ > 0, d > 0 and h > 0, 00 −g + σ(−1 + g 2 + V 2 )g = 0 in ] − 1/2, 1/2[ , 0 g (±1/2) = 0 , −V 00 + d2 g 2 V = 0 in ] − 1/2, 1/2[ , V 0 (±1/2) = η , (g, V ) ∈ (H 2 (] − 1/2, 1/2[))2 , with σ = κd and η = hd.
(5.17)
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C. BOLLEY and B. HELFFER
Let us prove, using the new parameter ¯ = d−1/2 , √ √ Proposition 5.9 [Bifurcation from the solution (0, 12u; 12)]. There , σ)) for (5.17), exists a C ∞ family of bifurcating solutions (g(., ¯, σ), V (., ¯, σ); η(¯ parametrized by ¯ and σ = κd (with κ > 0 and d > 0), which bifurcates from √ √ (0, 12u; 12). | ≤ ¯0 and for 0 < σ ≤ There exist constants ¯0 , σ0 and γ0 such that for 0 < |¯ , σ)) is defined and is the unique solution in σ0 , the solution (g(., ¯, σ), V (., ¯, σ); η(¯ H 2 (] − 1/2, 1/2[) × H 2 (] − 1/2, 1/2[) × R of (5.17) s.t. (i) (ii) (iii)
kg(., ¯, σ)kH 2 (]−1/2,1/2[) ≤ γ0 , √ kV (., ¯, σ) − 12ukH 2 (]−1/2,1/2[) ≤ γ0 , Z 12 g(u, ¯ , σ)g0 (u)du = ¯ . − 12
The function g0 (., η, σ) is the first eigenfunction of the Neumann problem (2.12) ¯ with g0 = φ. Proof. In our asymptotic study of the bifurcating solutions when κ tends to 0, in Sec. 2.2 of [8], we have considered all the starting normal solutions and proved that the only possible bifurcating solutions are the symmetric ones. Here, we write, when κd tends to 0, a simpler proof by using Theorem 3.4 which still gives that, for small κd, the bifurcating solutions are symmetric. Let us define the spaces, H 2,N e (] − 1/2, 1/2[) = {g ∈ H 2 (] − 1/2, 1/2[); g 0(±1/2) = 0} , ( H 2,od (] − 1/2, 1/2[) =
V ∈ H 2 (] − 1/2, 1/2[); V 0 (−1/2) = V 0 (1/2),
Z
)
1 2
V (u)du = 0
.
− 12
We consider the map Ψ from H 2,N e (] − 1/2, 1/2[) × R × H 2,od(] − 1/2, 1/2[) × R into L2 (] − 1/2, 1/2[) × R × L2 (] − 1/2, 1/2[) × R3 , defined by, (g, g+ , V, σ) → Ψ(g, g+ , V, v+ , σ) = (z, ¯, B, η, ρ) , and
(a) (b) (c) (d) (e)
z = −g 00 + σ(−g + g 3 + V 2 g) + φ¯1 g+ , Z 12 g0 gdu , ¯ = − 12
B = −V 00 + d2 g 2 V , η = V 0 (−1/2) , ρ =σ.
(5.18)
STABILITY OF BIFURCATING SOLUTIONS
...
603
We verify that Ψ is a local diffeomorphism in a neighborhood of (g, g+ , V, σ) = (0, 0, ηu, 0) by proving that the derivative of Ψ at (0, 0, ηu, 0) is invertible. We have, indeed, at (0, 0, ηu, 0), with V0 = ηu, 00 2 ¯ δz = −δg + σ(−1 + V0 )δg + φ1 δg+ , 1 Z 2 g0 δg du , δ¯ = −1 2
δB = −δV 00 , δη = δV 0 (−1/2) , δρ = δσ . So, the injectivity and the surjectivity are easy. If we denote by Φ = (Φi )i=1,...,4 the local inverse of Ψ, the solutions of the Ginzburg–Landau equations are given by the solutions of Φ2 (0, ¯, 0, η, ρ) = 0 . Let ζ(¯ , η, ρ) = Φ2 (0, ¯, 0, η, ρ) , 2 be the function defined from R ×√[0, +∞[ √ to R. We know that (g, V ; η) = (0, 12u; 12) is a solution of (5.17), so that
ζ(0, η, ρ) = 0 . We search a function ηˆ(¯ , ρ) such that ( ζ(0, ηˆ(¯ , ρ), ρ) = 0 , √ ηˆ(0, 0) = 12 . We shall then get a solution of the GL equations as function of ¯ and σ. For this, we apply the implicit function theorem to the function ˜ , η, ρ) = 1 ζ(¯ , η, ρ) . ζ(¯ ρ¯ We remark, indeed, that ∂g+ ¯, (0, 0, 0, η, ρ) = −λ ∂¯ ¯ is the principal eigenvalue of (2.12). where λ ¯ = 0 and ∂g+ (0, 0, 0, η, 0) = 0. When σ = 0, then λ ∂¯ We get, √ ∂ζ (0, 12, 0) = 0 . ∂¯ ¯ Moreover, we have λ = σ[τ (σ, 1, 0, η) − 1], where τ (κ, d, e, h) is the principal eigenvalue of (2.2). Therefore, ¯ λ lim = 0 , σ→0 σ
604
C. BOLLEY and B. HELFFER
and then
√ ∂2ζ (0, 12, 0) = 0 . ∂σ∂¯ In order to apply the implicit function theorem, we have now to prove that √ 1 ∂2 ζ (0, 12, 0) 6= 0 . ∂η∂¯ σ But, ∂2 ∂η∂¯
√ Z 12 √ ∂ 3 1 1¯ 2 2 ζ (0, 12, 0) = − λ = 2η , u g0 (u) du = σ ∂η σ 3 − 12
where we have used that η = Proposition 5.9 follows.
√ 12 and g0 (u) ≡ 1 when σ = 0.
5.2.3. Proof of Theorem 5.8 Let us now study the bifurcation starting from the normal solution (0, h0 x; h0 ) uniformly with respect to κd in ]0, a1 ]. We calculate, for fixed small σ, the first terms of the partial expansion in powers of at (σ, 0), of the C ∞ solutions given by this theory. If we let σ = κd, the bifurcating solutions (f (., σ, ), A(., σ, ); h(σ, )) satisfy, in a neighborhood of (0, A0 (., σ); h0 ) with A0 (., σ) = h0 x and h0 = h0 (σ), an expansion in powers of . In the scaling (5.16) and with ¯ = d−1/2 we get a partial expansion in powers of ¯ at (σ, 0), of (g(., σ, ¯), V (., σ, ¯); η(σ, ¯)) with η = hd (see Proposition 5.9) g(., σ, ¯) = ¯g0 (., σ) + ¯g˜(., σ, ¯) , 3 ) , V (., σ, ¯) = V0 (., σ) + ¯2 V1 (., σ) + O(¯ V 0 (±1/2, σ, ¯) = dh(σ, ¯) = dh0 + ¯2 dh1 (σ) + O(¯ 3 ) , with (g0 , g˜)L2 (]−1/2,1/2[) = 0 , g˜(., σ, ¯) =
p X
¯2j gj (., σ) + O(¯ 2p+1 ) ,
j=1
and where the elements gi are in H 2 (]− 1/2, 1/2[), Vi ∈ H 2 (]− 1/2, 1/2[) and hi ∈ R for the various indices i. They depend on σ and d, but we omit in the following the d-dependence. Using the regularity of the problem, we get immediately that the O are uniform in σ for 0 < σ ≤ a1 . We determine the terms of these expansions by equating powers of ¯ in the Ginzburg–Landau equations (5.17). First using the uniqueness of the bifurcating solutions, we get that the functions gi are even functions and that the Vi are odd functions (see (3.11)). We remark also that the expansion is actually in powers of κ2 d2 .
STABILITY OF BIFURCATING SOLUTIONS
...
605
We have, in particular, see (3.1), √ dh0 = 2 3 + O(σ 2 ) .
(5.19)
The cancellation of the ¯ terms give, in the first GL equation, ( −g000 + σ 2 (−1 + V02 )g0 = 0 in ] − 1/2, 1/2[ , g00 (±1/2) = 0 , therefore, g0 (u, σ) = g0 (u, 0) + O(σ 2 ) with g0 (u, 0) ≡ const. Moreover, the normalization (2.13) implies that g0 (u, σ) ≡ 1 + O(σ 2 ) .
(5.20)
The ¯2 -terms give (
−V100 + d2 g02 V0 = 0 in ] − 1/2, 1/2[ , V10 (±1/2) = dh1 (σ) .
We get, by integration, using (5.19) and (5.20), ! ! √ √ 3 3 2 2 2 3 2 2 + O(σ ) u + dh1 (σ) − d + d O(σ ) u , V1 (u, κ) = d 3 4
(5.21)
The term h1 (σ) will be determined later. Now, the ¯3 terms give ( −g100 + σ 2 (−1 + V02 )g1 + σ 2 [g03 + 2V0 V1 ]g0 = 0 , g10 (±1/2) = 0 . So, the compatibility condition implies Z 12 1 2 4 4 2 2 4 g0 + h0 d (1 + O(σ ))g0 u du 3 − 12 Z + 2h0 d
1 2
− 12
d3 h0 (1 + O(σ 2 )) u2 g02 du = 0 . dh1 − 8
We then get, using (5.19) and (5.21), √ d2 2 2 2 + (1 + d )O(κ d ) . dh1 (κ) = − 3 1 − 5 √ So that, when |d − 5| ≥ η and κd is small enough, √ sign (h1 ) = sign (d − 5) . When ¯ is small enough, we have √ d2 2 2 2 + (1 + d )O(κ d ) ¯2 + Oκ,d (¯ 3 ) . dh(κ, ¯) = dh0 (κ) − 3 1 − 5
(5.22)
(5.23)
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C. BOLLEY and B. HELFFER
With the parameter , we get, √ d2 3 + (1 + d2 )O(κ2 d2 ) 2 + Oκ,d (3 ) . h(κ, ) = h0 (κ) − 2 1 − d 5
(5.24)
Theorem 5.8 follows. 6. The Sign of h1 in the Asymmetric Case Let us now study the asymmetric case. We consider bifurcating solutions starting from normal solutions (0, h0 (x + e); h0 ) when e is different from zero and ¯ d, e). h0 = h(κ, 6.1. Main results It results from Theorem 3.3 and Theorem 3.4 the existence, for κd large, of a unique positive value of e = e¯(κ, d) defining a bifurcation point. We shall show, in that section, the following theorems: Theorem 6.1 [κd large]. Let a1 be the constant defined in Theorem 3.3 and Σ0 > 0 defined in (2.34). For any η > 0, there exists a constant a2 ≥ a1 , such that for (κ, d) satisfying |κ − Σ0 | ≥ η and κd ≥ a2 , for e = e¯(κ, d) and h0 satisfying (2.5), we have: (a) (Σ0 − κ) · h1 > 0 . (b) There exists 0 = 0 (κ, d) such that, if (f (., ), A(., ); h() is a curve of bifurcating solutions starting from the normal solution (0, h0 (x+ e¯(κ, d)); h0 ) then, for 0 < || ≤ 0 , (Σ0 − κ) · [h() − h0 ] > 0 ,
(6.1)
(Σ0 − κ) · ∆G(f (), A(); h()) > 0 .
(6.2)
and
When d is fixed, we get Theorem 6.2 [d fixed, κ large]. Let d > 0. There exists a constant κ ˜0 = ˜ 0 (d), for e = e¯(κ, d) and for h0 satisfying (2.5), κ ˜ 0 (d) > Σ0 such that, for κ ≥ κ we have: (i) h1 < 0. (ii) There exists 0 = 0 (κ, d) such that, if (f (., ), A(., ); h() is a curve of e(κ, d)); h0 ) bifurcating solutions starting from the normal solution (0, h0 (x+¯ then, for 0 < || ≤ 0 , h() < h0
and
∆G(f (), A(); h()) < 0 .
There exists a constant C > 0 such that κ ˜ 0 can be chosen verifying κ ˜ 0 (d) = Σ0 + O(exp(−Cd2 )
when
d → +∞ .
(6.3)
STABILITY OF BIFURCATING SOLUTIONS
...
607
Remark 6.3. This suggests that, for large d, there exists a curve d → κ(d), whose asymptot is Σ0 , determining the change of sign of h1 (κ, d) for large κd. Before giving the proofs, let us recall some previous results on the linearized problem around a normal asymmetric solution. 6.2. Previous results on the principal eigenfunction We used, in [4], the scaling (2.7) and then considered, for κ > 0, d > 0, e > 0, ¯ d, e), the problem (2.8)–(2.9). h0 = h(κ, In [4], we have constructed quasimodes for this problem when a tends to +∞ and when c verifies (6.4) c > −ρ2 · a for some fixed ρ2 satisfying 0 < ρ2 < 1. Let us recall the principal ideas for this construction. We consider the basis φ1 (., µ), φ2 (., µ), defined in (2.29) and (2.30), of the set of the solutions of the problem (2.28). Then, for α > 0, we consider, as in Subsec. 2.4, the first eigenvalue µ1 (α) of the harmonic oscillator P in ] − α, +∞[ with the Neumann condition at −α. The associated normalized eigenfunction is given by γφ1 (., µ1 (α)), where γ is a constant. Let us now consider the spectral problem P f¯ = µf¯ on the bounded interval ] − α, β[, with α = a − c, β = a + c. We have obtained the following result: Proposition 6.4 [see Proposition 5.3 in [4]]. There exists a0 > 0 such that, if (a, c) verifies (6.4) with a > a0 , the principal eigenvalue µ = µ(a − c, a + c) for the Neumann problem satisfies: 11 (6.5) |µ − µ1 (a − c)| ≤ C(a0 ) exp − (a + c)2 , 16 and the corresponding eigenfunction f¯ verifies 5 kf¯ − ΨkC 1 ([−a+c,a+c]) ≤ C(a0 ) exp − (a + c)2 , 8 where
˜ Ψ(y, a, c) = γ(a, c)φ1 (y, µ1 (a − c)) + δ(a, c)φ2 (y, µ1 (a − c))Ξ
y (a + c)
(6.6) ,
(6.7)
with: (a)
δ(a, c) = γ(a, c) exp(−(a + c)2 ) · (a + c)µ1 · (1 + O(a−2 ))
(b)
γ(a, c) = g(a − c) + O exp − 14 (a + c)2 ,
(6.8)
˜ verifies (uniformly for (a, c) verifying (6.4) with a > a0 ), where the cutoff function Ξ ˜ Ξ(y) =0
if y ≤
1 , 2
˜ Ξ(y) =1
if y ≥
3 , 4
608
C. BOLLEY and B. HELFFER
and where α → g(α) = kφ1 (y, µ1 (α))k−1 L2 (]−α,+∞[) is a bounded, continuous, strictly positive function such that: (6.9) g(0) = 21/2 · π −1/4 . ˜ the second term with φ2 , in the definition By the choice of the cutoff function Ξ, of Ψ in (6.7), appears as a corrective term for large y which makes the Neumann condition at y = a + c satisfied. The Neumann condition at y = −a + c is also satisfied due to the choice of the function φ1 which is an eigenfunction for the Neumann problem on ] − a + c, +∞[. It results immediately from (6.5) and from (2.31), that, for a ≥ a0 and c satisfying (6.4), we have, because h0 = µκ (see (2.7)), h0 =
11 κ + O exp − (a + c)2 . µ1 (a − c) 16
(6.10)
Let us now consider the problem (2.8) when the normal solution is a bifurcation point. As we recalled above, the positive parameter e = e¯(κ, d) is then uniquely determined (in the initial units) as a function of κ and d (for κd large enough). It is in fact proved in [4] (see Lemma 3.2), that, in the scaling (2.7), the positive value of c for which we have a bifurcation is uniquely determined as a function c¯ of a. Our study, in [4], shows also that α(a) = a − c¯(a) and µ1 (a − c¯(a)) tend exponentially fast respectively to the constants α0 and µ01 , when a tends to +∞, where α0 and µ01 are defined in Subsec. 2.4 (see (2.31)). Lemma 6.5 in [4] gives 1 2 , (6.11) |a − c¯(a) − α0 | ≤ C · exp − a 2 and we have
1 |µ1 (a − c¯(a)) − µ01 | ≤ C · exp − a2 2
as a → +∞ .
Therefore, when a tends to +∞, κ 1 2 . h0 = 0 + O exp − a µ1 2
(6.12)
We have also verified in that paper that, for all a > 0, h0 >
κ . µ01
(6.13)
6.3. Structure of the bifurcating solutions Our purpose, here, is to prove Theorem 6.1 and Theorem 6.2 which give sign(h1 ) as functions of κ and d. Several steps are needed for the proofs. The first three ones are dealing with the scaling (2.7) and only ask for a large and c satisfying (6.4)
STABILITY OF BIFURCATING SOLUTIONS
...
609
Step 1. Rewriting of the relation (2.27) giving h1 We rewrite the relation (2.27) which gives h1 , in the new variables introduced in (2.7). We get, as in (5.7), Z h1 a+c ¯ ¯ 2 dy 2 B0 (y)2 f(y) h0 −a+c Z a+c Z a+c p 4 −2 0 2 ¯ ¯ (6.14) f (y) dy + 2κ B1,0 (y) dy , = κh0 − −a+c
−a+c
√ ¯ 1,0 (y) = κA1,0 (x) with y = κh0 (x + e). ¯0 (y) = A0 (x) and B where B ¯1,0 verifies (cf. (5.8)) The function B −1 00 ¯ 2 ¯ 00 B1,0 (y) = h0 A1,0 (x) = y · f (y) for y ∈] − a + c, a + c[ , ¯ 0 (±a + c) = 0 , B 1,0 B ¯1,0 (0) = 0 . ˜ where Λ ˜ is defined by From (6.14), sign(h1 ) will be given by sign(Λ), Z a+c Z a+c 4 −2 0 2 ¯ ˜ ¯ Λ(a) ≡ − f (y) dy + 2κ B1,0 (y) dy . −a+c
(6.15)
(6.16)
−a+c
Let us prove, for c verifying (6.4), the following lemma: Lemma 6.5. There exists a0 > 0 such that for (a, c) verifying a ≥ a0 and (6.4), and for κ > 0, ˜ Λ(a) = (g(a − c))4 · −ρ(α(a)) + κ−2 σ(α(a)) 1 , + (1 + κ−2 )O exp − (a + c)2 4
(6.17)
where ρ(α) and σ(α), defined resp. in (2.32) and (2.33), depend only on α. The proof of Lemma 6.5 is given in the following two steps, corresponding to ˜ the analysis of the two terms appearing in the right-hand side of (6.16) defining Λ. ˜ Step 2. Analysis of the first integral in Λ 4 Using (6.6) and the decomposition of Ψ given by (6.7), we have, for a > a0 , Z a+c Z a+c 5 4 4 4 2 ¯ , (6.18) φ1 (y) dy + O a exp − (a + c) f(y) dy = γ(a, c) 8 −a+c −a+c uniformly for c verifying (6.4). Here, we have used thate the product φ1 φ2 is ˜ and ˜ and that the terms (δ(a, c)3 φ22 Ξ) bounded, for a large, on the support of Ξ, 4 4˜ (δ(a, c) φ2 Ξ) are small in comparison with the preceding error given by (6.6). e In the following, we omit the reference to the parameter µ for φ and φ . It is given by µ = µ (α). 1 2 1
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C. BOLLEY and B. HELFFER
Then, (6.8)(b) implies Z a+c Z 4 4 ¯ (f (y)) dy = g(a − c)
1 2 . φ1 (y) dy + O a exp − (a + c) 4 −a+c −a+c (6.19) We need however to verify that the last term in (6.19) is small in comparison with the preceding integral, when a tends to +∞. This is given by the following lemma: a+c
4
Lemma 6.6. There exists a0 > 0 s.t. for (a, c) verifying (6.4) with a ≥ a0 , Z a+c g(a − c)4 φ1 (y)4 dy ≥ Ca−1 , −a+c
for some strictly positive constant C. Proof of Lemma 6.6. Let us first recall that the function g is defined by Z +∞ 2 φ1 (y)2 dy = 1 . g(a − c) −a+c
Therefore, for every η > 0, there exists a0 > 0 s.t. for a ≥ a0 , Z a+c ∀ a > a0 , 1 − η ≤ g(a − c)2 · φ1 (y)2 dy ≤ 1 . −a+c
We then get, using the Cauchy–Schwarz inequality, Z g(a − c)
4
a+c
1 φ1 (y) dy ≥ 2a −a+c 4
≥
Z g(a − c)2
2
a+c 2
−a+c
φ1 (y) dy
1 (1 − η)2 , 2a
which gives Lemma 6.6. Now, using that for large β (see (2.29) and the upper bound µ ≤ 1 in (2.31)), Z Z ∞ 2 ∞ 1 exp(−2β 2 ) , φ1 (y)4 dy ≤ y exp(−2y 2 )dy = β 2β β β we can approximate in the right-hand side of (6.19) the integral on the bounded interval [−a + c, a + c] by an integral on [−a + c, +∞[. Therefore, (6.19) becomes Z a+c Z ∞ 1 . (6.20) φ1 (y)4 dy + O exp − (a + c)2 f¯(y)4 dy = g(a − c)4 4 −a+c −a+c ˜ Step 3. Analysis of the second integral in Λ ˜ as above, we have, Using (6.15) and the same properties for φ1 , φ2 and Ξ Z y 5 0 2 2 2 ¯ . t · φ1 (t) dt + O (a + c) exp − (a + c) B10 (y) = γ(a, c) 8 −a+c
STABILITY OF BIFURCATING SOLUTIONS
Therefore, using also that Z
a+c
−a+c
R a+c −a+c
...
611
γ 2 φ21 dy = O(1) for large a, we get
¯ 0 (y)2 dy = γ(a, c)4 B 1,0
Z
a+c
Z
2
y
t · φ1 (t) dt 2
−a+c
−a+c
dy
5 . + O (a + c)2 exp − (a + c)2 8
(6.21)
¯ 0 are only defined on [−a + c, a + c], but the functions ¯ 1,0 and B The functions B 1,0 φ1 and φ2 are defined on the interval [−a + c, ∞[, with, Z y 1 2 2 . t · φ1 (t) dt ≤ C exp − (a + c) for y > a + c, 2 a+c Therefore, Z
a+c
−a+c
¯ 0 (y)2 dy B 1,0 "Z
= γ(a, c) · 4
∞
−a+c
Z
2
y
t · φ1 (t) dt 2
−a+c
# 1 2 , dy + O exp − (a + c) 2
and then, using again (6.8)(b), Z
a+c −a+c
¯ 0 (y)2 dy = g(a − c)4 · B 1,0
Z
∞
−a+c
Z
2
y
−a+c
t · φ1 (t)2 dt
1 . + O exp − (a + c)2 4
dy (6.22)
Lemma 6.5 results from (6.22) and (6.20). ˜ and of h1 Step 4. The sign of Λ We suppose now that c = c¯(a) (or equivalently e = e¯(κ, d)). Using the regularity C 1 of φ1 and the strict positivity of the function g (see Lemma 4.1 in [7]) we get that g(α), σ(α) and ρ(α) tend respectively to g(α0 ), σ(α0 ) and ρ(α0 ) as α tends to α0 . More precisely, the function g(α), σ(α) and ρ(α) are lipschitzian in a neighborhood of α0 , so that the fast convergence of α(a) to α0 when a → +∞ (see (6.11)), gives that g(α(a)), σ(α(a)) and ρ(α(a)), tend respectively to g(α0 ), σ(α0 ) and ρ(α0 ) like exp(− 21 a2 ) when a tends to +∞. These results are completely independent of κ. Therefore, we get, from (6.17) and the definition (2.34) of Σ0 , the existence of some a2 > 0, independent of κ, such that, for a ≥ a2 , 1 ˜ Λ(a) = g(α0 )4 ρ(α0 ) · κ−2 (Σ20 − κ2 ) + (1 + κ2 )O exp − a2 . 4
612
C. BOLLEY and B. HELFFER
We have proved: Lemma 6.7. For any η > 0, there exists a constant a2 > 0 such that, for a ≥ a2 , and for |κ − Σ0 | ≥ η, ˜ = sign(Σ2 − κ2 ) . sign(Λ) 0 Let us now come back to the initial units. Using (6.12) and (6.13), we get κd , therefore, for κd large a + c¯(a) ≥ √ 0 2
µ1
1 κ2 d2 4 −2 2 2 2 ˜ Σ0 − κ + (1 + κ )O exp − . Λ(a) = g(α0 ) ρ(α0 ) · κ 4 µ01
(6.23)
We get, combining (6.14), (6.16) and (6.23), the following proposition: Proposition 6.8. Let a1 be the constant defined in Theorem 3.3 and Σ0 > 0 defined in (2.34). For any η > 0, there exists a constant a ˜2 > a1 such that, for (κ, d) ˜2 , and for h0 satisfying (2.5) with e = e¯(κ, d), satisfying |κ − Σ0 | ≥ η and κd ≥ a then h1 · (Σ0 − κ) > 0 . Consequently, using Theorem 3.4, we get Theorem 6.1. Theorem 6.2 follows from Proposition 6.8 and Theorem 3.4. The asymptotic relation (6.3) results from (6.23). The critical value κ ˜ 0 (d) plays, for the asymmetric bifurcating solutions, the same role as the constant κ0 (d) in Theorem 4.2 for the symmetric solutions. 7. Stability of the Bifurcating Solutions 7.1. The spectral problem when is small The local stability of a bifurcating solution (f (., ), A(., ), h()), given by Theorem 2.1, is obtained by studying for h = h() the hessian of (∆G)h calculated at the point (f (., ), A(., )). It will be denoted by Hessf (.,),A(.,)(∆G)h or more shortly by Q . It is defined on (H 1 (] − d/2, d/2[))2 × (H 1 (] − d/2, d/2[))2 by g (g, b)|Q | b L2 (]−d/2,d/2[) =κ
−2
Z
d 2
02
Z
d 2
g dx + −d 2
Z
−d 2
Z
d 2
+ 4 −d 2
(A20 − 1 + 22 A0 A1 + 32 f02 )g 2 dx
A0 f0 gbdx +
d 2
−d 2
b02 dx + 2
Z
d 2
−d 2
f02 b2 dx + o(2 ) .
(7.1)
The study of the local stability is then reduced to the analysis of the spectrum of the self-adjoint operator attached to this quadratic form. Because the resolvent is
STABILITY OF BIFURCATING SOLUTIONS
...
613
compact, the spectrum is discrete and we shall deduce the local stability from the analysis of the sign of the lowest eigenvalue. This leads us to study the corresponding linearized GL equations at (f (., ), A(., ); h()): ( (a) ( (b)
−κ−2 g 00 + (A(., )2 + 3f (., )2 − 1)g + 2A(., )f (., )b = λ()g , g 0 (±d/2) = 0 , −b00 + f (., )2 b + 2A(., )f (., )g = λ()b
(7.2)
in ] − d/2, d/2[) ,
b0 (±d/2) = 0 .
¯ d, e), with e ∈ R, the lowest eigenvalue for this problem By the choice h(0) = h(κ, when = 0 is equal to zero with multiplicity two (see (2.2)). The corresponding eigenspace is generated by f0 0 and u2 = . u1 = 0 1 When → 0, (7.2) can be written in the form g g = λ() , (M0 + M1 + 2 M2 + O(3 )) b b with
2 −2 d 2 −κ + A − 1 0 0 dx2 ; M0 = 2 d 0 − 2 dx 2A0 A1 + 3f02 0 . M2 = 0 f02
M1 =
0 2A0 f0
2A0 f0 0
;
Because the problem (7.2) is self-adjoint and is regular with respect to as obtained in Theorem 2.1, we can apply the general theory of perturbation for self-adjoint operators (see T. Kato [20]). Therefore, there exist two C 3 functions λ of , denoted (1) = λ(2) (0) = 0 and such that (7.2) has non-zero λ(1) and λ(2) , such that λ (0) g 3 solutions b which are also C with respect to . It is then sufficient to make a formal study in order to compute explicitly the first terms. The eigen-elements (g(., ), b(., ), λ()) will then be calculated such that ! ! 0 g(., ) f0 g2α,β g1α,β 2 + + O(3 ) , +β + =α (7.3) α,β 0 1 b(., ) bα,β b 1 2 and λ() = λα,β + 2 λα,β + O(3 ) . 1 2 with (α, β) ∈ R2 − {(0, 0)}, giα,β , bα,β in H 2 (] − d/2, d/2[) and λα,β in R for i = 1, 2. i i Let us prove
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C. BOLLEY and B. HELFFER
Proposition 7.1. Let d > 0, κ > 0 and (e, h0 ) satisfying (2.5). Then, there exists 0 s.t. for 0 < || ≤ 0 , the eigen-elements corresponding to the two lowest eigenvalues of (7.2) can be chosen in the set described by (1) 2 (1) 3 λ () = λ2 + O( ) (7.4) (a) g (1) (x, ) = f0 + O(2 ) b(1) (x, ) = 2 A1,0 + O(3 ) and
(2) 2 (2) 3 λ () = λ2 + O( ) g (2) (x, ) = f0 + O(3 ) b(2) (x, ) = 1 + O(2 ) ,
(b)
(7.5)
with (1) λ2
h1 = −4 h0
Z
d 2
−d 2
A20
f02
dx ;
(2) λ2
1 = d
2 1+ h0
Z
!
d 2
−d 2
A0 f0 ψ0 dx
and where ψ0 is the unique solution in H 2 (] − d/2, d/2[) of −2 00 2 −κ ψ0 + (A0 − 1) ψ0 = −2 h0 A0 f0 ψ00 (±d/2) = 0 , (ψ0 , f0 )L2 (]−d/2,d/2[) = 0 .
,
(7.6)
f
in ] − d/2, d/2[ , (7.7)
Remark 7.2. We remark that, using Lemma 2.3 (b) (ii), (2)
λ2 =
1 ∂2τ ¯ d, e)) , (κ, d, e, h(κ, 2 d ¯h2 ∂e2
(7.8)
where τ is defined in (2.2). Proof of Proposition 7.1. We calculate the first terms of the expansion in powers of (whose existence is proved in [20]). We expand the equations in (7.2) by using (7.3), (2.2) and (2.6) and equal the corresponding powers of . Cancellation of the terms. −κ−2 (g1α,β )00 + (A20 − 1) g1α,β + 2β A0 f0 = α λα,β f0 in ] − d/2, d/2[ , 1 (g α,β )0 (±d/2) = 0 , 1 α,β 00 α,β 2 −(b , 1 ) + 2A0 f0 = β λ1 α,β 0 (b ) (±d/2) = 0 . 1 f The function ψ is the partial derivative of f with respect to the parameter e (see [3]). 0 0
(7.9)
STABILITY OF BIFURCATING SOLUTIONS
...
615
Compatibility equations give then the necessary conditions for the existence of a solution, Z d2 2β A0 f02 dx = α λα,β , 1 −d 2
Z
d 2
2α −d 2
A0 f02 dx = β λα,β . 1
¯ d, e) can be written (see But, the condition (2.5)(b) which is satisfied by h0 = h(κ, Lemma 2.3 (a) (ii)), Z d2 A0 f02 dx = 0 . −d 2
Therefore, we get
(
α λα,β = 0, 1 β λα,β = 0, 1
and then, because (α, β) 6= (0, 0), = 0. λα,β 1 We then solve the system (7.9) in such a way that all the elements g1 and b1 are normalized by Z d2 Z d2 g1α,β f0 dx = 0 and bα,β dx = 0 . 1 −d 2
−d 2
We get β ψ0 and bα,β = 2α A1,0 , 1 h0 is defined in (2.26). The constants α and β remain free at this stage. g1α,β =
where A1,0
Cancellation of the 2 terms −κ−2 (g2α,β )00 + (h20 (x + e)2 − 1) g2α,β + 2α A0 A1 f0 + 3α f03 + 2A0 f0 bα,β = α λα,β f0 1 2 α,β 0 (g2 ) (±d/2) = 0 , α,β β 00 2 −(bα,β , 2 ) + βf0 + 2 h0 A0 f0 ψ0 = β λ2 α,β 0 (b2 ) (±d/2) = 0 .
in ] − d/2, d/2[ , (7.10)
Using the relation (2.27), the compatibility conditions give then the new necessary conditions, Z d2 αλα,β = α 2A0 A1 f02 dx 2 −d 2
Z
d 2
+ 3α −d 2
Z f04 dx + 2α
d 2
−d 2
A0 A1,0 f02 dx .
(7.11)
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C. BOLLEY and B. HELFFER
and
Z βλα,β 2 d =β
d 2
−d 2
f02 dx + 2
β h0
Z
d 2
A0 f0 ψ0 dx ,
−d 2
(7.12)
If α 6= 0, the condition (7.11) then becomes Z (1) λ2
d 2
=4 −d 2
Z A0 A1,0 f02
and using (2.27), (1) λ2
h1 = −4 h0
Z
d 2
dx + 2
d 2
−d 2
−d 2
f04 dx
(7.13)
A20 f02 dx .
(7.14)
When β 6= 0, the condition (7.12) gives, (2) λ2
1 = d
2 1+ h0
Z
!
d 2
−d 2
A0 f0 ψ0 dx
.
(7.15)
We will see that in most of the limiting problems these two quantities are not equal. In any case, we get the first solution with β = 0, the second with α = 0. In particular, g11,0 ≡ 0 ; b0,1 1 = 0. and, if as before (g2α,β , f0 )L2 = 0 and (bα,β 2 , 1)L2 (]−d/2,d/2[) = 0, g20,1 ≡ 0 ;
b1,0 2 = 0.
Proposition 7.1 follows. 7.2.
The stability and instability of symmetric solutions
7.2.1. Proof of Theorems 4.3 and 4.5 For showing the stability or the instability of the bifurcating symmetric solutions, (1) (2) we only have to calculate the two first eigenvalues λ2 and λ2 . Let us first remark, using (7.8) and Theorem 3.1 (b), that (2)
Lemma 7.3. (i) There exists a constant a0 such that, for κd ≤ a0 ,λ2 > 0. (2) (ii) We suppose e = 0. There exists a constant a2 such that, for κd ≥ a2 , λ2 < 0. (2)
The positivity of λ2 for κd small can also be proved as follows. Starting from the expansions of g0 (u) = f0 (x) and V0 (u) = A0 (x) in powers of σ = κ2 d2 written, in the scaling u = xd (see Subsec. 5.2), we search an expansion for ψ¯0 (u) = ψ0 (x), solution of (7.7), as ψ¯0 = ψ¯0,0 + κ2 d2 ψ¯0,1 + O(κ4 d4 ) . We can easily justify this expansion by regular perturbation theory.
STABILITY OF BIFURCATING SOLUTIONS
...
617
The cancellation of the κ0 -terms in (7.7) gives ¯00 ψ0,0 = 0 , 0 ψ¯0,0 (±1/2) = 0 , (ψ¯ , g ) 2 0,0 0 L (]−1/2,1/2[) = 0 . Therefore, ψ¯0,0 ≡ 0, and, from (7.15), (2)
λ2 =
1 (1 + O(κ2 d2 )) . d
Let us prove the first part of Theorem 4.3 which concerns κd small. From (1) (2) Theorem 7.1, it is sufficient to verify that the both λ2 and λ2 are positive. (1) From Proposition 5.8, we get that, for any η > 0, λ2 is positive when κd ≤ a0 √ (2) and d ≤ 5 − η. From Lemma 7.3 (i), we obtain λ2 > 0 in the same case. The stability of the bifurcating solutions, for 6= 0 in a neighborhood of 0, follows. Proof the first part. When √ of Theorem 4.3 (ii). We proceed as in the proof of (1) d > 5 and κd small, we use Theorem 5.8, which gives that λ2 is stricly negative. Part (ii) in Proposition 4.3 follows. 7.2.2. The instability for large κd: proof of Theorem 4.5 Theorem 4.5 results from Lemma 7.3 (ii), because, for large κd, one eigenvalue (2) (here λ2 ) is strictly negative. The instability of the symmetric bifurcating solutions, for 6= 0 in a neighborhood of 0 is then proved for κd large. Remark 7.4. When e = 0, we have shown (see Proposition 5.4) that h1 < 0 when κ > 2−1/2 and κd is large, and similarly that h1 > 0 when κ < 2−1/2 and κd is large. Theorem 4.3 (ii) shows that the local stability of the symmetric solutions is not always given by the sign of h1 as generally admitted. (1)
(2)
The asymptotics given in Sec. 5 permit to analyze the behavior of λ2 and λ2 when e = 0 and κd tends to +∞. (2) The asymptotics for d f¯(a)−2 λ2 , calculated in [3, Proof of Proposition 2.21] when e = 0 and κd tends to +∞, give in fact more information than those of Lemma 7.3. We get 2 2 1 d κ (2) 1 + O(d−2 κ−2 ) . (7.16) λ2 = − π −1/2 d2 κ3 exp − 2 4 (1)
Moreover, the coefficient λ2 is given by (7.6) and, with the notations of Subsec. 5.1, (1) by λ2 = −2 κ Λ. Therefore, using the computations of Sec. 5 (see (5.15)), we get when κd tends to +∞, 2 2 d κ (1) −1/2 −1/2 −1 2 2 . (7.17) 1 − 2κ + (1 + κ ) O exp − π κ λ2 = −2 8
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C. BOLLEY and B. HELFFER
7.2.3. Local stability for a reduced symmetric problem In the study [6], and as was done in previous articles of [15] or [9], we have restricted the research of the solutions of the Ginzburg–Landau equations to the set of the symmetric solutions. This assumption leads us to restrict the domain of the GL functional to a subset Hsym of (H 1 (] − d/2, d/2[))2 corresponding to these solutions. This subset is defined by Hsym = {(f, A) ∈ (H 1 (] − d/2, d/2[))2 ; ∀ x ∈] − d/2, d/2[ (f (−x), A(−x)) = (f (x), −A(x))} . This symmetric problem is equivalent to a problem restricted to the half interval ]0, d/2[), where we consider a functional Φh defined on {(f, A) ∈ (H 1 (]0, d/2[))2 ; A(0) = 0} by Z d/2 1 κ−2 f 02 − f 2 + f 4 + A2 f 2 + (A0 − h)2 dx . Φh (f, A) = 2 0 This last point of view was used in particular in [6]. We remark that, by considering the symmetric problem, the stability of a GL solution can be different than in the initial problem. This is in particular the case for the symmetric bifurcating solutions when κd is large. We have proved in Theorem 4.5 that the symmetric bifurcating solutions are unstable for κd large enough (with respect to the GL functional (∆G)h ). We prove now the following theorem. Theorem 7.5. Let e = 0. For any η > 0, there exists a constant a1 > 0 such that, for (κ, d) satisfying κd ≥ a1 and |κ − 2−1/2 | ≥ η, and for h0 satisfying (2.5), there exists 1 > 0 such that for 0 < || ≤ 1 , the following properties are verified. (i) When κ > 2−1/2 , the bifurcating solutions (f (., ), A(., ); h()) starting from (0, h0 x; h0 ) are locally stable by respect to the symmetric problem. (ii) When κ < 2−1/2 , they are unstable. Proof. It is sufficient to prove that the bifurcating solutions give local minima for (∆G)h in restriction to Hsym , when h is fixed. So, as in the proof of Theorem 4.5, we study the hessian of the functional at a bifurcating solution, but we restrict this hessian to Hsym × Hsym . Because (f (.; ), A(.; )) ∈ Hsym , the operator defined by the left-hand side of (7.2) leaves Hsym stable, so that the new spectral problem is a classical one. The restriction to Hsym eliminates the eigenvalue λ(2) () and the lowest eigenvalue µ(1) () of the hessian reduced to the symmetric solutions is simple and equal to λ(1) (). We get, for small, (7.18) λ(1) () = µ(1) () . For proving Theorem 7.5, it is then sufficient to apply Proposition 5.4 which gives that h1 < 0 when κ > 2−1/2 and h1 > 0 when κ < 2−1/2 , then to apply Proposition 7.1 which proves that for small enough, λ(1) has the sign of h1 .
STABILITY OF BIFURCATING SOLUTIONS
...
619
7.3. The stability or instability of asymmetric solutions (2)
In view of proving Theorem 4.6, we first estimate λ2 (2)
7.3.1. The study of λ2
when κd tends to +∞.
for κd large
Let us show: Proposition 7.6. Let a1 > 0 be defined as in Theorem 3.3. There exists a2 ≥ ¯ d, e¯(κ, d)) satisfying a1 such that, for (κ, d) satisfying κd ≥ a2 and for h0 = h(κ, (2.5)(a, b), 1 1 2 2 (2) 2 2 . (7.19) λ2 = α0 (g(α0 )) (φ1 (−α0 )) + O exp − d κ d 16 (2)
Proof. According to (7.8), the sign of λ2
is given by the sign of
∂2τ ¯ d, e¯)) . (κ, d, e¯(κ, d), h(κ, ∂e2 For getting an estimate of this term, we use a characterization of the variation of the eigenvalues of the operator P with respect to the boundaries of the domain which is given in [12]. In the scaling used in Sec. 6 and with µ ˜(a, c) = µ(a − c, a + c), we get ∂µ ˜ (a, c) = Φ(c) − Φ(−c) , ∂c
(7.20)
with Φ(c) = ((a + c)2 − µ ˜(a, c)) · (f¯(a + c, a, c))2 , and τ (κ, d, e, h) =
h d µ ˜ (κh)1/2 , (κh)1/2 e . κ 2
Formula (7.20) has also been used for proving Lemma 7.3 in the symmetric case. We remark that 2 ∂2τ ˜ ¯ d, e¯)) = h ¯2 ∂ µ (κ, d, e¯(κ, d), h(κ, (a, c) , 2 ∂e ∂c2
(7.21)
and compute the last term. Using the property ∂∂cµ˜ (a, c) = 0 when c = c¯(a), the symmetry properties and the boundary conditions f¯0 (±a + c, a, c) = 0, we get by differentiation of (7.20), ˜ ∂2µ (a, c) = 2(a + c)(f¯(a + c, a, c))2 ∂c2 ∂ f¯ (a + c, a, c) + 2[(a + c)2 − µ ˜ (a, c)] · f¯(a + c, a, c) · ∂c + 2(a − c)(f¯(−a + c, a, c))2 ∂ f¯ (−a + c, a, c) . (7.22) − 2[(a − c)2 − µ ˜ (a, c)] · f¯(−a + c, a, c) · ∂c
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C. BOLLEY and B. HELFFER
In the symmetric case, the functions f¯(., a, c), ∂2µ ˜ ∂c2 (a, c)
∂ f¯ ∂c (., a, c)
(with c = 0) and then
are exponentially small near x = ±a, when a is large, so that we needed accurate estimates on each term in order to get the result. Let us prove, in the asymmetric case: Proposition 7.7. There exists a constant A > 0 such that for a ≥ A and c = c¯(a), 1 ˜ ∂2µ 2 2 2 (a + c ¯ (a)) . (7.23) (a, c ¯ (a)) = 2α (g(α )) (φ (−α )) + O exp − 0 0 1 0 ∂c2 4 Proof. We first deduce from Proposition 6.4 (using also (2.29) and (2.30)) and from (6.11) that, when a is large and c = c¯(a), 1 2 ¯ , f (a + c, a, c) = O exp − (a + c) 2 1 2 2 ((a − c) − µ . ˜ (a, c)) = O exp − (a + c) 2 Let us estimate f¯(−a + c, a, c) when c = c¯(a) and a tends to +∞. Using (6.6) and (6.7), 5 2 ¯ f (−a + c, a, c) = γ(a, c)φ1 (−a + c) + O exp − (a + c) , 8 then, from (6.8)(b) and (6.11)
1 2 ¯ f (−a + c, a, c) = g(−α0 )φ1 (−α0 ) + O exp − (a + c) , 4
(7.24)
with g(−α0 )φ1 (−α0 ) > 0 . ∂2 µ ˜ ¯(a)) ∂c2 (a, c
when a is large, we only need a control of the So, in order to estimate ∂ f¯ term ∂c (±a + c, a, c). We will use the following lemma. Lemma 7.8. There exist constants C > 0 and A > 0 such that for a ≥ A and c = c¯(a), ∂ f¯ (±a + c, a, c) ≤ C(a + c)4 . ∂c Proof. Let us use the following translated function: v(x, a, c) = f¯(x + c, a, c) for x ∈ [−a, a] , in order to differentiate more easily f¯. Then, v is solution of −v 00 + (x + c)2 v = µv in ] − a, a[ 0 v (±a) = 0 , kvkL2 (]−a,a[) = 1 , v ∈ H 2 (] − a, a[) .
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Its first partial derivative w ≡ ∂c v with respect to c satisfies ∂µ 00 2 −w + (x + c) w − µw = −2(x + c)v + ∂c v in ] − a, a[ w0 (±a) = 0 , (w, v)L2 (]−a,a[) = 0 , w ∈ H 2 (] − a, a[) , and it is studied at a point where
∂µ ∂c
= 0. Moreover,
∂c f¯(x + c, a, c) = ∂c v(x, a, c) − ∂x v(x, a, c)
for x ∈] − a, a[ .
Using the boundary conditions, we now have ∂c f¯(±a + c, a, c) = ∂c v(±a, a, c) . So, we can estimate ∂c v instead of ∂c f¯. We prefer to calculate this term with the variable y = x + c. The function z defined by z(y, a, c) = ∂c v(x, a, c) satisfies −z 00 + y 2 z − µz = −2y f¯ in ] − a + c, a + c[ z 0 (±a + c) = 0 , (7.25) ¯ L2 (]−a+c,a+c[) = 0 , (z, f) z ∈ H 2 (] − a + c, a + c[) , If we introduce the spaces, Fa,c = {v ∈ L2 (] − a + c, a + c[) ; (v, f¯)L2 (]−a+c,a+c[) = 0 \ Ga,c = {v ∈ H 2 (] − a + c, a + c[) Fa,c ; v 0 (±a + c) = 0} then (P − µ) is an isomorphism from Ga,c onto Fa,c . More precisely, we get, as in [4], using classical estimates, k(P − µ)−1 kL(Fa,c ,Ga,c ) ≤ C(a + c)2 . Consequently, it results from (7.25) that, kzkH 2 (]−a+c,a+c[) ≤ C(a + c)3 . Using the control with respect to a of the norm of the continuous injection from H 1 in C 0 , we get the lemma. In the initial units, it results from (7.8) and (7.21), that (2)
λ2 = Then, from (7.23) with a = (6.13)), we get (7.19).
d 2
1 ∂2µ ˜ (a, c¯(a)) . 2d ∂c2
√ κh0 and using once again that ¯h ≥ µ01 > 0 (see
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As a consequence, we obtain the following corollary: Corollary 7.9. There exists a constant a2 > 0 such that for (κ, d) satisfying κd ≥ a2 and for h0 satisfying (2.5)(a, b) with e = e¯(κ, d), then (2)
λ2 > 0 . We then get that (2.5)(c) is satisfied. This result, combined with (7.8), has been announced in the proof of Theorem 3.4 and permits to complete the proof of the theorem. 7.3.2. Stability or instability for κd large Let us prove Theorem 4.6 which gives the stability of the asymmetric bifurcating solutions when κd is large. Part (i) results from Theorem 6.1 because, under the hypothesis of the theorem, the eigenvalue λ(1) of (7.2) is negative. The second part of Theorem 4.6 results from Theorem 6.1 which gives that λ(1) is strictly positive for κ > Σ0 , and from Corollary 7.9 which gives that the eigenvalue λ(2) is then also strictly positive. 8. Conclusion Let us summarize the stability results in three pictures as function of the parameters κ and d. We distinguish different domains. For each of them we give, when they are known, the sign of the first eigenvalues λ(1) and λ(2) , and write S when the bifurcating solutions are stable and U when they are unstable. 8.1. Stability of symmetric solutions The parameters d and κd are the main parameters of this study (see Theorems 4.3 and 4.5). From Proposition 4.3, Lemma 7.3, (7.16) and (7.17), we √ get 7 , κd = a , d = {(κ, d) ; d = 5−η}, different domains limited by the curves κd = a 0 1 1 √ d2 = {(κ, d) ; d = 5 + η}, κ1 = {(κ, d) ; κ = 2−1/2 − η} and κ2 = {(κ, d) ; κ = 2−1/2 + η} for any η > 0 and unknown constants a0 and a1 . Domain Domain Domain Domain Domain Domain Domain
(1) (2) (3) (4) (5) (6) (7)
: : : : : : :
S, λ(1) > 0, λ(2) > 0. unknown unless λ2 > 0. U, λ(1) < 0, λ(2) > 0. unknown. U, λ(1) < 0, λ(2) < 0. U, λ(1) unknown, but λ(2) < 0. U, λ(1) > 0, λ(2) < 0.
One can probably replace, changing also a0 and a1 , the domains (2) and (6) by curves separating the domains (1) and (3) on one part and (5) and (7) on the other part.
STABILITY OF BIFURCATING SOLUTIONS
...
623
Fig. 1. Stability of symmetric bifurcating solutions: theoretical results.
8.2. Stability of asymmetric solutions (Fig. 2) We recall (see Theorem 3.3), that in that case, the asymmetric solutions exist when κd ≥ a1 (for some positive constant a1 large enough). Theorem 6.2 gives the stability results. We get 5 different domains limited by the curves κd = a1 , κd = a2 , κ1 = {(κ, d); κ = Σ0 − η} and κ2 = {(κ, d); κ = Σ0 + η} for any η > 0 and some constant a2 verifying a2 ≥ a1 . Domain Domain Domain Domain Domain
(1) (2) (3) (4) (5)
: : : : :
no asymmetric solutions. unknown. U, λ(1) < 0, λ(2) > 0. unknown, unless λ(2) > 0. S, λ(1) > 0, λ(2) > 0.
One can also probably replace the domain (4) by a curve separating the domains (3) and (5), with other a1 and a2 . 8.3. Stability for the symmetric problem (Fig. 3) Using Subsec. 7.2.3 and formula (7.18), we only need to consider, in this case, the sign of the eigenvalue µ(1) , or of the eigenvalue λ(1) for (7.2). From Theorems 7.1, and 7.5, we get, as in the first limited by curves κd = a0 , √ case, 7 different domains √ κd = a1 , d1 = {(κ, d); d = 5 − η}, d2 = {(κ, d); d = 5 + η}, κ1 = {(κ, d) κ =
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Fig. 2. Stability of asymmetric bifurcating solutions: theoretical results.
Fig. 3. Stability of bifurcating solutions for the symmetric problem: theoretical results.
STABILITY OF BIFURCATING SOLUTIONS
...
625
2−1/2 − η} and κ2 = {(κ, d); κ = 2−1/2 + η} for any η > 0 and unknown constants a0 and a1 . We get Domain Domain Domain Domain Domain Domain Domain
(1) (2) (3) (4) (5) (6) (7)
: : : : : : :
S, µ(1) > 0. unknown. U, µ(1) < 0. unknown. U, µ(1) < 0. unknown. S, µ(1) > 0.
The picture representing these results (see Fig. 3) appears similar as Fig. 1 with possibly different constants. But it is clear that the domain where the solutions are stable are no more the same as before. 8.4. Epilogue This study justifies mathematically a great part of the stability results, in various asymptotics, which were generally admitted in the literature but not proved before. But, the instability of the symmetric bifurcating solutions, when κd is large, is also established. Because in that case the normal solutions are also unstable, this result means that there exists another superconducting solution which is stable. We conjecture that, for κd large and for h near ¯h(κ, d, 0) with h < ¯h(κ, d, 0), bifurcating ¯ d, 0)), solutions belonging to the two curves starting, one from (0, ¯h(κ, d, 0)x; h(κ, ¯ ¯ the other from (0, h(κ, d, e¯(d))(x + e¯(d)); h(κ, d, e¯(d))) exist simultaneously and that the stable one is the asymmetric solution. On the other hand, we have proved the stability of the symmetric bifurcating solutions for κd and κ large enough when we restrict the problem to symmetric solutions. Acknowledgements This study was partially motivated by questions and numerical computations of J. Chapman and private discussions with him a few years ago. We were also motivated by the paper by S. P. Hastings–W. C. Troy [17] and correspondence with S. P. Hastings on the existence of asymmetric solutions. References [1] H. Berestycki, A. Bonnet and S. J. Chapman, “A semi-elliptic system arising in the theory of type-II superconductivity, ” Comm. App. Nonlinear Anal. 1 (3) (1994) 1–21. [2] C. Bolley, “Familles de branches de bifurcations dans les ´ equations de Ginzburg– Landau,” M2 AN 25 (3) (1991) 307–335. [3] C. Bolley, “Mod´elisation du champ de retard a ` la condensation d’un supraconducteur par un probl`eme de bifurcation,” M2 AN 26 (2) (1992) 235–287. [4] C. Bolley and B. Helffer, “An application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material,” Annales de l’Institut Henri Poincar´e (Section Physique Th´eorique) 58 (2) (1993) 189–233.
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[5] C. Bolley and B. Helffer, “Rigorous results on the Ginzburg–Landau models in a film submitted to an exterior parallel magnetic field,” preprint Ecole Centrale de Nantes, 1993. [6] C. Bolley and B. Helffer, “Rigorous results for the Ginzburg–Landau equations associated to a superconducting film in the weak κ-limit, Rev. Math. Phys. 8 (1) (1996) 43–83. [7] C. Bolley and B. Helffer, “Rigorous results on the Ginzburg–Landau models in a film submitted to an exterior parallel magnetic field. Part I,” Nonlinear Studies 3 (1) (1996) 1–29. [8] C. Bolley and B. Helffer, “Rigorous results on the Ginzburg–Landau models in a film submitted to an exterior parallel magnetic field. Part II.” Nonlinear Studies 3 (2) (1996) 1–32. [9] S. J. Chapman, “Nucleation of superconductivity in decreasing fields. I and II. “European J. Appl. Math. 5 (4) (1994) 449–468, 469–494. [10] S. J. Chapman, S. D. Howison, J. B. McLeod and J. R. Ockendon, “Normalsuperconducting transitions in Ginzburg–Landau theory,” Proc. Roy. Soc. Edin. 119A (1991) 117–124. [11] S. J. Chapman, S. D. Howison and J. R. Ockendon, “Macroscopic models for superconductivity,” SIAM review, 344 (1992) 529–560. [12] M. Dauge and B. Helffer, “Eigenvalues variation I, Neumann Problem for Sturm– Liouville operators,” J. Differential Eqs., 104 (2) (1993) 243–262. [13] B. Dugnoille, “Etude th´eorique et exp´erimentale des propri´et´es magn´etiques des couches minces supraconductrices de type 1 et de κ faible,” thesis, Mons, 1978. [14] V. L. Ginzburg “On the theory of superconductivity,” Nuovo Cimento 2, (1995) 1234. [15] V. L. Ginzburg, “On the destruction and the onset of superconductivity in a magnetic field,” Soviet Phy. JETP 7 (1958) 78. [16] V. L. Ginzburg and L. D. Landau, “On the theory of superconductivity,” Zh. Eksperim. i teor. Fiz. 20 (1950) 1064–1082; English translation L. D. Landau Men of Physics ed. D. Ter Haar, Pergamon Oxford, (1965) 138–167. [17] S. P. Hastings and W. C. Troy, “There are asymmetric minimizers for the onedimensional Ginzburg–Landau model of superconductivity,” preprint, 1996. [18] B. Helffer and J. Sj¨ ostrand, “Multiple wells in the semiclassical limit I, Comm. in P.D.E., 9 (4) (1984) 337–408. [19] D. St. James and P. G. de Gennes, “Onset of superconductivity in decreasing fields, Phys. Lett. 7 (1963) 306. [20] T. Kato, Perturbation Theory for Linear Operators, Springer, 1980. [21] M. H. Millman and J. B. Keller, “Perturbation theory of nonlinear boundary-value problems,” J. Math. Phys. 10 (2) (February 1969). [22] F. Odeh, “Existence and bifurcation theorems for the Ginzburg–Landau equations,” J. Math. Phys. 8 (12) (December 1967). [23] F. Odeh, “A bifurcation problem in superconductivity,” in Bifurcation Theory and Nonlinear Eigenvalue Problems, eds. J. B. Keller, S. Antman and W. A Benjamin, Inc. (1969) 99–112. [24] Y. Sibuya, “Global theory of a second linear differential equation with a polynomial coefficient,” North-Holland (1975).
CLUSTER PROPERTIES OF ONE PARTICLE ¨ SCHRODINGER OPERATORS. II V. KOSTRYKIN Institut f¨ ur Reine and Angewandte Mathematik Rheinisch-Westf¨ alische Technische Hochschule Aachen D-52056 Aachen, Germany
R. SCHRADER Institut f¨ ur Theoretische Physik Freie Universit¨ at Berlin Arnimallee 14, D-14195 Berlin, Germany Received 17 May 1997 We continue the study of cluster properties of spectral and scattering characteristics of Schr¨ odinger operators with potentials given as a sum of two wells, begun in our preceding article [Rev. Math. Phys. 6 (1994) 833–853] and where we determined the leading behaviour of the spectral shift function and the scattering amplitude as the separation of the wells tends to infinity. In this article we determine the explicit form of the subleading contributions, which in particular show strong oscillatory behaviour. Also we apply our methods to the critical and subcritical double well problems.
1. Introduction We consider Schr¨odinger operators with double well potentials when the distance between the wells tends to infinity. More precisely the main subject of our study is the asymptotic behaviour as |d| → ∞ of Hamiltonians in L2 (R3 ) of the form H(d) = −∆ + Vd ,
d ∈ R3 ,
(1.1)
with Vd = V1 + V2 (· − d) .
(1.2)
Here ∆ is the Laplace operator, and V1 and V2 are real valued functions in the Rollnik class R, acting as multiplication operators on L2 (R3 ). Recall that V ∈ R iff Z |V (x)kV (y)| dxdy < ∞ . |x − y|2 The present work is a direct continuation of our previous paper [27], where we established the limiting interrelations (as |d| → ∞) between the resolvent, scattering matrix and the spectral shift function of H(d) and those of the single well Hamiltonians Hi = −∆ + Vi , i = 1, 2 . (1.3) 627 Reviews in Mathematical Physics, Vol. 10, No. 5 (1998) 627–683 c World Scientific Publishing Company
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Following the terminology of many particle scattering theory [16, 19, 20] we call the aforementioned relations cluster properties. The main aim of the present article is to exhibit the explicit structure of the subleading terms in the asymptotic expansion of the spectral shift function and the scattering amplitude when |d| → ∞. Cluster properties of the discrete spectrum of H(d) have been the subject of intensive study above all due to the method of Born and Oppenheimer in the theory of molecules. This property can be described as follows [15]: every eigenvalue of H(d) (1.1) in the limit |d| → ∞ tends to some eigenvalue of H1 or H2 . If H1 and H2 have a common eigenvalue (say E0 ) then H(d) has a pair of eigenvalues E± (d), which are asymptotically degenerated, i.e. E± (d) → E0 for |d| → ∞. The splitting between asymptotically degenerated eigenvalues has been studied in [15, 25, 9]. Cluster properties of the discrete spectrum for long-range potentials which are not in R (in particular, Coulomb potentials, Vi (x) = |x|−1 ) were discussed in [26, 30, 15, 3, 8, 31]. It is interesting to note that for such potentials the limit |d| → ∞ is essentially equivalent to the semiclassical (large coupling constant) limit, since in the case of the Coulomb potential, say, H(d) is unitarily equivalent (up to a scaling factor |d|−2 ) to the Hamiltonian ! 1 1 , (1.4) + −∆ − |d| ˆ |x| |x − d| with dˆ = d|d|−1 . The study of the so-called critical double well problem was initiated by Ovchinnikov and Sigal [34] in order to investigate the Efimov effect [12] and by Klaus and Simon [24]. These authors considered the potential Vd (1.2) where Vi have compact support and the Hamiltonians Hi (1.3) have no negative bound states, but have zero energy resonances (see below for the precise definition of this notion). Under these assumptions Klaus and Simon [24] showed that for large |d| the operator H(d) has the only eigenvalue E(d) = −α2 /d2 + O(|d|−3 ) ,
(1.5)
where α is the (unique) real solution of the equation e−α = α. Later Høegh–Krohn and Mebkhout [17] found an infinite sequence of resonances En (d), n = 1, 2, . . ., tending to zero as |d| → ∞, such that En (d) = −γn2 /d2 + O(|d|−3 ). The constants γn do not depend on the potentials Vi and are the complex solutions of e−α = α. On the other hand Tamura [43] extended the results of [24] to the case of potentials with noncompact supports satisfying |Vi (x)| ≤ C(1 + |x|)−2− , > 0 (however with a lack of uniqueness, i.e. the proof in [43] does not prohibit the existence of other eigenvalues different from E(d) (1.5) and tending to zero as |d| → ∞). The low-energy scattering properties of the Hamiltonian H(d) (1.1) as |d| → ∞ were studied in [18], displaying the connection between (1.1) and Hamiltonians with point interaction. Let h0 be the Hamiltonian of the two-fixed-center point interaction, which is formally ˆ . h0 = −∆ + ν1 δ(x) + ν2 δ(x − d)
(1.6)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
629
The rigorous meaning of h0 and a detailed investigation of its properties can be found in [2]. Let Wr (r > 0) be the unitary scaling in L2 (R3 ) given by (Wr ψ)(x) = −1 = |d|−2 h , where r3/2 ψ(rx). Then W|d| H(d)W|d| h = −∆ + −2
V1
x
+ V2
! x − dˆ ,
with = |d|−1 . When → 0 for compactly supported Vi ∈ R, h converges to h0 (1.6) in the norm sense [2]. The limit is, however, very delicate since it depends very crucially on detailed properties of the spectral point zero for one center operators (1.3). One can expect that all zero-energy characteristics of H(d) in the limit |d| → ∞ are well described by the Hamiltonian h0 (1.6). Indeed, it is proven in [18] for compactly supported Vi ’s that Sd (E/d2 ) = s (E) , and s (E) = s0 (E) + O() ,
(1.7)
−1
again with = |d| . Here the operators Sd (E), s (E), and s0 (E) are the on-shell scattering matrices at energy E of H(d), h , and h0 respectively. A result of another type was proved by the present authors in [27]. Let S1 (E), S2 (E; d) and Sd (E) denote the on-shell scattering matrices of energy E for the pairs (H1 , H0 ), (H2 (d) = H0 + V2 (· − d), H0 ) and (H(d), H0 ) respectively. It was shown that both Sd (E) − S1 (E)S2 (E; d) and Sd (E) − S2 (E; d)S1 (E) tend to zero in Hilbert–Schmidt norm as |d| → ∞ uniformly in E on compact sets in (0, +∞). For Vi ∈ R ∩ L1 (R3 ) the result holds also in the trace norm sense. Since s0 (E) does not possess the cluster property, relation (1.7) shows that in general uniform convergence cannot hold on compact sets in [0, +∞). We turn to a description of the main results of the present article. For d ∈ R3 , let U (d) denote the unitary translation operator (U (d)f )(x) = f (x − d) ,
f ∈ L2 ,
(1.8)
such that U (d)V2 U (d)−1 = V2 (· − d) .
(1.9)
H2 (d) = U (d)H2 U (d)−1 = −∆ + V2 (· − d) .
(1.10)
Let We denote by R(z; d), R1 (z), R2 (z; d) and R0 (z) the resolvents of H(d), H1 , H2 (d) and H0 respectively and with the sign convention R0 (z) = (H0 − z)−1 , etc. We note that obviously R2 (z; d) = U (d)R2 (z)U (d)−1 , where R2 (z) is the resolvent of H2 . It is well known (see [41]) that for V1 and V2 in R ∩ L1 the differences R(z; d) − R0 (z), R1 (z) − R0 (z) and R2 (z; d) − R0 (z) are trace class for Im z 6= 0 and for all d ∈ R3 . Let Ei , i = 1, 2 be the set of z ∈ C for which the homogeneous equation φ = 1/2 1/2 Vi R0 (z)|Vi |1/2 φ has a solution in L2 (R3 ), where Vi (x) = |Vi (x)|1/2 sign Vi (x)
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1/2
(−)
such that Vi = Vi |Vi |1/2 . Note that Ei = Ei ∩ (−∞, 0) represents the negative (+) discrete spectrum of Hi . The sets Ei = Ei ∩ (0, +∞) are bounded closed sets of Lebesgue measure zero. More precisely, they are the unions of two subsets, belonging to the upper and lower lips of the cut (0, +∞) respectively. Let Π0 = C \ [0, +∞) and let Π0 be the closure of Π0 with the two lips of the cut added. We write E = E1 ∪ E2 . For real valued V1 and V2 in R ∩ L1 Krein’s spectral shift functions ξ1 (E), ξ2 (E; d), and ξ(E; d) for the pairs (H1 , H0 ), (H2 (d), H0 ) and (H(d), H0 ) respectively exist, satisfy (1 + |·|2 )−1 ξ ∈ L1 (R) and for each of them the following trace relations hold: Z φ0 (E)ξ1 (E)dE , tr φ(H1 ) − φ(H0 ) = R
tr φ(H2 (d)) − φ(H0 ) = tr φ(H(d)) − φ(H0 ) =
Z
R
Z
φ0 (E)ξ2 (E; d)dE ,
(1.11)
φ0 (E)ξ(E; d)dE
R
with φ being a function in a suitable class of continuously differentiable functions. For instance we may take φ to be in C0∞ (R), also φ(E) = e−tE , t > 0, φ(E) = (E − z)−1 , Im z 6= 0 are in this class. Note that ξ2 (E; d) is independent of d such that ξ2 (E; d) = ξ2 (E), where ξ2 (E) is spectral shift function for the pair (H2 , H0 ). The spectral shift functions are not uniquely defined by (1.11). They may be changed by an additive constant. We normalize the spectral shift functions by the conditions that ξ(E; d), ξ1 (E), and ξ2 (E) are identically zero for E below the spectra of H(d), H1 and H2 respectively (H(d) is bounded below uniformly in d by the Kato inequality). With this normalization for E < 0 one has ξi (E − 0) = −Ni (E) ,
ξ(E − 0; d) = −N (E; d) ,
where Ni (E) and N (E; d) are the counting functions for the Hamiltonians Hi and H(d) respectively. By the Birman–Krein theorem [4] the spectral shift function ξ(E) for the pair (H0 + V , H0 ) is related to the scattering matrix S(E) for fixed energy E: ξ(E) = −
1 log detS(E) , 2πi
E > 0.
(1.12)
The normalization of the specral shift function introduced above leads to the relation (see e.g. [32, 7]) ( ) √ Z E ξ(E) − 2 V (x)dx = 0 , (1.13) lim E→+∞ 4π which fixes the branch of the logarithm in (1.12) uniquely. Theorem 1. Let V1 and V2 be in L1 ∩ R. Let the spectral shift functions be normalized as above. Then
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
Z lim
|d|→∞
R
ξ(E; d) − ξ1 (E) − ξ2 (E) φ(E) dE = 0
631
(1.14)
for arbitrary φ ∈ C0∞ (R). This result extends results in [27], where we only established that ξ 0 (E; d) − − ξ20 (E) tends to zero in the sense of distributions as |d| → ∞. For potentials satisfying slightly more restrictive conditions we will provide a local information on the convergence of ξ(E; d)−ξ1 (E)−ξ2 (E) in the limit |d| → ∞. We impose the following additional condition on the potentials Vi : ξ10 (E)
Property A. The potential V satisfies (1 + | · |)2 V ∈ L1 (R3 ) ∩ R. There are constants R0 > 0 and C > 0 such that |V (x)| ≤ C for all |x| ≥ R0 and |xkV (x)| → 0 for |x| → ∞. The assumption (1 + | · |)2 V ∈ L1 (R3 ) ∩ R implies that V is an Agmon potential, and therefore σsc (H) = ∅ [38]. The fact that V is bounded outside some ball and |xkV (x)| → 0 for |x| → ∞ guarantees that σp (H) ∩ (0, ∞) = ∅ [38]. Hence for Vi ’s having Property A the sets Ei satisfy Ei ∩ (0, ∞) = ∅. Let Ai (ω, ω 0 ; E), i = 1, 2; ω, ω 0 ∈ S2 , E ∈ R+ be the scattering amplitude for the Hamiltonian Hi . We recall that it can be expressed in terms of the integral kernel Si (ω, ω 0 ; E) of the on-shell S-matrix Si (E) (see e.g. [37]): 2πi Ai (ω, ω 0 ; E) = − √ (Si (ω, ω 0 ; E) − δ(ω − ω 0 )) . E We recall the notation dˆ = d|d|−1 ∈ S2 . Theorem 2. Let the potentials Vi , i = 1, 2 have Property A. Then the function ξ12 (E; d) = ξ(E; d) − ξ1 (E) − ξ2 (E)
(1.15)
for sufficiently large |d| is jointly continuous in d and E ∈ (0, +∞) and has the asymptotic representation √ √ cos(2 E|d|) ˆ + sin(2 E|d|) Re a(E; d) ˆ + o(|d|−2 ) , (1.16) Im a(E; d) ξ12 (E; d) = |d|2 |d|2 where
ˆ = − 1 A1 (d, ˆ −d; ˆ E)A2 (−d, ˆ d; ˆ E) . a(E; d) π
The error term o(|d|−2 ) is uniform in E on compact sets in (0, +∞). Remarks. (1) For C0∞ -potentials V1 and V2 the functions ξ(E; d), ξ1 (E) and ξ2 (E) are infinitely differentiable in E on (0, +∞) [39]. Moreover, one can show that for any E ∈ (0, +∞) ξ 0 (E; d) → ξ10 (E) + ξ20 (E) as |d| → ∞. The statement of Theorem 2 shows in particular that the second derivative ξ 00 (E; d) does not converge pointwise as |d| → ∞.
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(2) Under additional restrictions on the potentials (for instance, the Vi ’s have compact support) the error term in (1.16) can be replaced by O(|d|−3 ). (3) The formulas (1.15), (1.16) remain valid for the Hamiltonians with point interactions centered at x = 0 and x = d. The behaviour of ξ12 (E; d) for E = 0 as |d| → ∞ is more delicate and strongly depends on the properties of the spectral point zero for H1 and H2 . We recall that zero is said to be a resonance for H, if the differential equation Hu = 0 has a nontrivial solution u ∈ L2s (R3 ) for some s such that −3/2 ≤ s < −1/2 and u ∈ / L2 (R3 ). Here L2s (R3 ) = {u ∈ L2loc(R3 )|(1 + x2 )s/2 u ∈ L2 (R3 )}. A detailed discussion of zero energy resonances can be found in [21]. Following Jensen and Kato [21] we call the point E = 0 regular for H if it is neither an eigenvalue nor a resonance of H. If E = 0 is a resonance but not an eigenvalue, it is said to be an exceptional point of the first kind. We note that the last case is in no way pathological: even for a square well potential there are isolated values of the coupling constant for which E = 0 is an exceptional point of the first kind. Also one can construct examples of square wells for which E = 0 is an eigenvalue but not a resonance (see e.g. [38]). In this case E = 0 is said to be an exceptional point of the second kind . If E = 0 is both a resonance and an eigenvalue, it is said to be an exceptional point of the third kind . We note that if H has no negative eigenvalues but has a resonance at zero energy, then E = 0 cannot be an eigenvalue of H (see [44]). Let c0 (V ) denote the scattering length of the Hamiltonian H0 + V (see Sec. 5). Then we have Theorem 3. Let the potentials V1 and V2 satisfy Property A. (i) Let E = 0 be either a regular point or an exceptional point of the first kind for both H1 and H2 . Then ξ12 (0+; d) = 0 for all sufficiently large |d|. (ii) Let E = 0 be a regular point for one of the Hamiltonians Hi , i = 1, 2 (say H1 ) and an exceptional point of the first kind for H2 . Then ξ12 (0+; d) = −signc0 (V1 )/2 for all sufficiently large |d|. We note that our definition of the scattering length (5.2) agrees with that used by Kato and Jensen [21] and differs by a sign from the definition customary used in physical literature [1, 6]. Now we apply Theorem 3 to an analysis of the discrete spectrum of H(d). The value of the spectral shift function at zero energy gives information on the multiplicity of the non-positive point spectrum of the Hamiltonian. By the generalized Levinson theorem [32, 10, 11, 6] ξ(0+) equals minus the number of nonpositive eigenvalues counting their multiplicities minus 1/2, if zero is a resonance for H = H0 +V . Thus, we have: Corollary 4. Let the potentials V1 and V2 have Property A. Let E = 0 be either an exceptional point of the first kind for both H1 and H2 , or a regular point for one of the Hamiltonians Hi with positive scattering length and an exceptional point
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
633
of the first kind for the other one. Moreover, let the negative spectrum of both H1 and H2 be empty. Then for all sufficiently large |d| the operator H(d) has a unique (nonpositive) eigenvalue. Now we comment on the case when E = 0 is an exceptional point of the first kind for (say) H2 and the regular one for H1 . For the case of compactly supported potentials we can easily prove that E(d) satisfies the equation √
√ e−2 −E|d| −E = c0 (V1 ) + O(|d|−3 ) + O(−E) . |d|2
(1.17)
Without explicitly pointing out the connection with the scattering length the Eq. (1.17) was proved by Klaus and Simon [24] for the case when V1 ≤ 0 and H1 is subcritical (i.e. E = 0 is a regular point for H1 and the operator H0 + (1 + )V for all −1 ≤ < δ has no bound states). It is easy to show (see Appendix A below) that if V1 ≥ 0 and is not identically zero then the scattering length c0 (V1 ) < 0. If V1 ≤ 0 and the discrete spectrum of H0 + V1 is empty then c0 (V1 ) > 0. In this case Eq. (1.17) has a solution. For stronger attractive potentials V1 which bind several states the scattering length can be positive or negative (for rotationally symmetric potentials see the discussion in [33]). The existence of the solution to (1.17) is controlled by signc0 (V1 ) in agreement with the statement of Theorem 3. Neither Eq. (1.17) nor Theorem 3 give an information on the low-energy spectrum of H(d) if c0 (V1 ) = 0. To avoid a possible reader’s question whether this is possible or not, we show in the Appendix A how to construct the potentials V for which c0 (V ) = 0. Consider now potentials Vi satisfying the inequality |Vi (x)| ≤ C(1 + |x|)−5− for some C > 0 and > 0. Combining Corollary 4 with the results of Tamura [43] we can conclude that for the case of critical double well potentials (the negative spectra of both H1 and H2 are empty and E = 0 is an exceptional point of the first kind for both H1 and H2 ) the Hamiltonian H(d) has a unique negative eigenvalue with asymptotics for |d| → ∞ given by formula (1.5). We note that our methods also can be applied to study other characteristics of H(d) in the limit |d| → ∞. As an example we consider the scattering amplitude 2πi Ad (ω, ω 0 ; E) = − √ (Sd (ω, ω 0 ; E) − δ(ω − ω 0 )) , E where now Sd (ω, ω 0 ; E) (ω, ω 0 ∈ S2 , E ∈ R+ ) is the integral kernel of the on-shell S-matrix Sd (E) for the pair of Hamiltonians (H(d), H0 ). Let us note that for Vi ∈ R ∩ L1 and for all E ∈ R+ \ E the scattering amplitude is a bounded function of its arguments. Let S2 (E; d) be the on-shell S-matrix for the Hamiltonian H2 (d). By (1.10) we obviously have S2 (E; d) = U (d)S2 (E)U (d)−1 . Note that since U (d) commutes with H0 , it has a corresponding spectral decomposition inducing translations in the spaces of fixed energy of H0 , also denoted by U (d). Hence for the corresponding scattering amplitude A2 (ω, ω 0 ; E; d) we have the relation
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V. KOSTRYKIN and R. SCHRADER
√
A2 (ω, ω 0 ; E; d) = e−i
Ehω−ω 0 ,di
A2 (ω, ω 0 ; E) .
(1.18)
Property B. The potential V has Property A and satisfies (1+|·|)4 V ∈ L1 (R3 ). We will prove the following analogue of Theorem 2. Theorem 5. Let the Vi ’s have Property B. Then the function A12 (ω, ω 0 ; E; d) = Ad (ω, ω 0 ; E) − A1 (ω, ω 0 ; E) − A2 (ω, ω 0 ; E; d) for all ω, ω 0 ∈ S2 and for sufficiently large |d| is continuous with respect to d and has the asymptotic representation A12 (ω, ω 0 ; E; d) =
1 ˆ E; d)A1 (d, ˆ ω 0 ; E) + A1 (ω, −d; ˆ E)A2 (−d, ˆ ω 0 ; E; d)] + o(|d|−1 ) . (1.19) [A2 (ω, d; |d|
The error term o(|d|−1 ) is uniform in ω, ω 0 ∈ S2 and E on compact sets in R+ . Intuitively this result is clear (see Fig. 1). By construction of A12 to A12 , both scattering centers have to contribute. Therefore to leading order in |d|−1 , the parˆ if it first has hit center 1, in order to hit center 2 ticle has to move in direction d, (and in direction −dˆ if it first hits center 2 and then center 1). Below we will also give (besides a rigorous proof) a formal proof based on this geometric picture and the Born series for the scattering amplitude. The scattering amplitude A12 also exhibits strong oscillations in the limit |d| → ∞, since by means of (1.18) we can rewrite (1.19) in the form 0
A12 (ω, ω ; E, d) =
√ E|d| h
ei
|d|
√
e−i
√ ˆ Ehω 0 ,di
+ ei
1
i ˆ E)A2 (−d, ˆ ω 0 ; E) + o(|d|−1 ) . A1 (ω, −d; 2
u
SS SS SSd SoS SS SSu
ˆ E)A1 (d, ˆ ω 0 ; E) A2 (ω, d;
S SS
2
ω
ˆ Ehω,di
u
ω0
SS SwS−d SS S
ω0
Fig. 1. Dominant contributions to A12 .
SS u ω 1
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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We remark that the present results can be easily generalized to potentials which are the sum of n > 2 terms by considering the situation where all the separations of their respective centers tend to infinity. The paper is organized as follows. In Sec. 2 we prove several auxiliary lemmas. The Secs. 3, 4 and 5 are devoted to the proofs of Theorems 2, 3 and 5, respectively. All the proofs can be made much easier, when considering the case of compactly supported potentials. The major efforts are made to accomodate the case of potentials having Properties A and B. Throughout the paper we freely do not distinguish between the formal expressions of the form V α R0 (z)|V |1−α defined on D(V 1−α ) and their closures defined on the whole L2 (R3 ). All formal manipulations with the operators of this form can be easily justified. Here after we will use the notations Jp (p ≥ 1) for the trace ideals of p-summable compact operators. In particular, J1 denotes the ideal of trace class operators, and J2 stands for the set of all Hilbert–Schmidt operators. The corresponding norms are denoted by k · kJp . 2. Auxiliary Results Hereafter we assume that Vi ∈ R, i = 1, 2. We start with the formulation of some technical results from our preceeding paper [27] (Theorem 1.1 and Lemma 2.2), which we recall here for the reader’s convenience. Theorem T. Let V1 , V2 ∈ R. Then 1/2
(i) the operator K12 (z; d) = V1 R0 (z)|V2 |1/2 (· − d) tends to zero in Hilbert– Schmidt norm uniformly in z ∈ Π0 as |d| → ∞, (ii) there is a constant c0 > 0 such that the Hamiltonians H1 , H2 , and H(d) defined in the form sense are bounded below by the constant −c0 for all d ∈ R3 . Moreover, if V1 , V2 ∈ R ∩ L1 the operator R(z; d) − R1 (z) − R2 (z; d) + R0 (z)
(2.1)
tends to zero as |d| → ∞ in trace norm uniformly in z on compact subsets of Π0 \ E. We note that a result similar to (ii) for H0 -form compact potentials, for which the convergence of (2.1) to zero holds in the operator norm, was proved earlier by Klaus [25, Appendix]. Lemma 2.1. Let V1 , V2 ∈ R. Then for all z ∈ Π0 the operator K12 (z; d) is continuous in d ∈ R3 in Hilbert–Schmidt norm. Proof. We recall that the translation operator U (d) is strongly continuous in all Lp -spaces with 1 ≤ p < ∞. By means of simple limiting arguments and by the inequality 2/3
1/3
kV kR ≤ 31/2 (2π)1/3 kV kL2 kV kL1 ,
V ∈ L1 (R3 ) ∩ L2 (R3 )
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V. KOSTRYKIN and R. SCHRADER
(see [41]) one can easily prove that U (d) is also strongly continuous in the Rollnik norm, Z 1/2 |V (x)kV (y)| dxdy . kV kR = |x − y|2 We recall that with this norm R is a complete normed vector space. For arbitrary d1 and d2 we estimate 1/2
kK(z; d1 ) − K(z; d2 )kJ2 ≤ kK(z; d1 ) − Vd1 R0 (z)|Vd2 |1/2 kJ2 1/2
+ kVd1 R0 (z)|Vd2 |1/2 − K(z; d2 )kJ2 ≤ kVd1 kR k(|Vd1 |1/2 − |Vd2 |1/2 )2 kR 1/2
1/2
+ kVd2 kR k(Vd1 − Vd2 )2 kR .
(2.2)
Now we use the inequality (|a + b|1/2 − |a|1/2 )2 ≤ |b| ,
(2.3)
which is valid for all a, b ∈ R and which is a simple consequence of |a + b|1/2 ≤ |a|1/2 + |b|1/2 . Due to (2.3) we have (|Vd1 (x)|1/2 − |Vd2 (x)|1/2 )2 = (|Vd1 (x) + V2 (x − d2 ) − V2 (x − d1 )|1/2 − |Vd1 (x)|1/2 )2 ≤ |V2 (x − d2 ) − V2 (x − d1 )| . Therefore the r.h.s. of (2.2) can be bounded by (kVd1 kR + kVd2 kR )kV2 (x − d2 ) − V2 (x − d1 )kR . Since the shift operator U (d) is strongly continuous in Rollnik norm this completes the proof of the lemma. Now for potentials having Property A we provide a more detailed information on the decay of K12 (z; d) as |d| → ∞. For z ∈ Π0 we set √ ˆ = |V1 (x)|1/2 exp{−i zhx, di} ˆ , Φ1 (x; z, d) (2.4) √ ˆ = |V2 (x)|1/2 exp{−i zhx, di} ˆ . (2.5) Φ2 (x; z, d) ˆ are in L2 (R3 ). However for non-compactly supported The functions Φi (·; z, d) loc potentials they are generally not in L2 (R3 ). This is a main obstacle for extending the results of [24, 25] to the case of non-compactly supported potentials. For E ≥ 0 we define √ (±) ˆ = |Vi (x)|1/2 exp ∓ i Ehx, di ˆ , i = 1, 2, (2.6) Φi (x; E, d) ˆ ∈ ˆ is the inner product of x and dˆ = d|d|−1 in R3 . Obviously Φ(±) (·; E, d) where hx, di i 2 3 (±) ˆ defined by (2.6) are the limiting values L (R ), i = 1, 2. The functions Φ (·; E, d) of (2.4), (2.5) on the upper and lower lips of the cut [0, +∞) respectively.
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Lemma 2.2. Let the potentials Vi have Property A. Then there is a constant C > 0 such that kK12 (z; d)kJ2 ≤
C , |d|
kK21 (z; d)kJ2 ≤
C |d|
for all sufficiently large |d| and for all z ∈ Π0 . Moreover for all real E ≥ 0 the operators √
e±i E|d| (±) ˆ (±) (· − d; E, d), ˆ ·) , sign V1 Φ1 (·; E, d)(Φ |d|K12 (E ± i0; d) − 2 4π
(2.7)
√
e±i E|d| (∓) ˆ (∓) (·; E, d), ˆ ·) (2.8) sign V2 (· − d)Φ2 (· − d; E, d)(Φ |d|K21 (E ± i0; d) − 1 4π tend to zero in Hilbert–Schmidt norm as |d| → ∞. The convergence is uniform in E on compact sets in (0, +∞). Remark. For compactly supported potentials Vi ∈ R ∩ L1 the second claim of Lemma 2.2 can be extended to the complex plane. For instance, one can then easily prove that the operator √
|d|K12 (z; d) −
ei
z|d|
4π
ˆ 2 (· − d; z, d), ˆ ·) → 0 sign V1 Φ1 (·; z, d)(Φ
(2.9)
in the Hilbert–Schmidt norm uniformly in z on compact sets in Π0 . Moreover, there is a constant C > 0 such that
√
ei z|d| C
ˆ ˆ sign V1 Φ1 (·; z, d)(Φ2 (· − d; z, d), ·) ≤
|d|K12 (z; d) −
4π |d| J2
uniformly in d for all large |d| and in z on compact sets in Π0 . Formula (2.9) was used by Klaus and Simon [24, 25] to study the critical double well problem. Our approach heavily uses the formula Z
1 f (x)g(y) dxdy = 2 |x − y| 4π
Z b f (p)b g (p) dp , |p|
(2.10)
which is valid for all f, g ∈ R. Here the hat b denotes the Fourier transform, b = f(p)
Z
e−ihp,xi f (x) dx .
The proof of the formula (2.10) can be found in [41].
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V. KOSTRYKIN and R. SCHRADER
To prove Lemma 2.2 we use the following asymptotics: Lemma 2.3. Let f ∈ C 2 (R3 ) and | · |−1 ∂ α f ∈ L1 (R3 ) for all |α| ≤ 2 (α is multiindex). Then for all E ≥ 0 and all large d ∈ R3 one has Z 4πf (0) f (p) −ihd,pi e dp = + o(|d|−2 ) , (2.11) (i) |p| |d|2 3 R Z (ii) R3
√ 2π 2 ±i√E|d| f (p)e−ihd,pi ˆ + o(|d|−1 ) , dp = e f (∓ E d) 2 p − E ∓ i0 |d|
(2.12)
where the error term is uniform in E on compact sets in (0, ∞). Remark. Here (x ∓ i0)−1 is understood in the sense of distributions, such that (x ∓ i0)−1 = v.p.
1 ± iπδ(x) . x
Since the distributions (x ∓ i0)−1 have order 1 they can be extended to linear continuous functionals on C k (R) with k ≥ 1. Proof. The claim (i) follows from the results of [40]. Asymptotics (2.12) is also well known. Its proof for analytic f is quite elementary. However, since we could not find a proof for the case f ∈ C 2 (R3 ) in the literature, we give it in Appendix B. Proof of Lemma 2.2. We consider only the operator K12 (z; d), since K21 (z; d) can be considered similarly. By means of the formula (2.10) one can easily show that Z |d|2 |V1 (x)kV2 (y − d)| dxdy |d|2 kK12 (z; d)k2J2 ≤ 2 (4π) |x − y|2 =
|d|2 (4π)3
Z d d |V1 |(p)|V2 |(p)e−ihd,pi dp . |p|
2 3 α d d d The Assumption A guarantees that |V i |(p) ∈ C (R ) and | · |∂ (|V1 |(p)|V2 |(p)) ∈ 1 3 L (R ). Applying Lemma 2.3 (i) we find that
|d|2 (4π)3
Z d d |V1 |(p)|V2 |(p)e−ihd,pi dp |p|
=
1 d d |V1 |(0)|V2 |(0) + o(1) (4π)2
=
1 kV1 kL1 kV2 kL1 + o(1) , (4π)2
(2.13)
which proves the first part of the claim. Now let z = E + i0. To prove the second part of the claim we consider the squared Hilbert–Schmidt norm of the operator (2.7), which is given by
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Z 1 |V1 (x)kV2 (y − d)| |V1 (x)kV2 (y − d)|dxdy dxdy + |x − y|2 (4π)2 Z |d| −i√E|d| |V1 (x)kV2 (y − d)| i√E|x−y| i√Ehx−y+d,di ˆ e − e e dxdy (4π)2 |x − y| Z |d| i√E|d| |V1 (x)kV2 (y − d)| −i√E|x−y| −i√Ehx−y+d,di ˆ e − e e dxdy . (2.14) (4π)2 |x − y|
|d|2 (4π)2
Z
We have already shown that the first term coincides asymptotically with the second one. Now we prove that for all real fi ∈ R ∩ L1 (R3 ) and all real E ≥ 0 1 4π
Z
f1 (x)f2 (y) ±i√E|x−y| e dxdy = |x − y|
Z
fb1 (p)fb2 (p) dp . p2 − E ∓ i0
(2.15)
First we note that the integral on the l.h.s. of (2.15) is well defined. Indeed, Z |f1 (x)kf2 (x)| dxdy ≤ k |f1 |1/2 R0 (0)|f2 |1/2 kJ2 k|f1 |1/2 kL2 k |f2 |1/2 kL2 4π|x − y| 1/2
1/2
= k |f1 |1/2 R0 (0)|f2 |1/2 kJ2 kf1 kL1 kf2 kL1 .
(2.16)
Now consider the integral 1 4π
Z
f1 (x)f2 (y) i√z|x−y| e dxdy |x − y|
(2.17)
for z ∈ Π0 . First let us suppose that fi ∈ L2 ∩ L1 . Then by the convolution formula (see e.g. [41]) Z Z 1 fb(p)b h(p)b g (p)dp , f (x)h(x − y)g(y)dxdy = (2π)3 where f, h ∈ L2 , g ∈ L1 we have that (2.17) equals Z b f1 (p)fb2 (p) dp . p2 − z Since the integrand in (2.17) is dominated by |f1 (x)kf2 (x)kx−y|−1 , by the Lebesgue dominated convergence theorem the limit Imz → ±0 exists. Thus (2.15) is proven for f ∈ L2 ∩L1 . A simple limiting procedure and (2.16) proves (2.15) for fi ∈ R∩L1 . Consider now the third and fourth terms of (2.14). By (2.15) we have Z √ |d| ∓i√E|d| |V1 (x)kV2 (y − d)| ±i√E|x−y| ±i√Ehx,di ˆ ∓i Ehy−d,di ˆ e e e e dxdy (4π)2 |x − y| =
|d| 1 ∓i√E|d| e (4π)2 2π 2
√ √ Z d d ˆ |V ˆ −ihd,pi |V1 |(p ± E d) E d)e 2 |(p ± dp . 2 p − E ∓ i0
(2.18)
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V. KOSTRYKIN and R. SCHRADER
Applying Lemma 2.3 (ii) we see that the r.h.s. of (2.18) has the asymptotics 2 −1 d d kV1 kL1 kV2 kL1 + o(1) , (4π 2 )−1 |V 1 |(0)|V2 |(0) + o(1) = (4π )
where the error term is uniform in E on compact sets in R+ . Summing up all contributions we obtain the claim of the lemma. The case z = E − i0 can be considered in exactly the same way. To proceed further we need the following technical Lemma 2.4. Let the Vi ’s have Property A. Then for every α ∈ (0, 1) both integrals Z |V1 (x)|α |V2 (x − d)|1−α |V1 (y)| dxdy |x − y|2 and
Z
|V1 (x)|α |V2 (x − d)|1−α |V2 (y − d)| dxdy |x − y|2
are o(|d|−2 ) as |d| → ∞. Proof. First we note that Vi ∈ R, i = 1, 2 implies |V1 |α |V2 |1−α ∈ R for any α ∈ (0, 1) (see [27]). Moreover by H¨older inequality k(1 + | · |)2 |V1 |α |V2 |1−α (· − d)kL1 ≤ k(1 + | · |)2α |V1 |α kL1/α · k(1 + | · |)2(1−α) |V2 |1−α (· − d)kL1/(1−α) 1−α 2 = k(1 + | · |)2 |V1 | kα L1 · k(1 + | · |) |V2 |(· − d)kL1 .
Therefore if the potentials Vi have Property A, then |V1 |α |V2 |1−α (· − d) has Property A also for all α ∈ (0, 1) and all d ∈ R3 . By means of formula (2.10) we obtain |d|2 4π
Z
= |d|2 4π
|V1 (x)|α |V2 (x − d)|1−α |V1 (y)| dxdy |x − y|2 |d|2 (4π)2
Z
d F1 (p; d)|V 1 |(p) dp , |p|
(2.19)
Z
|V1 (x)|α |V2 (x − d)|1−α |V2 (y − d)| dxdy |x − y|2 Z |d|2 |V1 (x + d)|α |V2 (x)|1−α |V2 (y)| = dxdy 4π |x − y|2
=
|d|2 (4π)2
Z
d F2 (p; d)|V 2 |(p) dp , |p|
(2.20)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
641
where Z F1 (p; d) = =e
e−ihp,xi |V1 (x)|α |V2 (x − d)|1−α dx
−ihp,di
e−ihp,xi |V1 (x + d)|α |V2 (x)|1−α dx , (2.21)
Z F2 (p; d) =
Z
e
−ihp,xi
Z
|V1 (x + d)| |V2 (x)| α
1−α
dx
e−ihp,xi |V1 (x)|α |V2 (x − d)|1−α dx .
= eihp,di
Consider first the r.h.s. of (2.19). With the help of Lemma 2.3 we get that in the limit |d| → ∞ |d|2 (4π)2
Z
d 1 d F1 (p; d)|V 1 |(p) dp = |V1 |(0) |p| 4π
Z |V1 (x + d)|α |V2 (x)|1−α dx + o(1) .
In the case α = 1/2 by Riemann–Lebesgue lemma we have Z Z \ \ 1/2 (p)|V 1/2 (p) = o(1) . |V1 (x + d)|1/2 |V2 (x)|1/2 dx = e−ihd,pi |V 1| 2|
(2.22)
Consider now the general case α ∈ (0, 1). Without loss of generality we can assume that α < 1/2. Then by H¨ older inequality Z |V1 (x + d)|α |V2 (x)|1−α dx Z =
|V1 (x + d)|α |V2 (x)|α |V2 (x)|1−2α dx Z
≤
|V1 (x + d)|
1/2
|V2 (x)|
1/2
2α Z 1−2α |V2 (x)|dx dx ,
which is o(1) by (2.22). The r.h.s. of (2.20) can be considered similarly.
3. Proof of Theorem 1 As already mentioned in the introduction the spectra of H(d), H1 and H2 are bounded below by a common constant [27]. Pick a real positive number c0 such that −c0 < min{inf σ(H(d)), inf σ(H1 ), inf σ(H2 )}. Then for Vi ∈ R∩L1 the differences Vd = R(−c0 ; d) − R0 (−c0 ) , Vi = Ri (−c0 ) − R0 (−c0 ) , V2 (d) = R2 (−c0 ; d) − R0 (−c0 ) with V2 (d = 0) = V2 are trace class for all d ∈ R3 .
i = 1, 2,
(3.1)
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V. KOSTRYKIN and R. SCHRADER
By Krein’s theorem [28] (see also [5]) for self-adjoint operators A2 and A1 such that A2 − A1 is trace class the spectral shift function ξ(λ; A2 , A1 ) exists and is given by ξ(λ; A2 , A1 ) = π −1 lim arg det(I + (A2 − A1 )(A1 − λ − i)−1 ) . (3.2) →+0
For all z ∈ C with Imz 6= 0 it has the property Z ξ(λ; A2 , A1 ) dλ = log det(I + (A2 − A1 )(A1 − z)−1 ) . λ−z R The branch of the logarithm is uniquely fixed by the condition log det(I + (A2 − A1 )(A1 − z)−1 ) → 0 , when Imz → ∞. We have Z R
ξ(λ; A2 , A1 )dλ = tr(A2 − A1 ) , (3.3)
Z R
|ξ(λ; A2 , A1 )|dλ = kA2 − A1 kJ1 .
In particular this guarantees the existence of the spectral shift functions ˜ d) = ξ(λ; R(−c0 ; d), R0 (−c0 )) , ξ(λ; ξ˜1 (λ) = ξ(λ; R1 (−c0 ), R0 (−c0 )) , ξ˜2 (λ; d) = ξ(λ; R2 (−c0 ; d), R0 (−c0 )) . Note that ξ˜2 (λ; d) = ξ˜2 (λ) = ξ(λ; R2 (−c0 ), R0 (−c0 )) by the unitarity of the translations (1.8). By the invariance principle (see e.g. [5, 22]) we can define the spectral shift functions ξ(E; d), ξ1 (E) and ξ2 (E; d) for the pairs (H(d), H0 ), (H1 , H0 ), and (H2 (d), H0 ) by ˜ ξ(E; d) = −ξ((E + c0 )−1 ; d) , ξ1 (E) = −ξ˜1 ((E + c0 )−1 ) ,
(3.4)
ξ2 (E; d) = −ξ˜2 ((E + c0 )−1 ; d) . Obviously, ξ(E; d) = ξ1 (E) = ξ2 (E; d) = 0 for any E ≤ −c0 and for all d ∈ R3 . Remark. Note that from (3.3) it follows that if A2 ≥ A1 then ξ(λ; A2 , A1 ) ≥ 0 for all λ ∈ R. Therefore by the monotonicity of the resolvent and by the invariance principle for arbitrary real W1 and W2 in R ∩ L1 satisfying W1 ≥ W2 pointwise we have ξ(E; H0 + W1 , H0 ) ≥ ξ(E; H0 + W2 , H0 ) for all E ∈ R. This yields an elementary proof of Kato’s monotonicity theorem [23] (see also [13]).
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Lemma 3.1. The limit relation Z ˜ Z ˜ ξ(λ; d) ξ1 (λ) + ξ˜2 (λ) dλ = dλ lim λ−z |d|→∞ R λ − z R holds for every z ∈ C with Imz 6= 0. Proof. Using the relation between the spectral shift function and the perturbational determinant [5] we have Z ˜ ξ(λ; d) dλ = log det(I + Vd r0 (z)) , R λ−z Z R
ξ˜i (λ) dλ = log det(I + Vi r0 (z)) , λ−z
(3.5) i = 1, 2,
where Vd and Vi , i = 1, 2 are defined by (3.1), and r0 (z) = [R0 (−c0 ) − z]−1 . In the representations (3.5) the branch of the logarithm is fixed uniquely by the conditions log det(I + Vd r0 (z)) → 0 , log det(I + Vi r0 (z)) → 0 ,
(3.6) i = 1, 2,
when Imz → ∞. Since ξ2 (λ) = ξ2 (λ; d) it suffices to prove that o n lim log det (I + Vd r0 (z))(I + V2 (d)r0 (z))−1 (I + V1 r0 (z))−1 = 0
(3.7)
for all nonreal z. If we now show that o n det (I + Vd r0 (z))(I + V2 (d)r0 (z))−1 (I + V1 r0 (z))−1 → 1
(3.8)
|d|→∞
as |d| → ∞ and that the convergence is uniform in Imz ∈ [, +∞) (or Imz ∈ (−∞, −]) for all > 0, then the conditions (3.6) guarantee (3.7). We note that the operators I + V2 (d)r0 (z) and I + V1 r0 (z) are invertible for nonreal z. The operator norms of (I + V1 r0 (z))−1 and (I + V2 (d)r0 (z))−1 can be bounded uniformly in Imz ∈ [, +∞) (Imz ∈ (−∞, −]) for all > 0. Therefore relation (3.8) will be proved once we show that kI + Vd r0 (z) − (I + V1 r0 (z))(I + V2 (d)r0 (z))kJ1 → 0 when |d| → ∞ uniformly in z ∈ Π0 . By Theorem T we have that Vd −V1 −V2 (d) → 0 in trace norm. Thus, it suffices to prove that lim kV1 r0 (z)V2 (d)kJ1 = 0 .
|d|→∞
(3.9)
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To establish (3.9) we note that by the second resolvent identity and due to the obvious relation zr0 (z)R0 (−c0 ) = −R0 (z −1 − c0 ) , we have V1 r0 (z)V2 (d) = [R1 (−c0 ) − R0 (−c0 )]r0 (z)[R2 (−c0 ; d) − R0 (−c0 )] = R1 (−c0 )V1 R0 (−c0 )r0 (z)R0 (−c0 )V2 (· − d)R2 (−c0 ; d) = R1 (−c0 )V1 R0 (−c0 )V2 (· − d)R2 (−c0 ; d) + zR1 (−c0 )V1 r0 (z)R0 (−c0 )V2 (· − d)R2 (−c0 ; d) = R1 (−c0 )V1 R0 (−c0 )V2 (· − d)R2 (−c0 ; d) − R1 (−c0 )V1 R0 (z −1 − c0 )V2 (· − d)R2 (−c0 ; d) .
(3.10)
Consider the first term on the r.h.s. of (3.10). We decompose R1 (−c0 )V1 R0 (−c0 )V2 (· − d)R2 (−c0 ; d) 1/2
= R1 (−c0 )|V1 |1/2 V1
1/2
R0 (−c0 )|V2 |1/2 (· − d)V2
(· − d)R2 (−c0 ; d) . 1/2
Note that the Hilbert–Schmidt norms of R1 (−c0 )|V1 |1/2 and V2 are uniformly bounded in d. Moreover, (i) of Theorem T yields 1/2
k V1
(· − d)R2 (−c0 ; d)
R0 (−c0 )|V2 |1/2 (· − d)kJ2 → 0
when |d| → ∞. The second term on the r.h.s. of (3.10) can be discussed in a similar way. In this case Theorem T says that the last term tends to zero in Hilbert– Schmidt norm as |d| → ∞ uniformly in z ∈ Π0 , thus proving (3.9), and by the previous remark also Lemma 3.1. Now we use the continuity of the Stieltjes transform (see e.g. [35, Appendix A]): Lemma 3.2. Let the sequence of measurable functions µn (t) on R (n ∈ N) satisfy the following properties: Z |µn (t)| dt < ∞ , R 1 + |t| (3.11) Z |µn (t)| dt = 0 . lim sup c→∞ n≥1 |t|≥c |t| R (t)dt If the sequence µnt−z converges to a function f (z) for every nonreal z, then there exists a measurable function µ(t) such that µn (t) → µ(t) in the sense of distributions and f is the Stieltjes transform of µ, i.e. Z µ(t)dt . f (z) = t−z
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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˜ d). Obviously, We show that the condition (3.11) is satisfied for ξ(λ; Z |t|≥c
˜ d)| 1 |ξ(t; dt ≤ |t| c
Z |t|≥c
˜ d)|dt ≤ |ξ(t;
1 kVd kJ1 . c
The norm kVd kJ1 is uniformly bounded in d since kR(−c0 ; d) − R0 (−c0 )kJ1 ≤ kR(−c0 ; d) − R1 (−c0 ) − R2 (−c0 ; d) + R(−c0 )kJ1 + kR1 (−c0 ) − R0 (−c0 )kJ1 + kR2 (−c0 ) − R0 (−c0 )kJ1 ≤ sup {kR(−c0 ; d) − R1 (−c0 ) − R2 (−c0 ; d) + R(−c0 )kJ1 } d∈R3
+ kR1 (−c0 ) − R0 (−c0 )kJ1 + kR2 (−c0 ) − R0 (−c0 )kJ1 . By Theorem T the norm kR(−c0 ; d) − R1 (−c0 ) − R2 (−c0 ; d) + R0 (−c0 )kJ1 can be bounded by a constant which is independent of d. Now, it follows from Lemmas 3.1 and 3.2 that Z ˜ d) − ξ˜1 (λ) − ξ˜2 (λ) ψ(λ)dλ = 0 lim ξ(λ; |d|→∞
R
for all ψ ∈ C0∞ (R). Using (3.4) we get Z (ξ(E; d) − ξ1 (E) − ξ2 (E))ψ((E + c0 )−1 ) lim |d|→∞
R
dE =0 (E + c0 )2
for arbitrary ψ ∈ C0∞ (R). Let us consider functions ψ ∈ C0∞ (R) with supp ψ ⊂ (0, +∞). Then obviously ψ((E + c0 )−1 ) (3.12) φ(E) = (E + c0 )2 is infinitely differentiable and has compact support lying in (−c0 , +∞). Conversely, an arbitrary C0∞ -function φ with suppφ ⊂ (−c0 , +∞) can be represented in the form (3.12) with ψ ∈ C0∞ (R). Since c0 can be taken arbitrary large this proves Theorem 1. 4. Proof of Theorem 2 In the proof of Theorem 2 we will use the representation of the spectral shift function for the pair of Hamiltonians H = H0 + V and H0 in terms of regularized Fredholm determinants. We recall that for an arbitrary Hilbert–Schmidt operator A the regularized Fredholm determinant det2 (I + A) is defined as the product Q −λj (A) , where the λj (A) are the eigenvalues of A. j (1 + λj (A))e
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Lemma 4.1. For V ∈ R ∩ L1 the spectral shift for the pair (H = H0 + V, H0 ) function can be represented in the form ξ(E; H, H0 )
" ( )# √ Z i E 1 det2 (I + V 1/2 R0 (E + i0)|V |1/2 ) log exp θ(E) V dx , (4.1) = 2πi 2π R det2 (I + V 1/2 R0 (E − i0)|V |1/2 )
where θ(E) is the Heaviside unit step function, θ(t) = 1, t > 0, and θ(t) = 0 otherwise. The branch of the logarithm is chosen so that lim|Imz|→∞ log det2 (I + V 1/2 R0 (z)|V |1/2 ) = 0. This formula has previously appeared in [32, 14, 7]. Since its proof for V ∈ R ∩ L1 still seems to be unpublished, for the reader’s convenience it will be given in Appendix C. Remark. We note that the operators V 1/2 R0 (E±i0)|V |1/2 are continuous in the Hilbert–Schmidt norm in E ∈ R+ \E [37]. Since the determinant det2 A is continuous with respect to A (see e.g. [42]), the functions det2 (I + V 1/2 R0 (E ± i0)|V |1/2 ) are both continuous in E ∈ R+ \ E. Since the operators I + V 1/2 R0 (E ± i0)|V |1/2 are invertible for all E ∈ R+ \ E and due to (4.1) the spectral shift function is continuous in E ∈ R+ \ E. For the potentials having Property A the intersection E ∩ (0, +∞) = ∅. Therefore in this case ξ(E; H, H0 ) is continuous on (0, +∞). Let 1/2
Ki (z) = Vi K2 (z; d) =
R0 (z)|Vi |1/2 ,
1/2 V2 (·
i = 1, 2,
− d)R0 (z)|V2 |1/2 (· − d) ,
1/2
R0 (z)|V2 |1/2 (· − d) ,
1/2
(· − d)R0 (z)|V1 |1/2 ,
1/2
R0 (z)|Vd |1/2 .
K12 (z; d) = V1 K21 (z; d) = V2
K(z; d) = Vd
We recall that for arbitrary Hilbert–Schmidt operators A, B, C the following identity holds (see e.g. [42]) det2 ((I + A)(I + B)(I + C)) = det2 (I + A)det2 (I + B)det2 (I + C) · exp{−tr[AB + AC + BC + ABC]} . (4.2) We can write I + K(z; d) = (I + K1 (z))(I + K2 (z; d))(I + L(z; d)) ,
(4.3)
with L(z; d) = (I + K2 (z; d))−1 (I + K1 (z))−1 · (K(z; d) − K1 (z) − K2 (z; d) − K1 (z)K2 (z; d)) .
(4.4)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Applying the formula (4.2) to (4.3) we get det2 (I + K(z; d)) = det2 (I + K1 (z)) · det2 (I + K2 (z; d))det2 (I + L(z; d))e−t(z;d) , where t(z; d) = trK1 (z)K2 (z; d) + trK1 (z)L(z; d) + trK2 (z; d)L(z; d) + trK1 (z)K2 (z; d)L(z; d) .
(4.5)
Now by means of Lemma 4.1 we get the representation ξ(E; d) = ξ1 (E) + ξ2 (E) + ξ12 (E; d) , where ξ12 (E; d) =
1 det2 (I + L(E + i0; d)) log exp{−t(E + i0; d) + t(E − i0; d)} . (4.6) 2πi det2 (I + L(E − i0; d))
Due to (1.13) the branch of the logarithm can be fixed uniquely by the condition lim ξ12 (E; d) = 0 .
E→+∞
By Theorem T L(z; d) tends to zero in Hilbert–Schmidt norm uniformly in z on compact sets in Π0 when |d| → ∞. Therefore the operator I + L(E ± i0; d) is invertible for all sufficiently large |d|. From Lemma 2.1 it follows that the operator K2 (z; d) is J2 -continuous in d (special case V1 = 0). Thus the operator L(z; d) and the function t(z; d) are continuous in d. By the continuity of the determinant det2 , ξ12 (E; d) is continuous in d for all sufficiently large |d| and fixed E. Combining these arguments with the remark after Lemma 4.1 we can easily prove the joint continuity of ξ12 (E; d) in E and d. As an easy consequence of Lemma 2.4 we get: Lemma 4.2. Let the potentials Vi have Property A. Then for all z ∈ Π0 kK1 (z)K2 (z; d)kJ1 = o(|d|−2 ) . as |d| → ∞ Proof. It is easy to see that 1/2
kK1 (z)K2 (z; d)kJ1 ≤ kV1
R0 (z)|V1 |1/4 |V2 |1/4 (· − d)kJ2
· k |V1 |1/4 |V2 |1/4 (· − d)R0 (z)|V2 |1/2 (· − d)kJ2 ≤
1 (4π)2 ·
Z
1 (4π)2
Z
|V1 (x)kV1 (y)|1/2 |V2 (y − d)|1/2 dxdy |x − y|2
1/2
|V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy |x − y|2
which is o(|d|−2 ) due to Lemma 2.4.
1/2 ,
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e d) as We define the operator K(z; e d) = K(z; d) − K1 (z) − K2 (z; d) − K12 (z; d) − K21 (z; d) . K(z; e d)kJ2 tends Lemma 4.3. Let the potentials Vi have Property A. Then |d| kK(z; to zero as |d| → ∞ uniformly in z on compact subsets of Π0 . e d) is identically Remark. For compactly supported potentials the operator K(z; zero for sufficiently large |d|. Proof. Obviously,
1/2 1/2 1/2 e d)kJ2 ≤ kK(z;
Vd − V1 − V2 (· − d) R0 (z)|Vd |1/2
J2
1/2
+ V1 R0 (z) |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)
J2
1/2
+ V2 (· − d)R0 (z) |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)
J2
. (4.7)
Due to Lemma 2.3 of [27]
1/2
Vd
1/2
(x) − V1
1/2
(x) − V2
2 (x − d) ≤ 4|V1 (x)|1/2 |V2 (x − d)|1/2 .
(4.8)
Also Vi ∈ R ∩ L1 implies |V1 |1/2 |V2 |1/2 (· − d) ∈ R ∩ L1 . Let us consider the first term on the r.h.s. of (4.7). The two other terms can be discussed in a completely similar way. It is easy to see that
2
1/2
1/2 1/2 |d|2 Vd − V1 − V2 (· − d) R0 (z)|Vd |1/2
J2
≤
|d|2 4π 2
≤
|d|2 4π 2
Z
|V1 (x)|1/2 |V2 (x − d)|1/2 |Vd (y)| dxdy |x − y|2
Z
|V1 (x)|1/2 |V2 (x − d)|1/2 |V1 (y)| dxdy |x − y|2 Z |d|2 |V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy . + 2 4π |x − y|2
Due to Lemma 2.4 both these integrals are o(1), which completes the proof of the lemma. We can represent the operator L(z; d) (4.4) in the form L(z; d) = (I + K2 (z; d))−1 (I + K1 (z))−1 e d) , · [K12 (z; d) + K21 (z; d)] + L(z;
(4.9)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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where e d) = (I + K2 (z; d))−1 (I + K1 (z))−1 L(z; e d) − K1 (z)K2 (z; d) , · K(z; Due to Lemma 4.2 and Lemma 4.3 for all z ∈ C \ E and all large |d| the operator e d) can be estimated as L(z; e d)kJ2 = o(|d|−1 ) . kL(z; We estimate now the square of the operator L(z; d). Lemma 4.4. For all z ∈ C \ E and sufficiently large |d|, L2 (z; d) = (I + K1 (z))−1 K12 (z; d)(I + K2 (z))−1 K21 (z; d) + (I + K2 (z; d))−1 K21 (z; d)(I + K1 (z))−1 K12 (z; d) + o(|d|−2 ) , where the error term is understood in the sense of trace norm und uniform in z on compact sets in C \ E. Proof. Due to (4.9) and Lemma 4.3 for all z ∈ C \ E and all sufficiently large |d| we have L2 (z; d) = (I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) · (I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) + (I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) · (I + K2 (z; d))−1 (I + K1 (z))−1 K21 (z; d) + (I + K2 (z; d))−1 (I + K1 (z))−1 K21 (z; d) · (I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) + (I + K2 (z; d))−1 (I + K1 (z))−1 K21 (z; d) · (I + K2 (z; d))−1 (I + K1 (z))−1 K21 (z; d) + o(|d|−2 ) ,
(4.10)
where the error term o(|d|−2 ) is understood in the sense of trace norm. Consider the first term in the r.h.s. of (4.10) (the third term can be considered in exactly the same way). Let us estimate the operator |d|2 K12 (z; d)(I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) . For the case of compactly supported potentials this operator is identically zero for all large |d|. Obviously, for all z in compact sets in C \ E we have
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|d|2 kK12 (z; d)(I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d)kJ1 2 = |d|2 k(I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d)kJ1 2 ≤ C|d|2 kK12 (z; d)kJ1 1/2
≤ CkV1
1/4
R0 (z)|V1 |1/4 |V2 |1/4 (· − d)kJ2 kV1
|V2 |1/4 (· − d)R0 (z)|V2 |1/2 (· − d)kJ2 . (4.11)
for some C > 0. The r.h.s. of (4.11) can be bounded by C (4π)2 Z ·
Z
|V1 (x)kV1 (y)|1/2 |V2 (y − d)|1/2 dxdy |x − y|2
|V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy |x − y|2
1/2
1/2 ,
which is o(|d|−2 ) by Lemma 2.4. We turn to the discussion of the second term in (4.10) (the fourth term can be considered in exactly the same way). We show that |d|2 {(I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) · (I + K2 (z; d))−1 (I + K1 (z))−1 K21 (z; d) − (I + K1 (z))−1 K12 (z; d)(I + K2 (z; d))−1 K21 (z; d)}
(4.12)
tends to zero in trace norm as |d| → ∞. Again for the case of compactly supported potentials the expression (4.12) is identically zero for all large |d|. First we express the difference (4.12) in the following form |d|2 (I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) · (I + K2 (z; d))−1 {(I + K1 (z))−1 − I}K21 (z; d) + |d|2 {(I + K2 (z; d))−1 − I}(I + K1 (z))−1 K12 (z; d) · (I + K2 (z; d))−1 K21 (z; d) . We show that |d|2 k{(I + K1 (z))−1 − I}K21 (z; d)kJ1 → 0 as |d| → ∞. Clearly, (I + K1 (z))−1 − I = −(I + K1 (z))−1 K1 (z) . Thus, it suffices to prove that |d|2 kK1 (z)K21 (z; d)kJ1 → 0
(4.13)
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as |d| → ∞. To this end we use the estimate kK1 (z)K21 (z; d)kJ1 1/2
≤ kV1
R0 (z)|V1 |1/4 |V2 |1/4 (· − d)kJ2
1/4
· kV1 ≤
1 (4π)2
1/4
V2 Z
(· − d)R0 (z)|V1 |1/2 kJ2
|V1 (x)kV1 (y)|1/2 |V2 (y − d)|1/2 dxdy . |x − y|2
This integral is o(|d|−2 ) due to Lemma 2.4. Similarly we can show that |d|2 kK21 (z; d){(I + K2 (z; d))−1 − I}kJ1 → 0
(4.14)
as |d| → ∞. Consider the second term in (4.13). Obviously, |d|2 k{(I + K2 (z; d))−1 − I}(I + K1 (z))−1 · K12 (z; d)(I + K2 (z; d))−1 K21 (z; d)kJ1 = |d|2 kK21 (z; d){(I + K2 (z; d))−1 − I} · (I + K1 (z))−1 K12 (z; d)(I + K2 (z; d))−1 kJ1 . Due to (4.14) this tends to zero as |d| → ∞.
Consider now the function t(z; d) given by (4.5). Lemma 4.5. For all z ∈ C \ E and sufficiently large |d|, t(z; d) = o(|d|−2 ) uniformly in z on compact sets in C \ E. Remark. large |d|.
For Vi ’s with compact supports t(z; d) = 0 for all sufficiently
Proof. From (4.5) by Lemma 4.2 it follows that t(z; d) = trK1 (z)L(z; d) + trK2 (z; d)L(z; d) + o(|d|−2 ) . Consider trK1 (z)L(z; d). By (4.4) and again by Lemma 4.2 we have |trK1 (z)L(z; d)| = |trL(z; d)K1 (z)| = |tr[(I + K2 (z; d))−1 (I + K1 (z))−1 {K(z; d) − K1 (z)}K1 (z)]| + o(|d|−2 ) ≤ CkK2 (z; d)K1 (z)kJ1 + CkK12 (z; d)K1 (z)kJ1 e d)K1 (z)kJ1 + o(|d|−2 ) . + CkK21 (z; d)K1 (z)kJ1 + CkK(z;
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In the course of the proof of Lemma 4.4 we have already shown that kK1 (z)K21 (z; d)kJ1 = o(|d|−2 ) when d → 0. In exactly the same way we can show that kK12 (z; d)K1 (z)kJ1 is o(|d|−2 ) also. The norm kK2 (z; d)K1 (z)kJ1 is o(|d|−2 ) by Lemma 4.2. We consider e d)K1 (z)kJ1 . As in the course of the proof of Lemma 4.3 using (4.8) we now kK(z; can estimate e d)kJ1 kK1 (z)K(z;
1/2 1/2 1/2 ≤ K1 (z) Vd − V1 − V2 (· − d) R0 (z)|Vd |1/2
J1
1/2
+ V1 R0 (z) |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d) K1 (z)
J1
1/2
+ V2 R0 (z) |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d) K1 (z)
J1
1/2
≤ 2kV1
R0 (z)|V1 |3/8 |V2 |1/8 (· − d)kJ2 · kV1 |3/8 |V2 |1/8 (· − d)R0 (z)|Vd |1/2 kJ2
1/2
R0 (z)|V1 |3/8 |V2 |1/8 (· − d)kJ2 · kV1
1/2
(· − d)R0 (z)|V1 |3/8 |V2 |1/8 (· − d)kJ2
+ 2kV1 + 2kV2 3/8
· kV1
3/8
|V2 |1/8 (· − d)R0 (z)|V1 |1/2 kJ2
|V2 |1/8 (· − d)R0 (z)|V1 |1/2 kJ2 .
Due to Lemma 2.4 this is o(|d|−2 ). In exactly the same way one can estimate the remaining term trK2 (z; d)L(z; d), thus completing the proof of the lemma. To proceed further with the proof of Theorem 2 we use the obvious estimate log det2 (I + A) + 1 trA2 ≤ kA3 kJ1 , (4.15) 2 where A is an arbitrary Hilbert–Schmidt operator with operator norm satisfying kAk < 1/2. (This estimate easily follows from the definition of the modified Fredholm determinant.) We apply (4.15) to the operator L(E ± i0; d). Let us note that kL3 (E ± i0; d)kJ1 ≤ kL(E ± i0; d)k kL2 (E ± i0; d)kJ1 ≤ kL(E ± i0; d)k3J2 . From (4.9), Lemmas 2.2–2.4 and 4.3 it follows that kL(E ± i0; d)k3J2 ≤
C |d|3
for some C > 0. Therefore, it follows from (4.6) and Lemma 4.5 that ξ12 (E; d) = −
1 tr[L2 (E + i0; d) − L2 (E − i0; d)] + o(|d|−2 ) . 4πi
(4.16)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
653
Applying Lemmas 4.4 and 2.2 we get that √
e±2i E|d| |d| trL (E ± i0; d) − 2 (4π)2 (∓) ˆ (I + K1 (E ± i0))−1 sign V1 Φ(±) (·; E, d) ˆ · Φ1 (·; E, d), 1 2
2
(±) ˆ (I + K2 (E ± i0; d))−1 sign V2 (· − d)Φ(∓) (· − d; E, d) ˆ · Φ2 (· − d; E, d), 2 tends to zero when |d| → ∞. By translation invariance (±) ˆ (I + K2 (E ± i0; d))−1 sign V2 (· − d)Φ(∓) (· − d; E, d) ˆ Φ2 (· − d; E, d), 2 (±) ˆ (I + K2 (E ± i0))−1 sign V2 Φ(∓) (·; E, d) ˆ . = Φ2 (·; E, d), 2 Therefore to leading order in |d|−1 tr[L2 (E + i0; d) − L2 (E − i0; d)] √
2 e2i E|d| (−) ˆ (I + K1 (E + i0))−1 sign V1 Φ(+) (·; E, d) ˆ Φ (·; E, d), = 1 1 (4π)2 |d|2 (+) ˆ (I + K2 (E + i0))−1 sign V2 Φ(−) (·; E, d) ˆ · Φ2 (·; E, d), 2 √
2 e−2i E|d| (+) ˆ (I + K1 (E − i0))−1 sign V1 Φ(−) (·; E, d) ˆ Φ1 (·; E, d), − 1 2 2 (4π) |d| (−) ˆ (I + K2 (E − i0))−1 sign V2 Φ(+) (·; E, d) ˆ + o(|d|−2 ) . · Φ2 (·; E, d), 2 With the help of the well-known representation for the scattering amplitude (see e.g. [37, 1]) √ 0 1 i√Ehω,·i 1/2 e |Vi |1/2 , (I + Ki (E + i0))−1 Vi ei Ehω ,·i , (4.17) Ai (ω, ω 0 ; E) = − 4π the symmetry relation √ √ ˆ ˆ 1/2 |Vi |1/2 e−i Ehd,·i , (I + Ki (E − i0))−1 Vi ei Ehd,·i √ √ 1/2 ˆ ˆ , (I + Ki (E + i0))−1 Vi e−i Ehd,·i = |Vi |1/2 ei Ehd,·i and (2.6) one gets tr[L2 (E + i0; d) − L2 (E − i0; d)] √
e2i E|d| ˆ −d; ˆ E)A2 (−d, ˆ d; ˆ E) A1 (d, =2 |d|2 √
e−2i E|d| ˆ −d; ˆ E)A2 (d, ˆ −d; ˆ E) + o(|d|−2 ) , −2 A1 (d, |d|2 which together with (4.16) gives (1.16).
654
V. KOSTRYKIN and R. SCHRADER
5. Proof of Theorem 3 Let E = 0 be a regular point for both H1 and H2 . Hence the operators I +Ki (0), i = 1, 2 are invertible. Therefore all the arguments used to prove Theorem 2 can be repeated verbatim in the case E = 0, thus yielding ξ12 (0+; d) = where
1 ˆ + o(|d|−2 ) , Ima(0; d) |d|2
(5.1)
ˆ −d; ˆ 0)A2 (−d, ˆ d; ˆ 0) . ˆ = − 1 A1 (d, a(0; d) π
We recall that c0 (Vi ) := Ai (ω, ω 0 ; 0) = −
1 1/2 1/2 |Vi | , (I + Ki (0))−1 Vi , 4π
(5.2)
ˆ = 0. From is the scattering length, which is obviously real. Therefore Ima(0; d) (5.1) and the fact that ξ12 (0+; d) is integer or half-integer, it follows that for all sufficiently large |d| ξ12 (+0; d) = 0 . Now we consider the case where E = 0 is an exceptional point of the first kind at least for one of the operators H1 and H2 . It is well known (see e.g. [1]) that E = 0 is an exceptional point of the first kind for Hi iff the equation Ki (0)ϕi = −ϕi ,
(5.3)
has a unique solution ϕi ∈ L2 (R3 ) and (|Vi |1/2 , ϕi ) 6= 0. The corresponding eigenprojector we denote by Pi , i.e. Pi =
ϕi (ϕ˜i , ·) , (ϕ˜i , ϕi )
where ϕ˜i = (sign Vi )ϕi . Obviously ϕ˜i satisfies the equation Ki (0)∗ ϕ˜i = −ϕ˜i . It is / easy to see that (ϕ˜i , ϕi ) 6= 0 (see [2, pp. 21–22]). The function ψi = R0 (0)|Vi |1/2 ϕi ∈ L2 is called a zero energy resonance wave function and satisfies Hi ψi = 0 in the sense of distributions. For potentials having Property A we can expand the operators Ki (z) in the √ powers of z: √ i z 1/2 V (|Vi |1/2 , ·) + zNi + o(z) , Ki (z) = Ki (0) + (5.4) 4π i where Ni is a Hilbert–Schmidt operator with integral kernel Ni (x, y) = −
1 1/2 V (x)|x − ykVi |1/2 (y) . 8π i
The low-energy expansion obtained in [1] yields the operator relation 4π (I + Ki (z))−1 = √ Qi + Mi (z) , i z
(5.5)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
with
ϕi (ϕ˜i , ·) , |(|Vi |1/2 , ϕi )|2
Qi =
655
(5.6)
being a rank 1 operator and Mi (z) is a bounded operator in a neighborhood of z = 0. For z → 0 Mi (z) has a representation ϕi (ϕ˜i , Vi )(Ti∗ |Vi |1/2 , ·) Ti Vi (|Vi |1/2 , ϕi )(ϕ˜i , ·) − |(|Vi |1/2 , ϕi )|2 |(|Vi |1/2 , ϕi )|2 1/2
Mi (z) = Ti −
1/2
1/2
+
ϕi (|Vi |1/2 , Ti Vi )(ϕ˜i , ·) ϕi (ϕ˜i , Ni ϕi )(ϕ˜i , ·) + (4π)2 + o(1) , (5.7) 1/2 2 |(|Vi | , ϕi )| |(|Vi |1/2 , ϕi )|4
where 1/2
Ti = n − lim (I + Vi →0+
R0 (0)|Vi |1/2 + )−1 (I − Pi ) .
The error term in (5.7) is understood in the sense of the operator norm. Below we will use the formula 4π (I + Ki (z))Mi (z) = I − √ (I + Ki (z))Qi i z 1/2
= I − Vi
√ (|Vi |1/2 , ·)Qi + O( z) .
(5.8)
To prove this we multiply (5.5) by Ki (z) thus obtaining 4π Ki (z)(I + Ki (z))−1 = √ Ki (z)Qi + Ki (z)Mi (z) . i z
(5.9)
On the other hand Ki (z)(I + Ki (z))−1 = I − (I + Ki (z))−1 4π = − √ Qi + I − Mi (z) . i z
(5.10)
Comparing (5.9) and (5.10) proves the first part of (5.8). Now expanding Ki (z) √ around z = 0 in z and using Ki (0)Qi = −Qi we obtain (I + Ki (z))Qi =
√ i z 1/2 V (|Vi |1/2 , ·)Qi + O(z) , 4π i
thus proving (5.8). First let us consider the case when E = 0 is a regular point for H1 and an exceptional one of the first kind for H2 . In this case we represent the operator L(z; d) (4.4) in the form 4π L(z; d) = √ L(0) (z; d) + L(1) (z; d) , i z
656
V. KOSTRYKIN and R. SCHRADER
where L(0) (z; d) = Q2 (d)(I + K1 (z))−1 · (K12 (z; d) e d) − K1 (z)K2 (z; d)) , + K21 (z; d) + K(z; L(1) (z; d) = M2 (z; d)(I + K1 (z))−1 · (K12 (z; d) + K21 (z; d)
(5.11)
e d) − K1 (z)K2 (z; d)) . + K(z; Here Q2 (d) = U (d)Q2 U (d)−1 and M2 (z; d) = U (d)M2 (z)U (d)−1 . We note that the operator L(0) (z; d) has rank 1. Due to Theorem T for all sufficiently large |d| and all small z one has kL(1) (z; d)k < 1. Therefore I + L(1) (z; d) is then invertible. Hence, det2 (I + L(z; d))
4π (0) (1) −1 √ L (z; d)(I + L (z; d)) = det2 (I + L (z; d)) det2 I + i z 4π (0) (1) −1 (1) · exp − √ trL (z; d)(I + L (z; d)) L (z; d) i z 4π (0) (1) (1) −1 = det2 (I + L (z; d)) det I + √ L (z; d)(I + L (z; d)) i z 4π · exp − √ trL(0) (z; d) . i z (1)
Since for any rank 1 operator A, det(I + A) = 1 + trA, using (5.6) we can easily calculate 4π det I + √ L(0) (z; d)(I + L(1) (z; d))−1 i z 4π = 1 + √ trL(0) (z; d)(I + L(1) (z; d))−1 i z 4π e d) = 1 + √ tr Q2 (d)(I + K1 (z))−1 · (K12 (z; d) + K21 (z; d) + K(z; i z − K1 (z)K2 (z; d))(I + L(1) (z; d))−1 4π = 1 + √ (ϕ˜2 (· − d), (I + K1 (z))−1 · K12 (z; d) + K21 (z; d) i z e d) − K1 (z)K2 (z; d) · (I + L(1) (z; d))−1 ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2 . + K(z; Further we calculate t(z; d) defined in (4.5), 4π t(z; d) = √ t(0) (z; d) + t(1) (z; d) , i z
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
657
where t(0) (z; d) = trL(0) (z; d)(K1 (z) + K2 (z; d) + K1 (z)K2 (z; d)) = (ϕ˜2 (· − d), (I + K1 (z))−1 [K12 (z; d) + K21 (z; d) e d) − K1 (z)K2 (z; d)] · [K1 (z) + K2 (z; d) + K(z; + K1 (z)K2 (z; d)]ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2 , t(1) (z; d) = trL(1) (z; d)[K1 (z) + K2 (z; d) + K1 (z)K2 (z; d)] + trK1 (z)K2 (z; d) . Now we study the limit E → +0 of the expression 4π 4π ± √ trL(0) (E ± i0; d) ± √ t(0) (E ± i0; d) . i E i E
(5.12)
Simple calculations shows that (5.12) equals 4π ± √ (ϕ˜2 (· − d), (I + K1 (E ± i0))−1 [K12 (E ± i0; d) + K21 (E ± i0; d) i E e + K(E ± i0; d) − K1 (E ± i0)K2 (E ± i0; d)]ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2 4π ± √ (ϕ˜2 (· − d), (I + K1 (E ± i0))−1 [K12 (E ± i0; d) + K21 (E ± i0; d) i E e + K(E ± i0; d) − K1 (E ± i0)K2 (E ± i0; d] · [K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d)]ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2 . (5.13) Now we prove Lemma 5.1. Let the Vi ’s have Property A. Then for E → +0 1 √ [(K1 (E ± i0) + K2 (E ± i0; d) E + K1 (E ± i0)K2 (E ± i0; d))ϕ2 (· − d) + ϕ2 (· − d)] √ i 1/2 (I + K1 (0))V2 (· − d)(|V2 |1/2 , ϕ2 ) + O( E) =± 4π
(5.14)
in L2 -norm uniformly in d ∈ R3 .
in
Proof. We expand the operators K1 (E ± i0) and K2 (E ± i0; d) in Taylor series √ E at E = 0: √ i E 1/2 V (|V1 |1/2 , ·) + O(E) , (5.15) K1 (E ± i0) = K1 (0) ± 4π 1 √ i E 1/2 V (· − d)(|V2 |1/2 (· − d), ·) + O(E) , (5.16) K2 (E ± i0; d) = K2 (0; d) ± 4π 2
658
V. KOSTRYKIN and R. SCHRADER
where the error terms O(E) are understood in the sense of Hilbert–Schmidt norm. Equation (5.14) follows immediately from (5.15) and (5.16). Remark. The low energy expansion for the operator K12 (z; d) √ i z 1/2 V (|V2 |1/2 (· − d), ·) + zN12 (d) + o(z) K12 (z; d) = K12 (0; d) + 4π 1 is not uniform in d. Here N12 (d) is the Hilbert–Schmidt operator with integral kernel 1 1/2 N12 (x, y; d) = − V1 (x)|x − ykV2 |1/2 (y − d) . 8π It is easy to show that kN12 kJ2 increases linearly with d. For this reason the limit E → +0 of ξ12 (E; d) is not uniform in d. √ Now by Lemma 5.1 the r.h.s. of (5.13) equals C(d) + O( E) (independent of the sign in (5.13)), where C(d) is a d-dependent constant. Therefore 4π lim √ trL(0) (E + i0; d)− trL(0) (E − i0; d)+ t(0) (E + i0; d)− t(0) (E − i0; d) = 0 E→+0 i E for all fixed d ∈ R3 . Since the operator I + L(1) (0; d) is invertible for all sufficiently large |d|, due to (4.6) we have (" 1 4π lim log 1 + √ (ϕ˜2 (· − d), ξ12 (+0; d) = 2πi E→+0 i E e d) − K1 (0)K2 (0; d) (I + K1 (0))−1 K12 (0; d) + K21 (0; d) + K(0; # · (I + L(1) (0; d))−1 ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2 "
4π · 1 − √ (ϕ˜2 (· − d), (I + K1 (0))−1 i E e d) − K1 (0)K2 (0; d) · K12 (0; d) + K21 (0; d) + K(0; #) · (I + L(1) (0; d))−1 ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2
.
(5.17)
To proceed further with our calculations of ξ12 (+0; d) we prove the following: Lemma 5.2. For all sufficiently large |d| (ϕ˜2 (· − d), (I + K1 (0))−1 [K12 (0; d) + K21 (0; d) e d) − K1 (0)K2 (0; d)](I + L(1) (0; d))−1 ϕ2 (· − d)) + K(0; =
|(|V2 |1/2 , ϕ2 )|2 c0 (V1 ) + o(|d|−2 ) , 4π|d|2
where c0 (V1 ) is the scattering length (5.2).
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
659
Proof. First we show that k(I + K1 (0)∗ )−1 ϕ˜2 (· − d) − ϕ˜2 (· − d)kL2 = o(|d|−2 ) for sufficiently large |d|. To this end we use K2 (0; d)∗ ϕ˜2 (· − d) = −ϕ˜2 (· − d) and write (I + K1 (0)∗ )−1 ϕ˜2 (· − d) − ϕ˜2 (· − d) = −(I + K1 (0)∗ )−1 K1 (0)∗ ϕ˜2 (· − d) = (I + K1 (0)∗ )−1 K1 (0)∗ K2 (0; d)∗ ϕ˜2 (· − d) . Its L2 -norm can be bounded by CkK1 (0)K2 (0; d)kJ1 kϕ2 kL2 , which is o(|d|−2 ) by Lemma 4.2. Therefore it suffices to consider e d) − K1 (0)K2 (0; d) (ϕ˜2 (· − d), K12 (0; d) + K21 (0; d) + K(0; · (I + L(1) (0; d))−1 ϕ2 (· − d))
= ϕ˜2 (· − d), K12 (0; d)(I + L(1) (0; d))−1 ϕ2 (· − d) e d)(I + L(1) (0; d))−1 ϕ2 (· − d) + ϕ˜2 (· − d), K(0;
− ϕ˜2 (· − d), K1 (0)K2 (0; d)(I + L(1) (0; d))−1 ϕ2 (· − d) + ϕ˜2 (· − d), K21 (0; d)(I + L(1) (0; d))−1 ϕ2 (· − d) .
(5.18)
Here the first summand is o(|d|−2 ) since K12 (0; d)∗ ϕ˜2 (· − d) = −K12 (0; d)∗ K2 (0; d)∗ ϕ˜2 (· − d) 1/4
= −|V2 |1/2 (· − d)R0 (0)V1
1/4
V2
(· − d)
· |V1 |1/4 |V2 |1/4 (· − d)R0 (0)|V2 |1/2 (· − d)ϕ˜2 (· − d) . Its L2 -norm can be bounded by 1 (4π)2 Z ·
Z
|V2 (x − d)kV1 (y)|1/2 |V2 (y − d)|1/2 dxdy |x − y|2
|V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy |x − y|2
which indeed is o(|d|−2 ) by Lemma 2.4.
1/2
1/2 kϕ2 kL2 ,
660
V. KOSTRYKIN and R. SCHRADER
Consider the second term on the r.h.s. of (5.18). Since L(1) (0; d) → 0 in Hilbert– e d)ϕ2 (· − d)) only. Schmidt norm as |d| → ∞, it suffices to consider (ϕ˜2 (· − d), K(0; Mimicking the idea of the proof of Lemma 4.3 we can write e d)ϕ2 (· − d)) (ϕ˜2 (· − d), K(0; 1/2
= −(ϕ˜2 (· − d), K2 (0; d)(Vd
1/2
+ (ϕ˜2 (· − d), K2 (0; d)V1
1/2
− V1
1/2
− V2
(· − d))R0 (0)|Vd |1/2 ϕ2 (· − d))
R0 (0)(|Vd |1/2 − |V1 |1/2
− |V2 |1/2 (· − d)) · K2 (0; d)ϕ2 (· − d)) 1/2
− (ϕ˜2 (· − d), V2
(· − d)R0 (0)(|Vd |1/2 − |V1 |1/2
− |V2 |1/2 (· − d)) · K2 (0; d)ϕ2 (· − d)) .
(5.19)
With the help of inequality (4.8) the first term on the r.h.s. of (5.19) can be bounded by Z 1/2 2 |V2 (x − d)kV2 (y − d)|3/4 |V1 (y)|1/4 dxdy 4π 2 |x − y|2 Z ·
|V1 (x)|1/4 |V2 (x − d)|3/4 |Vd (y)| dxdy |x − y|2
1/2 kϕ2 k2L2 .
By Lemma 2.4 this is o(|d|−2 ). This completes the proof of that the first term on the r.h.s. of (5.19) is o(|d|−2 ). The two other terms can be treated analogously. Hence the second term on the r.h.s. of (5.18) is o(|d|−2 ). The third term on the r.h.s. of (5.18) is o(|d|−2 ) due to Lemma 4.2. Finally consider the fourth term on the r.h.s. of (5.18). We replace (I + L(1) (0; d))−1 by the first two terms of its Neumann series expansion. It follows from (5.11) and Lemma 2.2 that (I + L(1) (0; d))−1 = I − L(1) (0; d) + O(|d|−2 ) . Hence (ϕ˜2 (· − d), K21 (0; d)(I + L(1) (0; d))−1 ϕ2 (· − d)) = (ϕ˜2 (· − d), K21 (0; d)ϕ2 (· − d)) − (ϕ˜2 (· − d), K21 (0; d)L(1) (0; d)ϕ2 (· − d)) + O(|d|−3 ) . Obviously, |(ϕ˜2 (· − d), K21 (0; d)ϕ2 (· − d))| = |(ϕ˜2 (· − d), K21 (0; d)K2 (0; d)ϕ2 (· − d))| ≤
1 (4π)2 ·
|V2 (x − d)kV2 (y − d)|1/2 |V1 (y)|1/2 dxdy |x − y|2
|V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy |x − y|2
1/2
1/2 kϕk2L2 ,
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
661
which is o(|d|−2 ) due to Lemma 2.4. Now applying Lemma 2.2 we get − (ϕ˜2 (· − d), K21 (0; d)L(1) (0; d)ϕ2 (· − d)) =−
1 1/2 (ϕ˜2 , V2 )(|V1 |1/2 , L(1) (0; d)ϕ2 (· − d)) + o(|d|−2 ) . 4π|d|
(5.20)
Using the representation (5.11) we find that (|V1 |1/2 , (I+K1 (0))−1 K12 (0; d)ϕ2 (·−d)) describes the leading term of the asymptotics of (|V1 |1/2 , L(1) (0; d)ϕ2 (· − d)) as |d| → ∞ (here we omit the corresponding calculations). Applying Lemma 2.2 once more we obtain that (5.20) is given by −
1 1/2 1/2 (ϕ˜2 , V2 )(|V1 |1/2 , (I + K1 (0))−1 V1 )(|V2 |1/2 , ϕ2 ) + o(|d|−2 ) (4π)2 |d|2 =
|(|V2 |1/2 , ϕ2 )|2 c0 (V1 ) + o(|d|−2 ) , 4π|d|2
which proves the lemma. Now we apply Lemma 5.2 to (5.17). Then for sufficiently large |d| we have 1 lim log ξ12 (+0; d) = 2πi E→+0
1 · 1− √ i E
(
1 1+ √ i E
c0 (V1 ) −2 + o(|d| ) |d|2
−1 ) c0 (V1 ) −2 . + o(|d| ) |d|2
Noting that for E > 0 and c0 (V1 ) 6= 0 c0 (V1 ) √ sign c0 (V1 ) < 0 , arg 1 + i E we calculate the limit E → +0 thus obtaining that for all sufficiently large |d| ξ12 (+0; d) =
1 1 log exp{−iπsign c0 (V1 )} = − sign c0 (V1 ) . 2πi 2
Thus the claim (ii) of Theorem 3 is proved. Next let us consider the case when E = 0 is an exceptional point of the first kind for both H1 and H2 . In this case we decompose the operator L(z; d) into its singular and regular parts, L(z; d) = L(0) (z; d) + L(1) (z; d) ,
662
V. KOSTRYKIN and R. SCHRADER
L(0) (z; d) = −
(4π)2 Q2 (d)Q1 z
e d) − K1 (z)K2 (z; d) · K12 (z; d) + K21 (z; d) + K(z; 4π + √ Q2 (d)M1 (z) i z e d) − K1 (z)K2 (z; d) · K12 (z; d) + K21 (z; d) + K(z; 4π + √ M2 (z; d)Q1 i z e d) − K1 (z)K2 (z; d) , · K12 (z; d) + K21 (z; d) + K(z;
(5.21)
L(1) (z; d) = M2 (z; d)M1 (z) e d) − K1 (z)K2 (z; d) . · K12 (z; d) + K21 (z; d) + K(z; We note that the operator L(0) (z; d) has rank 2. Due to Theorem T for all sufficiently large |d| and all small z one has kL(1) (z; d)k < 1. Therefore the operator I + L(1) (z; d) is invertible. Hence, det2 (I + L(z; d)) = det2 (I + L(1) (z; d)) · det2 [I + L(0) (z; d)(I + L(1) (z; d))−1 ] · exp{−trL(0) (z; d)(I + L(1) (z; d))−1 L(1) (z; d)} = det2 (I + L(1) (z; d)) · det[I + L(0) (z; d)(I + L(1) (z; d))−1 ] · exp{−trL(0) (z; d)} .
(5.22)
It is easy to show that kM2 (z; d)ϕ1 − ϕ1 kL2 → 0 , (ϕ2 (· − d), ϕ1 ) → 0 as |d| → ∞. We note that M2 (z; d)ϕ1 and ϕ2 (· − d) for sufficiently large |d| depend on d continuously (in L2 -norm). Therefore for sufficiently large |d| the set {M2 (z; d)ϕ1 , ϕ2 (·− d)} forms a basis (in general non-orthogonal) in Ran(L(0) (z; d)). Let P be the projector onto Ran(L(0) (z; d)), such that P = M2 (z; d)ϕ1 (ψ1 (z; d), ·) + ϕ2 (· − d)(ψ2 (z; d), ·) , where ψ1 (z; d) = c11 (d)M2 (z; d)ϕ1 + c12 (z; d)ϕ2 (· − d) , ψ2 (z; d) = c21 (z; d)M2 (z; d)ϕ1 + c22 (z; d)ϕ2 (· − d) is the basis in Ran(L(0) (z; d)) dual with respect to {M2 (z; d)ϕ1 , ϕ2 (· − d)}. The coefficients cij are given by
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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c11 (d) = (ϕ2 , ϕ2 )D−1 , c12 (z; d) = −(ϕ2 (· − d), M2 (z; d)ϕ1 )D−1 , c21 (z; d) = −(M2 (z; d)ϕ1 , ϕ2 (· − d))D−1 , c22 (z; d) = (M2 (z; d)ϕ1 , M2 (z; d)ϕ1 )D−1 , D = (ϕ2 , ϕ2 )(M2 (z; d)ϕ1 , M2 (z; d)ϕ1 ) − (ϕ2 (· − d), M2 (z; d)ϕ1 )(M2 (z; d)ϕ1 , ϕ2 (· − d)) . We note that D is nothing but Gram’s determinant of the vectors ϕ2 (· − d) and M2 (z; d)ϕ1 . Since for sufficiently large |d| these vectors are linear independent, we have that D 6= 0. Due to the identity det(I + AB) = det(I + BA) one has det I + L(0) (z; d)(I + L(1) (z; d))−1 = det I + P L(0) (z; d)(I + L(1) (z; d))−1 = det I + P L(0) (z; d)(I + L(1) (z; d))−1 P 1 + a11 a12 , = det a21 1 + a22 where ai1 = ψi (z; d), L(0) (z; d)(I + L(1) (z; d))−1 M2 (z; d)ϕ1 , ai2 = ψi (z; d), L(0) (z; d)(I + L(1) (z; d))−1 ϕ2 (· − d) for i = 1, 2. Now elementary calculations give α1 (z; d) α1/2 (z; d) + det I + L(0) (z; d)(I + L(1) (z; d))−1 = +1, z iz 1/2
(5.23)
where the functions αi (z; d) (i = 1/2, 1) are regular at z = 0 and are given by α1 (z; d) =
(4π)2 |(|V1 |1/2 , ϕ1 )|2 |(|V2 |1/2 , ϕ2 )|2 e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) · (ϕ˜1 , [K12 + K21 + K e − K1 K2 ](I + L(1) )−1 M2 (z; d)ϕ1 ) · (ϕ˜2 (· − d), M1 (z)[K12 + K21 + K e − K1 K2 ](I + L(1) )−1 M2 (z; d)ϕ1 ) − (ϕ˜1 , [K12 + K21 + K e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) · (ϕ˜2 (· − d), M1 (z)[K12 + K21 + K
e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) − (ϕ˜2 (· − d), ϕ1 )(ϕ˜1 , [K12 + K21 + K
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V. KOSTRYKIN and R. SCHRADER
and α1/2 (z; d) =
4π |(|V1 |1/2 , ϕ1 )|2 +
e − K1 K2 ](I + L(1) )−1 M2 (z; d)ϕ1 ) (ϕ˜1 , [K12 + K21 + K
4π |(|V2 |1/2 , ϕ2 )|2
e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) . (ϕ˜2 (· − d), [K12 + K21 + K
e K1 , K2 and L(1) for K12 (z; d), Here we have used the abbreviations K12 , K21 , K, (1) e K21 (z; d), K(z; d), K1 (z), K2 (z; d) and L (z; d) respectively. Lemma 5.3. For sufficiently large |d| α1 (0; d) = |d|−2 + o(|d|−2 ) . Proof. The calculations of the expression e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) (ϕ˜1 , [K12 + K21 + K
(5.24)
are very similar to those used in the course of the proof of Lemma 5.2. First one can easily show that the dominant contribution to (5.24) is given by (ϕ˜1 , K12 (0; d)ϕ2 (·− d)). Its asymptotic can be calculated by means of Lemma 2.2 finally giving 1 (ϕ1 , |V1 |1/2 )(|V2 |1/2 , ϕ2 ) + o(|d|−1 ) . 4π|d| In almost the same way we can calculate the expression e − K1 K2 ](I + L(1) )−1 M2 (z; d)ϕ1 ) , (ϕ˜2 (· − d), M1 (z)[K12 + K21 + K thus obtaining 1 (ϕ2 , |V2 |1/2 )(|V1 |1/2 , ϕ1 ) + o(|d|−1 ) . 4π|d| Also the expressions e − K1 K2 ](I + L(1) )−1 M2 (z; d)ϕ1 ) (ϕ˜1 , [K12 + K21 + K and
e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) (ϕ˜2 (· − d), M1 (z)[K12 + K21 + K
are O(|d|−2 ). Finally we estimate (ϕ˜2 (· − d), ϕ1 ). To this end we write (ϕ˜2 (· − d), ϕ1 ) = (K2 (0; d)∗ ϕ˜2 (· − d), K1 (0)ϕ1 ) = (ϕ˜2 (· − d), K2 (0; d)K1 (0)ϕ1 ) , which is o(|d|−2 ) by Lemma 4.2.
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Hence we have lim log
E→+0
det[I + L(0) (E + i0; d)(I + L(1) (E + i0; d))−1 ] =0 det[I + L(0) (E − i0; d)(I + L(1) (E − i0; d))−1 ]
for all sufficiently large |d|. Now we turn to the estimate of the function ξ12 (E; d) as E → +0. By (4.6) and (5.22) we have ξ12 (E; d) =
1 det2 (I log 2πi det2 (I 1 det(I log · 2πi det(I ·
+ L(1) (E + i0; d)) + L(1) (E + i0; d))
+ L(0) (E + i0; d)(I + L(1) (E + i0; d))−1 ) + L(0) (E − i0; d)(I + L(1) (E − i0; d))−1 )
1 [−t(E + i0; d) + t(E + i0; d) 2πi
− trL(0) (E + i0; d) + trL(0) (E − i0; d)] .
(5.25)
Since the operator I + L(1) (0; d) is invertible for all large |d|, the first term on the r.h.s. of (5.25) equals zero in the limit E → +0 for all sufficiently large |d|. By our previous discussion the second term is also zero for E → +0 and for all sufficiently large |d|. Consider the third term of (5.25). Obviously t(E ± i0; d) + trL(0) (E ± i0; d) = trK1 (E ± i0)K2 (E ± i0; d) + trK1 (E ± i0)L(1) (E ± i0; d) + trK2 (E ± i0; d)L(1) (E ± i0; d) + trK1 (E ± i0)K2 (E ± i0; d)L(1) (E ± i0; d) + tr[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I]L(0) (E ± i0; d) . Here the first four terms are regular at E = 0. Consider the fifth term. Lemma 5.4. For every fixed d ∈ R3 lim tr[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I]
E→+0
· L(0) (E ± i0; d) = C(d) , where C(d) is a d-dependent constant.
(5.26)
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V. KOSTRYKIN and R. SCHRADER
Proof. By means of (5.21) we can write tr{[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I]L(0) (E ± i0; d)} =−
(4π)2 tr{[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I] E
· Q2 (d)Q1 · [K12 (E ± i0; d) + K21 (E ± i0; d) e + K(E ± i0; d) − K1 (E ± i0)K2 (E ± i0; d)]} 4π ± √ tr{[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I] i E · Q2 (d)M1 (E ± i0) · [K12 (E ± i0; d) + K21 (E ± i0; d) e + K(E ± i0; d) − K1 (E ± i0)K2 (E ± i0; d)]} 4π ± √ tr{[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I] i E · M2 (E ± i0; d)Q1 · [K12 (E ± i0; d) + K21 (E ± i0; d) e + K(E ± i0; d) − K1 (E ± i0)K2 (E ± i0; d)]} .
(5.27)
By Lemma 5.1 the second term on the r.h.s. of (5.27) is e d) − K1 (0)K2 (0; d)] (ϕ˜2 (· − d), M1 (0)[K12 (0; d) + K21 (0; d) + K(0; 1/2
· (I + K1 (0))V2
√ (· − d))(ϕ2 , |V2 |1/2 )−1 + O( E) .
Consider the last term on the r.h.s. of (5.27). Using the identity (5.8) we can write [K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I] · M2 (E ± i0; d)Q1 = (I + K1 (E ± i0))(I + K2 (E ± i0; d))M2 (E ± i0; d)Q1 = (I + K1 (E ± i0))Q1 4π ∓ √ (I + K1 (E ± i0))(I + K2 (E ± i0; d))Q2 (d)Q1 . i E Now we use the identity (I + Ki (z))Qi =
√ i z 1/2 V (|Vi |1/2 , ·)Qi + zNi Qi + o(z) , 4π i
which is a direct consequence of (5.4) and of Ki (0)Qi = −Qi . Thus we obtain
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I] · M2 (E ± i0; d)Q1 1/2
= −(I + K1 (0))V2 (· − d)(|V2 |1/2 (· − d), ·)Q2 (d)Q1 √ i E 1/2 V (|V1 |1/2 , ·)Q1 ± 4π 1 √ i E 1/2 1/2 V (|V1 |1/2 , V2 (· − d))(|V2 |1/2 (· − d), ·)Q2 (d)Q1 ∓ 4π 1 √ √ ±4πi E(I + K1 (0))N2 (d)Q2 (d)Q1 + o( E) , where N2 (d) = U (d)N2 U (d)−1 . Therefore the last term on the r.h.s. of (5.27) equals 4π e d) − K1 (0)K2 (0; d)] ∓ √ (ϕ˜1 , [K12 (0; d) + K21 (0; d) + K(0; i E 1/2
· (I + K1 (0))V2
(· − d))(ϕ˜2 (· − d), ϕ1 )(ϕ2 , |V2 |1/2 )−1 |(ϕ1 , |V1 |1/2 )|−2
+ C3 (d) + o(1) , where C3 (d) is a d-dependent constant. Now we consider the first term on the r.h.s. of (5.27). By Lemma 5.1 this term equals 4π e d) + K1 (0)K2 (0; d)] ± √ (ϕ˜1 , [K12 (0; d) + K21 (0; d) + K(0; i E 1/2
· (I + K1 (0))V2
(· − d))(ϕ˜2 (· − d), ϕ1 )(ϕ2 , |V2 |1/2 )−1 |(ϕ1 , |V1 |1/2 )|−2
+ C1 (d) + o(1) , Summing up all the contibutions completes the proof of the lemma.
Thus from (5.26) and Lemma 5.4 it follows that lim
E→+0
t(E + i0; d) + trL(0) (E + i0; d) − t(E − i0; d) − trL(0) (E − i0; d) = 0
for all d ∈ R3 . This completes the proof of Theorem 3. 6. Proof of Theorem 5 Before we present the rigorous proof of Theorem 5 we give a simple derivation based on the Born series for the scattering amplitudes (see e.g. [37]). Here we restrict ourselves to the leading terms in the Born series since the consideration of higher order terms is essentially the same.
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V. KOSTRYKIN and R. SCHRADER
The first contribution in the Born series of the scattering amplitude Ad (ω, ω 0 ; E) involving both potentials V1 and V2 (· − d) is given by the Born approximation 0 ABorn 12 (ω, ω ; E; d)
√ Z b √ V1 ( Eω − q)Vb2 (q − Eω 0 ) −ihd,qi e dq q 2 − E − i0 √ √ Z 2π 2 −i√Ehd,ωi Vb2 ( Eω − q)Vb1 (q − Eω 0 ) ihd,qi e e dq . + (2π)6 q 2 − E − i0
2π 2 i√Ehd,ω0 i = e (2π)6
Applying now Lemma 2.3 we find that 0 ABorn 12 (ω, ω ; E; d)
=
√ √ (2π 2 )2 −1 b √ ˆ 0i ˆ Vb2 ( E(−ω 0 − d))e ˆ i Ehd,d+ω |d| V1 ( E(ω + d)) 6 (2π)
+
√ √ (2π 2 )2 −1 b √ ˆ ˆ Vb1 ( E(−ω 0 + d))e ˆ i Ehd,d−ωi |d| V2 ( E(ω − d)) + o(|d|−1 ) . 6 (2π)
Noting that the first term in the Born series for the amplitudes A1 and A2 is given by the Born approximation (ω, ω 0 ; E) = − ABorn 1 ABorn (ω, ω 0 ; E; d) = − 2
2π 2 b √ V1 ( E(ω − ω 0 )) , (2π)3 √ 0 2π 2 b √ V2 ( E(ω − ω 0 ))e−i Ehd,ω−ω i , 3 (2π)
we arrive at the claim of Theorem 5 in the Born approximation. Theorem 5 then by Ai . “follows” by replacing ABorn i We now turn to the rigorous proof and start with: Lemma 6.1. Let the Vi ’s have the Property A. Then |d| (I + K(z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 + (I + K1 (z))−1 K12 (z; d)(I + K2 (z; d))−1 + (I + K2 (z; d))−1 K21 (z; d)(I + K1 (z))−1 → 0 as |d| → ∞ in Hilbert–Schmidt norm uniformly in z on compact sets in Π0 \ E. Proof. We start with a remark on the invertibility of the operator I + K(z; d). Let us denote by E(d) the set of all z ∈ Π0 for which I + K(z; d) has no bounded inverse. We proved in Lemma 2.7 in [27] that for an arbitrary compact set I ⊂ Π0 \E there is d(I) > 0 such that for all |d| > d(I) the set E(d) has an empty intersection with I, E(d) ∩ I = ∅. Therefore we fix a compact set I ⊂ Π0 \ E and suppose that |d| > d(I).
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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First we show that |d| k(I + K1 (z) + K2 (z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 kJ2 → 0 (6.1) as |d| → ∞. This follows immediately from the equality (I + K1 (z) + K2 (z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 = (I + K1 (z))−1 (I + K2 (z; d))−1 K2 (z; d)K1 (z)(I + K1 (z))−1 · [I − (I + K2 (z; d))−1 K2 (z; d)K1 (z)(I + K1 (z))−1 ]−1 (I + K2 (z; d))−1 and Lemma 4.2. Consider the difference (I + K(z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 = (I + K(z; d))−1 − (I + K1 (z) + K2 (z; d))−1 + (I + K1 (z) + K2 (z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 . Due to (6.1) the second term is o(|d|−1 ) in the sense of Hilbert–Schmidt norm. Due to the second resolvent identity the first term can be written as follows: (I + K(z; d))−1 − (I + K1 (z) + K2 (z; d))−1 = −(I + K1 (z) + K2 (z; d))−1 [K(z; d) − K1 (z) − K2 (z; d)](I + K(z; d))−1 e d)](I + K(z; d))−1 = −(I + K1 (z) + K2 (z; d))−1 · [K12 (z; d) + K21 (z; d) + K(z; = −(I + K1 (z) + K2 (z; d))−1 [K12 (z; d) + K21 (z; d)](I + K1 (z) + K2 (z; d))−1 + (I + K1 (z) + K2 (z; d))−1 [K12 (z; d) + K21 (z; d)](I + K1 (z) + K2 (z; d))−1 e d)](I + K(z; d))−1 · [K12 (z; d) + K21 (z; d) + K(z; e d)(I + K(z; d))−1 . − (I + K1 (z) + K2 (z; d))−1 K(z;
(6.2)
Due to Lemmas 2.2 and 4.3 the second and third terms on the r.h.s. of (6.2) are o(|d|−1 ) in the sense of Hilbert–Schmidt norm. The relation (6.1) shows that the first term on the r.h.s. of (6.2) up to corrections of order o(|d|−1 ) is given by −(I +K1 (z))−1 (I +K2 (z; d))−1 [K12 (z; d)+K21 (z; d)]·(I +K1 (z))−1 (I +K2 (z; d))−1 . Hence |d| k(I + K(z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 + (I + K1 (z))−1 (I + K2 (z; d))−1 [K12 (z; d) + K21 (z; d)] · (I + K1 (z))−1 (I + K2 (z; d))−1 kJ2 → 0 . To complete the proof of the lemma it remains to apply once more the arguments used in the proof of Lemma 4.4.
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Due to Lemma 6.1 and the representation (4.17) for the scattering amplitude to prove Theorem 5 it suffices to show that h √ √ 0 1/2 |d| ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i i + 4πA1 (ω, ω 0 ; E) + 4πA2 (ω, ω 0 ; E; d) → 0 ,
(6.3)
ˆ E)A2 (−d, ˆ ω 0 ; E; d) 4πA1 (ω, −d; √ − |d| ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 K12 (E + i0; d) √ 1/2 i Ehω 0 ,·i
· (I + K2 (E + i0; d))−1 Vd
e
→ 0,
(6.4)
ˆ E; d)A1 (d, ˆ ω 0 ; E) 4πA2 (ω, d; √ − |d| ei Ehω,·i |Vd |1/2 , (I + K2 (E + i0; d))−1 K21 (E + i0; d) √ 1/2 i Ehω 0 ,·i
· (I + K1 (E + i0))−1 Vd
e
→0
(6.5)
as |d| → ∞ uniformly in ω, ω 0 ∈ S2 and in E on compact sets in R+ . First we prove (6.3). We represent the l.h.s. of (6.3) h √ √ 0 1/2 |d| ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i i + 4πA1 (ω, ω 0 ; E) + 4πA2 (ω, ω 0 ; E; d) h √ √ 0 1/2 = |d| ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i √ √ 0 1/2 − ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 V1 ei Ehω ,·i i √ √ 0 1/2 (6.6) − ei Ehω,·i |V2 |1/2 (· − d), (I + K2 (E + i0; d))−1 V2 (· − d)ei Ehω ,·i as the sum of the following terms: √ |d| ei Ehω,·i (|Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)) , √ 1/2 i Ehω 0 ,·i
(I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 Vd
e
√ + |d| ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 i √ h 1/2 1/2 i Ehω 0 ,·i e · (I + K2 (E + i0; d))−1 Vd − V1 √ h + |d| ei Ehω,·i |V2 |1/2 (· − d), (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 1/2
·Vd
− (I + K2 (E + i0; d))−1 V2
1/2
i √ 0 (· − d) ei Ehω ,·i .
(6.7)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Since (I + K1 (E + i0))−1 and (I + K2 (E + i0; d))−1 are uniformly norm bounded in E in any compact set in R+ (say, by a constant C) and due to the Schwarz inequality the first term of (6.7) can be bounded by C 2 k |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)kL2 kVd kL1 ≤ C 2 k |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)kL2 (kV1 kL1 + kV2 kL1 ) . Due to the inequality (4.8) we have 1/2
k |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)kL2 ≤ 2k |V1 |1/2 |V2 |1/2 (· − d)kL1 Z =2
\ \ 1/2 (p)|V 1/2 (p)e−ipd dp |V 1| 2|
1/2 . (6.8)
\ 1/2 ∈ L2 (R3 ) ∩ C 2 (R3 ). For the potentials Vi having the Property B one has that |V i| Now we use the following asymptotics: Lemma 6.2. Let f ∈ L1 (R3 ) ∩ C 2 (R3 ). Then Z f (p)e−ipd dp = o(|d|−2 ) as |d| → ∞. This asymptotics can be directly derived from the asymptotic formula (2.11). Now due to Lemma 6.2 the first term of (6.7) is o(1). Consider the second term of (6.7). First we show that |d| k(I + K2 (E + i0; d))−1 Vd
1/2
− (I + K2 (E + i0; d))−1 V2
1/2
1/2
(· − d) − V1
kL2 (R3 ) → 0
(6.9)
as |d| → ∞. For this aim we represent the L2 -function on the l.h.s. of (6.9) in the form (I + K2 (E + i0; d))−1 Vd
1/2
− (I + K2 (E + i0; d))−1 V2
1/2
= (I + K2 (E + i0; d))−1 (Vd
1/2
1/2
− V1
1/2
− V2
− (I + K2 (E + i0; d))−1 K2 (E + i0; d)V1
1/2
(· − d) − V1
(· − d))
1/2
.
(6.10)
The first term on the r.h.s. of (6.10) is o(|d|−1 ) in the sense of L2 (R3 )-norm due to (6.8) and Lemma 6.2. The second term on the r.h.s. of (6.10) can be represented in the form (I + K2 (E + i0; d))−1 · V2
1/2
1/4
· V1
|V2 |1/4 (· − d)) .
(· − d))R0 (E + i0)|V1 |1/4 |V2 |1/4 (· − d))
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V. KOSTRYKIN and R. SCHRADER
Therefore its L2 -norm is bounded by 1/2
1/4
1/2
1/2
(· − d))R0 (E + i0)V1 |V2 |1/4 (· − d))kJ2 kV1 |V2 |1/2 (· − d))kL1 Z 1/2 C |V2 (x − d)kV1 (y)|1/2 |V2 (y − d)|1/2 ≤ dxdy 4π |x − y|2
CkV2
1/2
· kV1
1/2
|V2 |1/2 (· − d)kL1 ,
which is o(|d|−2 ) due to Lemma 2.4, (6.8) and Lemma 6.2. Thus (6.9) is proven. Now to prove that the second term of (6.7) is o(1), it suffices to show that √ |d| ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 · (I + K2 (E + i0; d))−1 V2
1/2
√
(· − d)ei
Ehω 0 ,·i
→0
as |d| → ∞ uniformly in ω, ω 0 ∈ S2 . The proof of this fact is similar to that of Lemma 2.9 in [27]. First we show that h √ |d| ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 · (I + K2 (E + i0; d))−1 V2
1/2
√
(· − d)ei
Ehω 0 ,·i
√ − ei Ehω,·i |V1 |1/2 (I + K2 (E + i0; d))−1 · (I + K1 (E + i0))−1 V2
1/2
√
(· − d)ei
Ehω 0 ,·i
i
→0
as |d| → ∞. This follows immediately from the identity (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 − (I + K2 (E + i0; d))−1 (I + K1 (E + i0))−1 = (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 · [K1 (E + i0)K2 (E + i0; d) − K2 (E + i0; d)K1 (E + i0)] · (I + K2 (E + i0; d))−1 (I + K1 (E + i0))−1 and Lemma 4.2. Now we show that h √ |d| ei Ehω,·i |V1 |1/2 , (I + K2 (E + i0; d))−1 (I + K1 (E + i0))−1 1/2
· V2
√
(· − d)ei
Ehω 0 ,·i
√ i √ 0 1/2 − ei Ehω,·i |V1 |1/2 , V2 (· − d)ei Ehω ,·i → 0
(6.11)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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as |d| → ∞. To this end we can represent the l.h.s. of (6.11) in the form √ − |d| ei Ehω,·i |V1 |1/2 , K2 (E + i0; d)(I + K2 (E + i0; d))−1 · (I + K1 (E + i0))−1 V2
1/2
√
(· − d)ei
Ehω 0 ,·i
√ √ 0 1/2 − |d| ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 K1 (E + i0)V2 (· − d)ei Ehω ,·i . Here the second term tends to zero since 1/2
|d| kK1 (E + i0)V2
1/2
(· − d)kL2 ≤ |d| kV1
R0 (E + i0)|V1 |1/4 |V2 |1/4 (· − d)kJ2 1/2
· k |V1 |1/2 |V2 |1/2 (· − d)kL1 . The first term can be rewritten in the form: √ 1/4 − |d| ei Ehω,·i |V1 |1/4 V2 (· − d), |V1 |1/4 |V2 |1/4 (· − d)R0 (E + i0)|V2 |1/2 (· − d) · (I + K2 (E + i0; d))−1 (I + K1 (E + i0))−1 V2
1/2
√
(· − d)ei
Ehω 0 ,·i
.
Therefore its absolute value can be bounded by 1/2
|d|C 2 kV1
1/2
V2
1/4
· k |V1 |1/4 V2
|d|C 2 ≤ 4π
1/2
(· − d)R0 (E + i0)|V2 |1/2 (· − d)k kJ2
Z
1/2
· kV1
1/2
(· − d)kL1 kV2 kL1
|V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy |x − y|2
1/2
V2
1/2
1/2
1/2
(· − d)kL1 kV2 kL1 ,
which is again o(1) as |d| → ∞. Now we note that √ √ 0 1/2 |d| ei Ehω,·i |V1 |1/2 , V2 (· − d)ei Ehω ,·i 1/2
≤ |d| kV1
1/2
V2
(· − d)kL1 ,
which is again o(1) due to Lemma 6.2. This estimate and (6.11) completes the proof of that the second term in (6.7) tends to zero as |d| → ∞. The third term in (6.7) can be considered in exactly the same way. Thus (6.3) is proven. Now we prove (6.4). (6.5) can be treated in exactly the same way. Due to Lemma 2.2 √ |d| ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 K12 (E + i0; d) √ 1/2 i Ehω 0 ,·i
· (I + K2 (E + i0; d))−1 Vd −
e
√ 1 i√Ehω,·i ˆ 1/2 e |Vd |1/2 , (I + K1 (E + i0))−1 V1 e−i Ehd,·i 4π
√ √ 0 ˆ 1/2 · e−i Ehd,·i |V2 |1/2 (· − d), (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i
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V. KOSTRYKIN and R. SCHRADER
tends to zero when |d| → ∞. Therefore to prove (6.4) it suffices to show that both √ √ ˆ 1/2 ˆ E) ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 V1 e−i Ehd,·i + 4πA1 (ω, −d; √ √ ˆ 1/2 = ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 V1 e−i Ehd,·i √ √ ˆ 1/2 − ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 V1 e−i Ehd,·i
(6.12)
and √ √ 0 ˆ 1/2 ˆ E; d) e−i Ehd,·i |V2 |1/2 (· − d), (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i + 4πA2 (ω, d; √ √ 0 ˆ 1/2 = e−i Ehd,·i |V2 |1/2 (· − d), (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i √ √ 0 ˆ 1/2 − e−i Ehd,·i |V2 |1/2 (· − d), (I + K2 (E + i0; d))−1 V2 (· − d)ei Ehω ,·i tend to zero. We give the proof of (6.12) only. First we represent (6.12) in the form √ √ ˆ 1/2 ei Ehω,·i (|Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)), (I + K1 (E + i0))−1 V1 ei Ehd,·i √ √ ˆ 1/2 + ei Ehω,·i |V2 |1/2 (· − d)), (I + K1 (E + i0))−1 V1 ei Ehd,·i . The first term is o(1) due to inequality (6.8) and Lemma 6.2. To consider the second term we represent it in the form √ √ ˆ 1/2 − ei Ehω,·i |V2 |1/2 (· − d)), K1 (E + i0)(I + K1 (E ± i0))−1 V1 ei Ehd,·i √ √ ˆ 1/2 + ei Ehω,·i |V2 |1/2 (· − d)), V1 ei Ehd,·i . We have already shown that the second term is o(1). The first term can be written as √ 1/4 − ei Ehω,·i V1 |V2 |1/4 (· − d), |V1 |1/4 |V2 |1/4 (· − d)R0 (E + i0)|V1 |1/2 √ ˆ 1/2 i Ehd,·i
· (I + K1 (E + i0))−1 V1
e
.
The absolute value of this expression can be bounded by 1/2
1/2
CkV1 kL1 kV1 1/4
· kV1
≤
1/2
|V2 |1/2 (· − d)kL1
|V2 |1/4 (· − d)R0 (E + i0)|V1 |1/2 kJ2 C 1/2 1/2 1/2 kV1 kL1 kV1 |V2 |1/2 (· − d)kL1 4π Z 1/2 |V1 (x)|1/2 |V2 (x − d)|1/2 |V1 (y)| · dxdy , |x − y|2
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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which is o(|d|−2 ) again due to Lemma 2.4, (6.8) and Lemma 6.2. This completes the proof of Theorem 5. Appendix A Here we prove the following statements: Theorem A.1. Consider the Hamiltonians H± = H0 ±V with V ∈ R∩L1 (R3 ) and V is nonnegative. Moreover suppose that E = 0 is a regular point for H− , which in addition has no negative bound states. Then the scattering lengths c0 (±V ) = ∓
1 1/2 V , (I ± V 1/2 R0 (0)V 1/2 )−1 V 1/2 4π
satisfy ∓c0 (±V ) > 0. Proof. The operator V 1/2 R0 (0)V 1/2 is self-adjoint compact and non-negative. Therefore (I +V 1/2 R0 (0)V 1/2 )−1 > 0. By the Birman–Schwinger Principle the total multiplicity of the discrete spectrum of H− equals the number of eigenvalues λk of −V 1/2 R0 (0)V 1/2 such that λk ≤ −1. Since by assumption the discrete spectrum is empty, we have that λk > −1 for all k, and therefore I − V 1/2 R0 (0)V 1/2 > 0. Hence (I − V 1/2 R0 (0)V 1/2 )−1 exists and is a positive operator. Theorem A.2. Let V satisfy the following conditions: (i) (ii) (iii) (iv)
1 V R ∈R∩L , V dx = 0, V is not identically zero, kV kR < 2π.
Let W ≥ 0 satisfy (i), (iii), and (iv). Then for any sufficiently small λ > 0 there is a ∈ (0, 1) such that c0 (λ(V + aW )) = 0. Proof. Denote Ua = V + aW . The assumptions of the theorem guarantee that the Neumann expansion (I + λUa1/2 R0 (0)|Ua |1/2 )−1 = I − λUa1/2 R0 (0)|Ua |1/2 + O(λ2 ) converges for all λ ∈ [0, 1] and a ∈ [0, 1]. Therefore Z
λ2 |Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 + O(λ3 ) 4π Z λ2 λa W dx + |Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 + O(λ3 ) . (A.1) =− 4π 4π
c0 (λUa ) = −
We note that
λ 4π
Ua dx +
|Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 > 0 .
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V. KOSTRYKIN and R. SCHRADER
Indeed,
Z U (x)U (y) a a dxdy . |Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 = 4π|x − y|
By (2.15) we have that Z
Ua (x)Ua (y) dxdy = 4π|x − y|
Z
ba (p)|2 |U dp > 0 . p2
Now consider the equation c0 (λUa ) = 0, which by (A.1) can be written in the form: Z λ a W dx + |Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 + O(λ2 ) = 0 . f (a, λ) := − 4π 4π Now fix λ0 > 0 so small that 0
0 and f (1, λ) < 0. Since f (a, λ) is continuous in a ∈ [0, 1], there is a = a(λ) such that f (a, λ) = 0. Appendix B This appendix is devoted to a proof of relation (2.12). Let us denote Z f (p)e−ihd,pi dp . I(d) = 2 R3 p − E ∓ i0
(B.1)
√ We define the function ηδ (q) ∈ C0∞ (R+ ) such that ηδ (q) ≡ 1 for |q − E| < δ and √ ηδ (q) ≡ 0 for |q − E| > 2δ. We represent the integral (B.1) as follows: Z I(d) =
f (p)ηδ (|p|)e−ihd,pi dp + p2 − E ∓ i0
Z
f (p)(1 − ηδ (|p|))e−ihd,pi dp . p2 − E ∓ i0
(B.2)
The integrand in the second term is in C 2 ∩ L1 . Therefore due to (2.11) the second integral is o(|d|−2 ). Consider the first integral on the r.h.s. of (B.2) for which we preserve the notation I(d). Also we write fδ (p) for f (p)ηδ (|p|). Integration by parts for an arbitrary differentiable function g(ω) on the unit sphere S2 gives Z o 2π n ˆ −i|p||d| ˆ i|p||d| g(d)e g(ω) exp{−i|p|hd, ωi}dω = − − g(−d)e i|p||d| S2 Z 1 ˆ ∇ig(ω)dω . (B.3) exp{−i|p|hd, ωi}hd, + i|p||d| S2 ˆ ∇i we mean the differential operator ∂/∂θ, if (θ, φ) are the polar coordinates By hd, of ω ∈ S2 with dˆ as polar axis.
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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First we consider the case E > 0. Let us fix δ such that E > 4δ 2 . With the help of the identity (B.3) we represent I(d) as the sum of two terms 2π I1 (d) = − i|d|
√
Z
√
E+2δ
E−2δ
and 1 I2 (d) = i|d|
√
Z
Z
ˆ −iq|d| qdq 2π fδ (q d)e + 2 q − E ∓ i0 i|d|
√
E+2δ
E−2δ
qdq q 2 − E ∓ i0
Z S2
√ E+2δ
√
E−2δ
ˆ iq|d| qdq fδ (−q d)e , q 2 − E ∓ i0
dωe−iqhd,ωi f1,δ (qω) ,
(B.4)
ˆ ∇ifδ (qω) ∈ C 1 (R3 ) by the assumption. Since (q 2 − E ∓ i0)−1 where f1,δ (qω) = hd, is the distribution of order 1, the integral (B.4) is well defined. Due to the distributional identities (q ∈ R) 1 1 = v.p. 2 ± iπδ(q 2 − E) , q 2 − E ∓ i0 q −E √ √ 1 δ(q − E) = (δ(q − E) + δ(q + E)) , 2|q|
(B.5)
2
one has (1)
(2)
I1 (d) = I1 (d) + I1 (d) , (1) I1 (d)
2π v.p. =− i|d|
Z
2π v.p. + i|d| (2)
I1 (d) = ∓
√ E+2δ
ˆ −iq|d| qdq fδ (q d)e q2 − E
√ E−2δ
Z
√ E+2δ
√ E−2δ
ˆ iq|d| qdq fδ (−q d)e , 2 q −E
√ √ π 2 √ ˆ −i√E|d| ˆ i E|d| . f ( E d)e − f (− E d)e |d|
(1)
Now we rewrite I1 (d) in the following way: (1) I1 (d)
√ ˆ − ηδ (q)f ( E d)]e ˆ −iq|d| qdq [fδ (q d) √ 2 q −E E−2δ √ √ Z E+2δ ˆ − ηδ (q)f (− E d)]e ˆ iq|d| qdq 2π [fδ (−q d) + √ i|d| E−2δ q2 − E
2π =− i|d|
Z
√
E+2δ
√
Z E+2δ 2π √ ˆ ηδ (q)e−iq|d| qdq f ( E d) v.p. √ − i|d| q2 − E E−2δ √ 2π ˆ v.p. f (− E d) + i|d|
Z
√
√
E+2δ
E−2δ
ηδ (q)eiq|d| qdq . q2 − E
(B.6)
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V. KOSTRYKIN and R. SCHRADER
Due to the fact that the functions √ ˆ − ηδ (q)f (± E d) ˆ fδ (±q d) 2 q −E have compact support and are bounded, by Riemann–Lebesgue lemma the first two terms on the r.h.s. of (B.6) are o(|d|−1 ). The integrands in the third and √ fourth terms on the r.h.s. of (B.6) are analytic for complex q in a small vicinity of E. Therefore due to the well-known property of integrals in the sense of principal value we have Z v.p.
√
√
E+2δ
E−2δ
Z
ηδ (q)e∓iq|d| qdq = q2 − E
(∓)
γ
√ iπ ηδ (q)e∓iq|d| qdq ∓ e∓i|d| E , 2 q −E 2
√ √ (±) , (0 < < 2δ) consists of the intervals [ E − 2δ, E − ], where the contour γ √ √ √ [ E + , E + 2δ] and the half circle |q − E| = , ∓Imq ≥ 0 respectively. (±) Since ηδ (q) and all its derivatives are zero at the ends of the contour γ and (±) since |e±i|d|q | ≤ 1 on γ respectively, by means of an integration by parts we can show that Z ηδ (q)e∓iq|d| qdq = O(|d|−∞ ) . 2−E (∓) q γ Thus we arrive at the asymptotic formula I1 (d) =
√ √ 2π 2 ˆ ±i E|d| + o(|d|−1 ) , f (∓ E d)e |d|
where the error term o(|d|−1 ) is uniform in E on compact sets in (0, +∞). Now we turn to the discussion of the integral I2 (d) given by (B.4). To complete the proof of (2.12) it suffices to show that √
Z
√
E+2δ
E−2δ
qdq 2 q − E ∓ i0
Z S2
e−iqhω,di f1,δ (qω)dω → 0
(B.7)
uniformly in E on compact sets in (0, ∞) as |d| → ∞. Let us note that f1,δ (p) = f1 (p)ηδ (|p|) ,
ˆ ∇if (p) . f1 (p) = hd,
Therefore the integral in (B.7) can be represented as the sum of two terms Z
√
E+2δ
√ E−2δ
Z +
qηδ (q)dq 2 q − E ∓ i0
√ E+2δ
√
E−2δ
Z S2
qηδ (q)dq 2 q − E ∓ i0
h i √ e−iqhω,di f1 (qω) − f1 ( Eω) dω
Z S2
√ e−iqhω,di f1 ( Eω)dω .
(B.8)
(B.9)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Since f1 is differentiable, the integrand in (B.8) is regular, and hence by Riemann– Lebesgue lemma the integral tends to zero as |d| → ∞. By (B.5) the integral (B.9) can be represented in the form Z v.p.
√
√
±
E+2δ
E−2δ
iπ 2
Z
S2
qηδ (q)dq q2 − E
Z S2
√ e−iqhω,di f1 ( Eω)dω
√ e−iqhω,di f1 ( Eω)dω ,
where the second term tend to zero again by Riemann–Lebesgue lemma. Consider the first term, Z √E+2δ qηδ (q) F (q, d)dq , (B.10) v.p. √ 2 E−2δ q − E where
Z F (q, d) = S2
√ e−iqhω,di f1 ( Eω)dω .
By (B.3) we have F (q, d) = F1 (q, d) + F2 (q, d) , o √ 2π n √ ˆ −iq|d| ˆ iq|d| , f1 ( E d)e F1 (q, d) = − − f1 (− E d)e iq|d| Z √ 1 e−iqhω,di f2 ( Eω)dω , F2 (q, d) = iq|d| S2 √ √ where f2 ( Eω) = hd, ∇if1 ( Eω). The arguments already used above allow one to calculate the contribution from F1 (q, d) thus giving Z v.p.
√
√
E+2δ
E−2δ
qηδ (q) F1 (q, d)dq q2 − E
√ √ π 2 √ ˆ −i|d|√E ˆ i|d| E + O(|d|−∞ ) . + f1 (− E d)e f1 ( E d)e = √ E|d|
Consider now the contribution from F2 (q, d), 1 v.p. i|d|
Z
√ E+2δ
√ E−2δ
ηδ (q)dq q2 − E
Z S2
√ e−iqhω,di f2 ( Eω)dω .
By the obvious estimate Z Z a ϕ(x) a ϕ(x) − ϕ(0) v.p. dx = dx ≤ 2a max |ϕ0 (x)| , x x∈[−a,a] −a x −a
(B.11)
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V. KOSTRYKIN and R. SCHRADER
we have that the absolute value of (B.11) can be bounded by 4δ
max√
√ q∈[ E−2δ, E+2δ]
Z √ 1 ∂ ηδ (q) e−iqhω,di f2 ( Eω)dω . |d| ∂q q + E S2
Note that 1 ∂ |d| ∂q
Z S2
Z √ √ ˆ 2 ( Eω)dω , e−iqhω,di f2 ( Eω)dω = −i e−iqhω,di hω, dif S2
which is o(1) as |d| → ∞ by Riemann–Lebesgue lemma. This completes the proof of (B.7). The proof of (2.12) in the case E = 0 follows from the results of [40]. Appendix C Here we give the proof of Lemma 4.1. According to (3.2) the spectral shift function for the pair of operators H = H0 + V and H0 can be written as ξ(E) = −ξ((E + c0 )−1 ; R(−c0 ), R0 (−c0 )) =− =
det[I + (R(−c0 ) − R0 (−c0 ))(R0 (−c0 ) − (E + c0 )−1 − i)−1 ] 1 lim log 2πi →+0 det[I + (R(−c0 ) − R0 (−c0 ))(R0 (−c0 ) − (E + c0 )−1 + i)−1 ]
det[I + (R(−c0 ) − R0 (−c0 ))(R0 (−c0 ) − ζ)−1 ] 1 lim log , 2πi →+0 det[I + (R(−c0 ) − R0 (−c0 ))(R0 (−c0 ) − ζ)−1 ]
(C.1)
where ζ = (E + i + c0 )−1 . Due to the resolvent equation we have R(−c0 ) − R0 (−c0 ) = −R0 (−c0 )|V |1/2 (I + V 1/2 R0 (−c0 )|V |1/2 )−1 V 1/2 R0 (−c0 ) . Obviously R0 (−c0 )|V |1/2 and V 1/2 R0 (−c0 ) are Hilbert–Schmidt operator. We recall that for an arbitrary trace class operator A the modified Fredholm determinant is given by det2 (I + A) = det(I + A)e−trA . Therefore det[I + (R(−c0 ) − R(−c0 ))(R0 (−c0 ) − ζ)−1 ] = det2 [I − (I + V 1/2 R0 (−c0 )|V |1/2 )−1 · V 1/2 R0 (−c0 )(R0 (−c0 ) − ζ)−1 R0 (−c0 )|V |1/2 ] · exp{−tr[(I + V 1/2 R0 (−c0 )|V |1/2 )−1 · V 1/2 R0 (−c0 )(R0 (−c0 ) − ζ)−1 R0 (−c0 )|V |1/2 ]} .
(C.2)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Now we note that R0 (−c0 )(R0 (−c0 ) − ζ)−1 R0 (−c0 ) = R0 (−c0 ) + ζ(R0 (−c0 ) − ζ)−1 R0 (−c0 ) = R0 (−c0 ) − R0 (ζ −1 − c0 ) = R0 (−c0 ) − R0 (E + i) . Therefore the r.h.s. of (C.2) equals n det2 I − (I + V 1/2 R0 (−c0 )|V |1/2 )−1 V 1/2 R0 (−c0 )|V |1/2 + (I + V 1/2 R0 (−c0 )|V |1/2 )−1 V 1/2 R0 (E + i)|V |1/2
o
· exp − tr(I + V 1/2 R0 (−c0 )|V |1/2 )−1
· (V 1/2 R0 (−c0 )|V |1/2 − V 1/2 R0 (E + i)|V |1/2 ) .
Since (I + V 1/2 R0 (−c0 )|V |1/2 )−1 V 1/2 R0 (−c0 )|V |1/2 = I − (I + V 1/2 R0 (−c0 )|V |1/2 )−1 , (C.3) we get det[I + (R0 (−c0 ) − R0 (−c0 ))(R0 (−c0 ) − ζ)−1 ] = det2 (I + V 1/2 R0 (−c0 )|V |1/2 )−1 (I + V 1/2 R0 (E + i)|V |1/2 ) · exp − tr(I + V 1/2 R0 (−c0 )|V |1/2 )−1 · (V 1/2 R0 (−c0 )|V |1/2 − V 1/2 R0 (E + i)|V |1/2 ) . (C.4) It is easy to see that for arbitrary Hilbert–Schmidt operators A and B such that I + A is invertible det2 [(I + A)−1 (I + B)] =
det2 (I + B) tr(I+A)−1 A(B−A) e . det2 (I + A)
(C.5)
Therefore using (C.3) and (C.5) the expression (C.4) can be represented as det2 (I + V 1/2 R0 (E + i)|V |1/2 ) det2 (I + V 1/2 R0 (−c0 )|V |1/2 ) · exp tr(V 1/2 R0 (E + i)|V |1/2 − V 1/2 R0 (−c0 )|V |1/2 ) . Finally we get det[I + (R(−c0 ) − R0 (−c0 )(R0 (−c0 ) − ζ)−1 )] det[I + (R(−c0 ) − R0 (−c0 )(R0 (−c0 ) − ζ)−1 )] =
det2 (I + V 1/2 R0 (E + i)|V |1/2 ) det2 (I + V 1/2 R0 (E − i)|V |1/2 ) · exp tr(V 1/2 R0 (E + i)|V |1/2 − V 1/2 R0 (E − i)|V |1/2 ) .
(C.6)
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V. KOSTRYKIN and R. SCHRADER
Taking the limit → 0 in (C.6) and using the relation lim tr(V 1/2 R0 (E + i)|V |1/2 − V 1/2 R0 (E − i)|V |1/2 )
→0
√ Z i E V dx , E ≥ 0 2π R3 = 0, E 1}\ Z, where Z is the closure of Z :=
[
Za ,
a∈N
with Za being the set of the 2a+1 − 1 solutions of the equation 1 = 0. τz ◦ · · · ◦ τz (1) + | {z } ζz a-times
Furthermore, Za are self-conjugated sets (i.e., z ∈ Za if z ∈ Za ) satisfying Za ∩ [0, ∞) = ∅ and Z 0 ≡ Z \Z ⊂ S 1 . There exists a ζ0 ∈ [0, 1/3] with ζ0 ' 0.29559a such that, for ζ ∈ [0, ζ0 ), we have the inclusion o n Za ⊂ z ∈ C : 1 < |z| < (1/ζ)1/(a+1) . We conjecture that Z 0 = S 1 for ζ ∈ [0, 1/3]. a See also Remark 3.13.
(1.16)
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J. C. A. BARATA and D. H. U. MARCHETTI
z=1
Fig. 2. The location of the set of poles Z and the unit circle S 1 . The points of Z accumulate on subsets Z 0 of the unit circle. Numerical computations indicate that Z 0 = S 1 in the ferromagnetic phase. (a+1)
The elements of the set Za are poles of τz (1). These singularities are removable in the magnetization (1.6). Theorem 1.1 states that the set of singular points of Ms is located in D>1 and accumulate on a subset Z 0 of S 1 . Figure 2 describes the singular set Z. We conjecture that the accumulation set Z 0 coincides with the accumulation set of the Lee–Yang singularities of the magnetization at the origin, studied in [1]. Numerical computations seem to confirm this idea. The inclusion (1.16) indicates how fast the sets Za tend to accumulate on S 1 for ζ ∈ [0, ζ0 ). As a consequence of the recurrence relation (1.9), we are also able to compute the two-point function explicitly. The next result shows that the fluctuation on the Cayley tree is bounded by that of the one-dimensional lattice model. Theorem 1.2 (Two-Point Function). The truncated two-point function hσ0 ; σx iN (ξ) := hσ0 σx iN (ξ) − hσ0 iN (ξ) hσx iN (ξ) can be written as hσ0 ; σx iN (ξ) = s(z∆0 )
n Y
t(ζj , z∆j )
j=1
where n = n(x) = dist(0, x) is the generation of the site x, s(x) := 1 −
1−x 1+x
2
(1.17)
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY
and t(x) = t(ζ, x) :=
x−1
...
757
ζ −1 − ζ . + x + ζ −1 + ζ
For all ζ ∈ [0, 1], z ∈ R+ and p ∈ πa , 0 ≤ a ≤ 1, the following identity holds for the expected value Eξ hσ0 ; σx i(ξ) = limN →∞ Eξ hσ0 ; σx iN (ξ): Eξ hσ0 ; σx i(ξ) =
∞ X
ak s(wk )
n Y
t(wk−j ) + a s(w)[t(w)]n ,
(1.18)
j=1
k=n
(n)
where w = w(ζ, z) is the limit point of the sequence wn = z τz (1), n ∈ N.
Remark 1.3. A simple upper bound on (1.18) for the paramagnetic phase can be obtained by using Perez’s self-avoiding random walk estimate [3]. Only one term, corresponding to the single self-avoiding path connecting 0 to x, contributes to the correlation functions in Ck . This fact shows that the Ising model on the Cayley tree behaves, on what concerns the asymptotic behavior of the correlations, as a one dimensional system. As a consequence, the correlation length is always finite, even at the transition point. (Note that |t(ζ, x)| ≤ 1/2 if ζ ∈ [1/3, 1] and x ∈ R+ .) Our last results shows that the quenched susceptibility at the origin χ := limN →∞ χN given by X Eξ hσ0 ; σx i(ξ) (1.19) χN := x∈C2,N
diverges as |ζ − ζc |γ at the critical point ζc = 1/3 with an exponent given by the classical theory γ = 1. More precisely, Theorem 1.4 (Susceptibility). The quenched susceptibility at origin χ = χ(ζ, z) is finite for all ζ ∈ [0, 1], z ∈ R+ \{1}. and p ∈ π. Moreover, limz→1 χ is also finite provided ζ 6= ζc . In addition, for p ∈ πa with 0 < a ≤ 1, the following asymptotic behavior χ(ζ, 1) ∼ Cη |η|γ
as
η := ζ − ζc → 0
(1.20)
holds with γ = 1, limη&0 Cη = 4a/3 and limη%0 Cη = 2a/3 provided the condition ∞ X
n an < ∞
n=0
is satisfied. The proofs of Theorems 1.2 and 1.4 will be given in Sec. 2.
(1.21)
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J. C. A. BARATA and D. H. U. MARCHETTI
2. The Two-Point Correlation Function This section is dedicated to the proof of Theorems 1.2 and 1.4. We shall consider the truncated two-point function hσ0 ; σx iM (ξ) = hσ0 σx iM (ξ) − hσ0 iM (ξ)hσx iM (ξ) , where hσ0 σx iM (ξ) =
X 1 σ0 σx e−βH(σ;ξ) ZM (ξ) σ
(2.1)
(2.2)
for all x ∈ C2, M with M large enough. 2.1. The one-point function Let us first recall the iteration procedure leading to expression (1.6) for the magnetization at the origin. We start by computing the partition function ZM (ξ) in a finite tree with M generations. Let Zj = (Zj+ , Zj− ), j = 0, 1, . . . , M , be a sequence of two-component vectors defined recursively by σ := Zj−1
!2
X
0
0
eβξj σσ eβhσ Zjσ
0
σ0 =±1
2 (1−σ)/2 + (1+σ)/2 = (ζj z)−1 ζj Zj + ζj z Zj−
(2.3)
+ − = ZM = 1. with ZM If the spin configurations are summed starting from the branches towards the root, the partition function ZM (ξ) can be written as
ZM (ξ) = z −1/2 Z0+ + z 1/2 Z0− .
(2.4)
To compute the one-point function hσ0 iM (ξ) =
X 1 σ0 e−βH(σ;ξ) , ZM (ξ) σ
(2.5)
we repeat the procedure leading to (2.4) for the numerator in (2.5). Except by the last summation on the spin at the root, all remaining ones give exactly the previous expressions. We thus have hσ0 iM (ξ) =
z −1/2 Z0+ − z 1/2 Z0− 1 − z∆0 , + − = −1/2 1/2 1 + z∆0 z Z0 + z Z0
where ∆j := Zj− /Zj+ . From (2.3), we have ∆j−1 =
ζj + z∆j 1 + ζj z∆j
2 = τj,z (∆j )
(2.6)
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY
...
759
with ∆M = 1. Recall that ∆j is a random variable since ζj = e−2βξj with ξj as given by (1.2). 2.2. The two-point function To compute the numerator of (2.2), we repeat the steps in the calculation of the partition function ZM (ξ). Our aim is to derive an expression analogous to (2.4). Let Z˜j = (Z˜ + , Z˜ − ), j = 0, . . . , M , be a sequence of two-component vectors j
j
defined recursively as in the following. For j = n0 + 1, . . . , M , with n0 =dist(0, x), we have Z˜jσ = Zjσ , i.e., Z˜jσ satisfy + − Eq. (2.3) with initial conditions Z˜M = Z˜M = 1. For j ≤ n0 , we consider a linear transformation of the form σ σ = Zj−1 Z˜j−1
ζ (1−σ)/2 Z˜j+ + ζ (1+σ)/2 z Z˜j− ζ (1−σ)/2 Zj+ + ζ (1+σ)/2 z Zj−
,
(2.7)
with Z˜nσ0 = σZnσ0 . We now observe that the sum over all spin configurations in the numerator of (2.2), except by spin at the origin, is determined by (2.7) and the sum over all spin configurations in the denominator is determined by (2.3). The two-point function (2.2) can thus be written in the following form: hσ0 σx iM (ξ) =
Z˜0+ − z Z˜0− . Z0+ + z Z0−
(2.8)
In the following lemma Eq. (2.8) will be reorganized and reexpressed in terms of the quantities Zjσ , j = 0, 1, . . . , M and σ = +, −. Lemma 2.1. The sequence of vectors Z˜n , n = 1, . . . , n0 , defined by (2.7), can be written as σ σ = (An + σBn )Zn−1 , (2.9) Z˜n−1 where An = An Bn+1 + · · · + An0 −1 Bn0 + An0
(2.10)
Bj = Bj Bj+1 . . . Bn0 ,
(2.11)
and with j = n, . . . , n0 . Here Aj = Aj (ζj , z) and Bj = Bj (ζj , z), j = 1, . . . , n0 , are random variables given by Aj = and Bj =
(z∆j )−1 − z∆j (z∆j )−1 + z∆j + ζj−1 + ζj ζj−1 − ζj (z∆j )−1 + z∆j + ζj−1 + ζj
(2.12)
.
(2.13)
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Proof. We shall prove Lemma 2.1 by induction. We let j = n0 and observe that, −(1−σ)/2 (z∆n0 )−1 + by multiplying the numerator and the denominator of (2.7) by (ζj −(1+σ)/2
ζj
)/Zn+0 , it can be written as (z∆n0 )−1 − z∆n0 + ζj−σ − ζjσ Z˜nσ0 −1 = Znσ0 −1 (z∆n0 )−1 + z∆n0 + ζj−σ + ζjσ =
(z∆n0 )−1 − z∆n0 + σ(ζj−1 − ζj ) (z∆n0 )−1 + z∆n0 + ζj−1 + ζj
= An0 + σBn0 .
(2.14)
Now, let j = n + 1. Assuming (2.9) valid, (2.7) can be written as (1−σ)/2 (1+σ)/2 ζj − ζj z∆n+1 Z˜nσ = A + Bn+1 . n+1 σ (1−σ)/2 (1+σ)/2 Zn ζj + ζj z∆n+1
(2.15)
Multiplying the numerator and the denominator of the second term on the right−(1−σ)/2 −(1+σ)/2 (z∆n+1 )−1 + ζj , gives hand side of this equation by ζj (z∆n+1 )−1 − z∆n+1 + σ(ζj−1 − ζj ) Z˜nσ = An+1 + Bn+1 (z∆n+1 )−1 + z∆n+1 + ζj−1 + ζj Zˆnσ = An+1 + (An + σBn )Bn+1 which, in view of (2.10) and (2.11), concludes the proof of Lemma 2.1.
Now we proceed with the proof of Theorem 1.2. Proof of Theorem 1.2. Using (2.9) to compute (2.8), gives hσ0 σx iM (ξ) =
(A1 + B1 )Z0+ − z(A1 − B1 )Z0− Z0+ + zZ0−
= A1
1 − z∆0 + B1 . 1 + z∆0
(2.16)
The one-point function at x can be computed by a procedure analogous to one described at the beginning of this subsection. The difference between this onepoint function and the two-point function is the spin at origin. As before, (2.7) with j = 0, . . . , n0 is of relevance for the description of the numerator of hσx i. The iteration gives an expression of the form (2.8) with the minus sign replaced by plus. The one–point function can thus be written as hσx iM (ξ) =
1 − z∆0 Z˜0+ + z Z˜0− . + − = A1 + B1 1 + z∆0 Z0 + z Z0
(2.17)
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY
...
761
Inserting (2.16) and (2.17) into (2.1) and taking into account (2.6), yields " 2 # 1 − z∆0 B1 . (2.18) hσ0 ; σx iM (ξ) = 1 − 1 + z∆0 When the thermodynamic limit, M → ∞, has been taken, the random variables ∆j ’s in (2.18) can be replaced by their limits limM→∞ ∆j (recall that B1 depends 0 , each of which defined by a recursion relation with initial condition on {∆j }nj=0 ∆M = 1 dependent on the generation M ). Since ∆j is bounded from above and below, the convergence in distribution is guaranteed by the convergence of their moments [4]. It follows from Theorem 1.1 that limM→∞ Eξ ∆j exists and is a real analytic function of z in z ∈ R+ \{1}. In order to take the expectation value of (2.18) we shall use the following property: for any bounded function f (ξ) = f (ξj , ξ 0 ) of the random variables ξ = (ξj , ξ 0 ), we have ζ −1 − ζ (2.19) Eξ f (ξ)Bj (ξ) = pj Eξ0 f (1, ξ 0 ) (z∆j )−1 + z∆j + ζ −1 + ζ since ζj−1 − ζj = e2βξj − e−2βξj = 0 if ξj = 0. Define 2 1−x s(x) := 1 − 1+x and ζ −1 − ζ . t(x) = t(ζ, x) := −1 x + x + ζ −1 + ζ Using (2.19) in the expected value of (2.18) gives n0 Y Eξ hσ0 ; σx iM (ξ) = p1 · · · pn0 Eξ0 s(z∆0 ) t(z∆j )
(2.20)
(2.21)
j=1
=
M−1 X k=n0
ak s(wk )
n0 Y
j=1
t(wk−j ) + p1 · · · pM s(wM )
n0 Y
t(wM−j ) ,
j=1
(2.22) (n) zτz (1), n
∈ N. Here, we have first taken partial expectations with where wn = respect to the variables ξ1 , . . . , ξn0 and have used, in the sequel, for all remaining expectations, that the sequence ∆j , j ∈ N, satisfies the recurrence relation z∆j−1 = zhj (z∆j ) with hj (1) = 1. Equation (1.18) then follows since s(x) and t(x) are continuous in R+ and wn converges to the solution w = w(ζ, z) of the fixed point equation w = zh(w) in this domain provided z ∈ R+ . This concludes the proof of Theorem 1.2. We now turn to the proof of Theorem 1.4 on the quenched susceptibility at origin χ. We note the following facts on the function s and t (the proof will be omitted):
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J. C. A. BARATA and D. H. U. MARCHETTI
Proposition 2.2. The function s : w ∈ R+ 7→ s(w) ∈ R+ and t : (ζ, w) ∈ [0, 1] × R+ 7→ t(ζ, w) ∈ R+ given by (2.20) and (2.21), respectively, have a maximum value at w = 1 with s(1) = 1 and t(ζ, 1) = (1 − ζ)/(1 + ζ), are monotonically increasing function of w in [0, 1] and satisfy s(w) = s(w−1 ) and t(ζ, w) = t(ζ, w−1 ). For z ∈ [0, 1) we recall that wn , n ∈ N, is a monotonically decreasing sequence with wn < 1 and for each n, wn = wn (z) is monotonically decreasing function of z in this domain. So, in view of Proposition 2.2, 2t(ζ, wn ) ≤ 2
1−ζ n0 2ζ 0 call Da := {w ∈ S 2 : |w| > a} .
(3.2)
For further purposes define also for a, b ∈ R+ , a < b, Da,b := {w ∈ S 2 : a < |w| < b} .
(3.3)
The following theorem has been proven in [1]: (n)
Theorem 3.1. The sequence τz (1), n ∈ N, of analytic functions on D1 \Z, where B is the closure of B. Then, analyticity of Fr on B follows from Lemmas 3.3 and 3.10. Since D>1 \Z can be covered by such open sets the theorem is proven. (n)
3.2. Estimates on the location of the poles of τz
(1)
To study of how fast the sets Zk accumulate on S 1 we will make use of a contraction theorem, described below, on the inverse mappings of h. First, some definitions. For w ∈ C, w 6= 1/ζ, define g(w) :=
ζ −w ζw − 1
(3.14)
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J. C. A. BARATA and D. H. U. MARCHETTI
and let u1/2 be some branch of the square root function in C. Define 1/2 ), h−1 + (u) := g(u
1/2 h−1 ). − (u) := g(−u
(3.15)
Then one has (h ◦ h−1 ± )(z) = z, ∀ z ∈ C. Theorem 3.12 (Contraction Theorem). There exists a number ζ0 ∈ [0, 1/3), whose approximate value is ζ0 ' 0.29559, such that for ζ ∈ [0, ζ0 ) there exists a strictly positive function e(ζ) such that for all u ∈ D1,a(ζ) with a(ζ) := ζ −1 + e(ζ) one has (3.16) |h−1 ± (u)| < |u| . Remark 3.13. Numerical computations indicate to be impossible to improve the region of validity of Eq. (3.16) to ζ ≥ ζ0 . Remark 3.14. As already observed, the inequality (3.16) becomes an equality in S 1 , which is the internal boundary of D1,a(ζ) . It is important to note also that the set D1,a(ζ) contains the pole z = −1/ζ, of h. We will present the proof of the Contraction Theorem in Appendix A. Let us now explore some of its consequences. Theorem 3.15. Let ζ0 as in Theorem 3.12. For all ζ ∈ (0, ζ0 ) and k ∈ N Zk ⊂ D1,rk ,
(3.17)
−1
holds with rk = ζ k+1 . This theorem illustrates explicitly that the accumulation points Z 0 lie in the unit circle and shows how fast the sets Zk converge to it, at least for ζ ∈ (0, ζ0 ). Proof. The proof of Theorem 3.15 makes use of the Contraction Theorem which requires the following technical lemma. We note that, from Eq. (3.11), there exists a finite sequence of signals {s(l) ∈ {−, +}, 1 ≤ l ≤ k} such that 1 1 1 −1 1 −1 1 hs(k) hs(k−1) · · · h−1 · · · = 1. (3.18) − z z z z s(1) zζ Lemma 3.16. Given z ∈ Zk , k ∈ N, k ≥ 1, consider a sequence of signals {s(l) ∈ {−, +}, 1 ≤ l ≤ k} satisfying (3.18) and define 1 −1 1 1 1 −1 1 h · · · h · · · , l ∈ {1, . . . , k} . − wl := h−1 z s(l) z s(l−1) z z s(1) zζ Then all wl ’s belong to D1,a(ζ) , except, of course, wk which is equal to 1.
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY
...
769
Proof. To prove Lemma 3.16 we take, without loss of generality, k > 1 and −1 /ζ) ∈ D1,1/ζ , by the Contraction note that, since z ∈ D1,1/ζ , one has h−1 s(1) (−z Theorem. Hence, w1 ∈ Dζ,1/ζ . On the other hand w1 cannot belong to Dζ,1 ∪ S 1 for the following reason: h−1 ± maps D 1 and, by (3), b+ > 1.
Before we prove this lemma, let us finish the proof of Theorem A.3 and, hence, of the Theorem A.1. According to Lemma A.4, P (t) < 0 if 1 < t < a+ . This follows from the localization of the roots and from the fact that P is a polynomial of even degree with a negative leading term. See Fig. 3.
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J. C. A. BARATA and D. H. U. MARCHETTI
P(t)
1
-1
a+
0
b+ t
a_ or b_
Fig. 3. The graphic of P (t).
√ The proof is completed by defining f (ζ) := a+ (ζ) − 1/ ζ, which, according with item 4 of Lemma A.4, is strictly positive for 0 ≤ ζ < ζ0 . This completes also the proof of the Contraction Theorem. Proof of Lemma A.4. We will prove each of the items of Lemma A.4 separately. Proof of item (1). The hypothesis that a− ≥ 0 is equivalent to (1 − ζ)2 ≥ (1 + ζ)(1 − 3ζ) for 0 ≤ ζ < 1/3. This last relation means 4ζ 2 ≥ 0, which is obviously verified. The hypothesis that a− ≤ 1 is equivalent to p 1 − (1 + ζ)(1 − 3ζ) ≤ 3ζ , which is equivalent to (1 − 3ζ)2 ≤ 1 − 2ζ − 3ζ 2 . This means 4ζ(3ζ − 1) ≤ 0, what is true for 0 ≤ ζ < 1/3, the equality holding only if ζ = 0. Proof of item (2). The hypothesis that b− ≥ 0 is equivalent to (1 + ζ)2 ≥ (1 − ζ)(1 + 3ζ), which is equivalent to 4ζ 2 ≥ 0, which is, of course, always true. The hypothesis that b− ≤ 1 is equivalent to (1 − ζ)2 ≤ (1 − ζ)(1 + 3ζ). This means that 4ζ(ζ − 1) ≤ 0, which is always true for 0 ≤ ζ ≤ 1. Proof of item (3). Since b + − a− = 1 +
p 1 p 1 + 2ζ − 3ζ 2 − 1 − 2ζ − 3ζ 2 , 2ζ
item (3) is proven, provided the term between parenthesis above is positive. This is implied by the inequality 1 + 2ζ − 3ζ 2 ≥ 1 − 2ζ − 3ζ 2 , which is always true for ζ ≥ 0.
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY
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773
√ Proof of item (4). The condition a+ > 1/ ζ means that p p (1 + ζ)(1 − 3ζ) > ζ + 2 ζ − 1 . (A.6) √ The right-hand side is strictly negative for 0 ≤ ζ < ( 2 − 1)2 ' 0.171. So, in this region the condition above is automatically satisfied. On the other hand, if the right-hand side of (A.6) is positive, we can square both sides and arrive to the equivalent condition (A.7) 4s(s3 + s2 + s − 1) < 0 √ where s = ζ. The polynomial s3 + s2 + s − 1 has one real root at s0 ' 0.543689 and two complex roots at s± ' −0.77 ± 1.115i. We call ζ0 := s20 , which gives ζ0 ' 0.295597. Thus, condition (A.7) is satisfied for 0 < ζ < ζ0 . With this the proof of Lemma A.4 is complete. References [1] J. C. A. Barata and D. H. U. Marchetti, Griffiths’ singularities in diluted Ising models on the Cayley Tree, J. Stat. Phys. 88 (1997) 231–268. [2] R. B. Griffiths, Nonanalytic behavior above the critical point in a random Ising ferromagnet, Phys. Rev. Lett. 23 (1969) 17–19. [3] J. F. Perez, Controlling the effect of Griffiths’ singularities in random ferromagnets: smoothness of the magnetization, Brazilian J. Phys. 23 (1993) 356–362. [4] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, John Wiley and Sons, second ed., 1971. [5] E. C. Titchmarsh, The Theory of Functions, Oxford Univ. Press, second ed., 1939.
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS OF THE SIMPLEST QUANTUM FILTERING EQUATION VASSILI N. KOLOKOL’TSOV Department of Mathematics, Statistics and Operational Research Nottingham Trent University Burton St., Nottingham, NG1 4BU, UK Received 20 November 1996 Revised 29 September 1997 The paper deals with the quantum Langevin equation describing a quantum particle with continuously observed position. Special Banach spaces of entire analytic functions are introduced and studied (including a theorem of Paley–Wiener type for them), which comprise all solutions of this equation and in which the uniform convergence (as time tends to infinity) of the solutions to the Gaussian function with a fixed dispersion (selflocalisation or continuous collapse) is proved. The asymptotic behavior at infinity of the mean position and momentum of the limit Gaussian wave packet (which satisfy classical Langevin equations) is also investigated.
1. Introduction 1.1. Quantum filtering equation The main equation of the theory of continuous quantum measurement (in the case of a measurement of diffusion type) has the form 1 ? i H + R R χ dt = Rχ dQ , (1.1) dχ + h 2 where χ is the unknown aposterior (non-normalized) wave function of the given continuously observed quantum system in a Hilbert space H, h is the Planck constant, the selfadjoint operator H = H ? in H is the Hamiltonian of a free (non-observed) quantum system, the vector-valued operator R = (R1 , . . . , Rd ) in H stands for the observed physical values, and Q is the standard (input) d-dimensional Brownian motion. In this general form this equation was first obtained by V. P. Belavkin in the framework of his quantum stochastic filtering theory [7, 8]. The Belavkin ˆ (which is observed in the theorem states also that the output diffusion process Q 2 process of measurement) has the density kχk with respect to the standard Wiener measure P of the input process Q, i.e. the mean EQˆ of some functional f (χ) of the ˆ has the form state χ over all realization of the output process Q EQˆ f (χ) = EQ (f (χ)kχk2 ) , 801 Reviews in Mathematical Physics, Vol. 10, No. 6 (1998) 801–828 c World Scientific Publishing Company
(1.2)
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V. N. KOLOKOL’TSOV
where EQ denotes the mean with respect to the Wiener measure P of the process Q. In particular, averaging the mean value hAi = hAiϕ = (Aϕ, ϕ) = (Aχ, χ)/(χ, χ)
(1.3)
of some operator A in H with respect to the normalized wave function ϕ = χ/kχk we have (1.4) EQˆ hAi = EQˆ (Aϕ, ϕ) = EQ (Aϕ, ϕ)(χ, χ) = EQ (Aχ, χ) . Rewriting Eq. (1.1) in terms of the so-called innovating process W , defined by the stochastic equation dW = dQ − 2hRidt , (1.5) and the normalised state vector ϕ = χ/kχk we obtain the equation [7] 1 i ? (H − hhReRiImR) + (R − hReRi) (R − hReRi) ϕ dt dϕ + h 2 = (R − hReRi)ϕ dW ,
(1.6)
where the symmetric operators ReR = (R + R? )/2 and ImR = (R − R? )/2i are the real and imaginary parts of R. For most natural physical examples, the operator R is selfadjoint and (1.1) and (1.6) reduce respectively to the stochastic equations 1 2 i H + R χ dt = Rχ dQ (1.7) dχ + h 2
and dϕ +
1 i H + (R − hRi)2 ϕ dt = (R − hRi)ϕ dW . h 2
(1.8)
A particular case of the last equation, when the Hamiltonian H in (1.8) is equal to zero, was proposed previously by N. Gisin [35], who followed the earlier ideas from [15, pp. 2 and 3]. In the theory of continuous quantum measurement, Eq. (1.6) was discovered in an attempt to avoid the so-called quantum Zeno paradox [28, 56, 16], which appears when trying to describe a continuous measurement as a limit of sequential discrete measurements with the interval of time between the latter tending to zero. In the first deduction of (1.6) [7–9], this equation was obtained as the equation of stochastic filtering applied to the Hudson–Parthasarathy [42] quantum stochastic evolution under the nondirect continuous nondemolition measurement of diffusion type, the nondemolition condition [10, 29] being simply the commutativity relation [X(t), Y (s)] = 0 ∀t ≥ s for the Heisenberg operators of the input and output process X(t) and Y (t) respectively. Another deduction proposed first in [22] for a particular case (1.14) below was based on the previous ideas from [5, 6, 34] and on the so-called master equation, which was obtained from different points of view in [20, 21] and [54] and was used in [5] to solve quantum Zeno’s paradox in the framework of the theory of nonideal measurements. The deduction of (1.14) from the theory of instruments see in [4] and [41]. In recent years there appeared many
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physical papers that present equation (1.6) (or some its modification, where, for instance, dW is considered to be a complex Brownian motion) as a new fundamental model of classical and quantum mechanics (state diffusion model), showing how this model can be deduced from some general physical principles, and discussing possible experimental justifications. Let us point out the papers [38, 39, 57] as more theoretically oriented and papers [40, 30, 37] as more experimentally oriented. In these papers one can find also a complete bibliography. Let us mention specially the papers [36, 33], where the connection with relativity is discussed, and the papers [24, 25, 32] dedicated to the possible applications to quantum gravity. Similar ideas were proposed also in [17, 18, 55]. The discussion of various points of view on Eq. (1.14) can be found in Proceedings [12], see also [61, 3, 53]. The important property of the model (1.1), (1.6) (and also one of the important motivation for its appearence) is the possibility to describe by means of it the process of sponteneous collapse (stochastically continuous selflocalization or reduction) of quantum systems. In fact, if H = 0 in (1.7), then its solution is χ(t) = exp{−tR2 + RQ(t)}χ0 and it tends (continuously collapses), as t → ∞, to the spectral subspace of the operator R2 corresponding to the minimum of its spectrum. In particular, if R2 has nondegenerate lowest eigenvalue, then χ/kχk tends to its lowest eigenfunction. When H 6= 0, the situation is surely more complicated, see discussion and some results in [11, 39, 57, 60]. In the present paper we consider an important nontrivial case of this situation, when, on the one hand, there is a collapse of the form of the solution (dispersion of its position and other central moments of higher orders tend to a fixed limit) and, on the other hand, there is scattering of position and momentum (the latter tend to infinity according to a classical Langevin equation). From the mathematical point of view, the linear Eq. (1.1) is distinguished from the general linear stochastic equation dχ + Aχ dt = Bχ dQ (with some linear operators A, B in a Hilbert space) by its norm conservation property. Namely, applying formally Ito’s formula to the squared norm kχk2 of the solution of (1.1), we get dkχk2 = 2(Rχ, χ) dQ = 2hRikχk2 dQ , which implies (again formally) that Z t Z t 2 2 kχk = exp 2 hRi dQ − 2 hRi dt . 0
(1.9)
(1.90 )
0
Rt Consequently, if for a solution of (1.1) almost surely 0 hRi2 dt is finite (with Rt respect to the measure defined by the Wiener process Q), then 0 hRi dQ is well defined and is a local martingale, which in turn implies that kχk2 is a positive supermartingale and a local martingale. In particular, EQ kχk2 (t) ≤ 1 for all t and EQ kχk2 (min(t, τk )) = 1 for a (so-called localizing) sequence of random Markov moments τk such that almost surely τk → ∞, as k → ∞. If some additional regularity properties are satisfied (see, for instance, [46]), which insure that a positive local martingale is, in fact, a martingale (and a proof of these properties for all physically meaningful situations is an important mathematical problem, whose
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V. N. KOLOKOL’TSOV
importance in the state diffusion model, or in quantum filtering theory, can be compared with that of the proof of the selfadjointness of a formally symmetric operator in the standard quantum mechanics), then EQ kχk2 (t) = 1
(1.10)
for all times t. Therefore, equation of type (1.1) describes natural stochastic generalization of the unitary evolution (it defines an evolution that preserves the expectation of the norm squared of solutions). The measures defined by the input process Q and the innovating process W are connected by the famous Girsanov formula. Namely, the Girsanov theorem states that if (1.10) holds for all times t, i.e. if kχk2 is a positive martingale (with respect to the Wiener measure of the process Q), then the innovating process W is a Wiener martingale with respect to the measure P˜ that has the density kχk2 with respect to the measure P : (1.11) dP˜ = kχk2 dP . Consequently, the mean (1.2) is equal to Z Z EQˆ f (χ) = EQ (f (χ)kχk2 ) = f (χ)kχk2 dP = f (χ)dP˜ = EW f (χ) .
(1.12)
In the sequel, when we speak that some property of the measured quantum process is satisfied for almost all (a.a.) Q, we mean the measure P , i.e. we consider the process Q to be the standard Brownian motion, and when we speak that some property is satisfied for a.a. W , we mean the measure P˜ , i.e. we consider the process W to be the standard Brownian motion. Let us stress that the difference in these two notions is quite essential, because although the measures generated by processes W and Q are equivalent on any finite interval of time (if (1.10) holds), they are not equivalent as measures defined on the space of continuous functions on [0, ∞) (see [52]). Due to the equation EQˆ = EW , the consideration of the innovating process is equivalent to the consideration of the output process, and consequently, from the point of view of the theory of measurement, all results should be formulated in terms of W (so, the process Q and linear equation (1.1) play only auxiliary role). 1.2. Plan of the paper and results This paper is mostly devoted to the investigation of the Belavkin equation dχ +
1 −ih∆ + x2 χ dt = xχ dQ , 2
(1.13)
describing the evolution of a “free” quantum particle with continuously observed coordinate. Equation (1.13) is a particular case of (1.7), where H = L2 (R), H = ˆ is the operator of the multiplication on x. As we have already −h2 ∆/2 and R = x mentioned, the corresponding normalized equation of form (1.8) dϕ +
1 −ih∆ + (x − hˆ xi)2 ϕ dt = (x − hˆ xi)ϕ dW , 2
(1.14)
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for this particular case was obtained also by L. Diosi [22] (in the same year that Belavkin’s paper [7] appeared with the general equations (1.1), (1.6)). In [22], L. Diosi has also written the system for the mean coordinate hˆ xi and momentum hˆ pi of the wave functions satisfying (1.14): (
xi2 ) dW , dhˆ xi = hˆ pi dt + 2(hˆ x2 i − hˆ dhˆ pi = (hˆ pxˆ + x ˆpˆi − 2hˆ xihˆ pi) dW .
(1.15)
These are classical Langevin equations for a free Newton particle disturbed by a random martingale force. Having this in mind, L. Diosi has referred to Eq. (1.14) as the quantum Langevin equation. For simplicity, we consider in Secs. 1–4 one-dimensional x, although all results and proofs hold for any dimensions. Only for the results of the last section the dimension is essential, which will be discussed therein. It turns out that analytic properties of the solution of the Cauchy problem for (1.13) are similar to those of the equation of the oscillator diffusion process ∂χ = ∂t
h∆ mΩ2 2 − x χ 2m 2h
(1.16)
with complex mass m and frequency Ω such that Re(mΩ) > 0 and ReΩ > 0. In Sec. 2 we describe in detail these analytic properties. For this purpose, we prove a theorem of Paley–Wiener type which seems to be of independent interest. Namely, we introduce a family of spaces of analytic functions (belonging to the Schwartz space S) that is invariant under Fourier transform (the classical Paley– Wiener duality between finite functions and their analytic Fourier transforms breaks the nice invariance property of the Schwartz space) and which comprises all solutions of (1.14) and (1.16). In Sec. 3 we prove the crucial asymptotic property of Eq. (1.14), namely, the uniform convergence (continuous collapse) in the spaces of entire analytic functions introduced in Sec. 2 of any solution to a Gaussian form with some fixed finite dispersion, as the time tends to infinity and for a.a. realizations of the innovating Wiener process W . In Sec. 4 we sketch another proof of this result based on improved arguments from [52], where this convergence was proved in L2 -norm (in fact, with a small error: roughly speaking, Lemma 4.1, see Sec. 4 below, was absent in [52]). The proven convergence property implies, in particular, that the solutions of (1.14) are not spreading at infinity (do not tend to zero in the uniform topology, as t → ∞), as are the solutions of the classical free Schr¨ odinger equation, which gives an explanation (in the framework of the stochastic theory of continuous quantum measurement) of the so-called watchdog effect (see, for instance, physical discussion in [14]). Moreover, it implies that the limit of the dispersion of the position for the wave functions solving (1.14) exists for a.a. realisations of the innovating process W (and does not depend on the initial function). This limit was called in [49, 50] the coefficient of the quality of measurement (the finiteness of this coefficient stands for the watchdog effect, and its positivity stands for the uncertainty principle).
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V. N. KOLOKOL’TSOV
In Sec. 5, we study the dynamics (1.15) of the means of position and momentum of the solution. We prove that for dimensions more than 2 the means of the position and of the momentum tend to infinity and give the estimates of its growth. This result allows one to speak about the (some special sort of) scattering described by dynamics (1.14). On the other hand, this result give interesting information about the behavior of the Brownian motion at infinity and seems to be interesting by itself. Let us note now that the well-posedness theorem for a natural generalization of Eq. (1.14) (when a bounded deterministic potential is also present) was given in [51, 52], and the ergodic properties of finite dimensional Eq. (1.1) (describing continuously observed spin systems) was investigated in [49, 50]. Some applications of the latter results in the study of computer graphic systems appeared afterwards in [44, 45]. 1.3. Local properties of the quantum Langevin equation We discuss here some known properties of Eqs. (1.13) and (1.16) that we need further. Proposition 1.1. For any χ0 ∈ L2 (R) there exists a unique solution χ of equation (1.13), which tends to χ0 in L2 , as t → 0. Moreover, kχk2 is a positive martingale (and therefore, the fundamental Eqs. (1.10) and (1.12) hold). This was proved by different methods in [31], and [13, 48] (see also [52]). In the paper [13], the explicit integral representation for the solution was also given (which allows one, in particular, to construct its solutions for rather general initial data). Namely, the following result holds. Proposition 1.2. The solution G(t, x, ξ) of (1.13) with initial data G(0, x, ξ) = δ(x − ξ) ,
(1.17)
(i.e. the Green function of the Cauchy problem for (1.13)) exists and has the form o n ω G (x2 + ξ 2 ) + βG xξ + aG x + bG ξ + γG , (1.18) G = CG exp − 2 where the coefficients CG , ωG , βG are deterministic (they do not depend on Q) and are equal to −1 −1/2 2t 2t 2t 2π sinh ωG = α coth , βG = α sinh , CG = , (1.19) α α α α and other coefficients are −1 Z t 2t 2τ dQ(τ ) , aG = sinh sinh α α 0 Z bG = ihα 0
t
aG (τ ) dτ , sinh(2τ /α)
γG =
ih 2
Z
t
a2G (τ ) dτ . 0
(1.20)
(1.21)
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There are two natural ways to obtain this result, one of them (presented in [13, 52],) is based on the investigation of Gaussian solutions of Eq. (1.13) (see below) and another being an application of the stochastic WKB method developed simultaneously by A. Truman, H. Zhao in [62, 63] (following some previous ideas from [27]) and by the author in [51] (following some ideas from [13]). Let us point out that these two approaches can be also used to obtain fifth and sixth proofs of the Mehler formula (see formula (1.35) below) for the Green function of the oscillator Eq. (1.16), in addition to the four methods of the proof described in [19]. An important role in the theory of Eqs. (1.13) and (1.14) is played by the Gaussian functions of the form ω i ω 2 (1.22) χ = cgq,p = c exp − (x − q) + px 2 h with some real q and p (being respectively the mean values of the operators R = x ˆ of the multiplication on the coordinate x and of the momentum operator pˆ = −ih∂/∂x on the Gaussian function (1.22)) and complex ω and c such that c 6= 0 and Reω > 0. It is easy to derive (see, [31, 14, 52]) the following Proposition 1.3. If the initial function for the Cauchy problem of Eq. (1.13) has the Gaussian form (1.22), then the solution will also have the Gaussian form (1.22), whose parameters will satisfy the system of ordinary stochastic differential equations dω = (2 − ihω 2 ) dt , 1 2q dt + dQ , dq = p − Reω (Reω
(1.23) (1.24)
Imω (2q dt − dQ) , Reω " ! 2 c 2 i Imω Imω dc = − q 1 + + p2 + 2i 2 Reω Reω h
dp = h
−ω
1 − ih (Reω)2
dt + cq
ω dQ . Reω
(1.25)
(1.26)
Remark. Similar results for the equation 1 dχ + (−ih∆ + x2 )χ dt = ixχ dQ 2 (which differs from (1.13) by the coefficient i in the r.h.s. and which defines with probability one a unitary evolution) was obtained in [62]. In terms of the complex variable z = ωq + ip/h Eqs. (1.24) and (1.25) can be rewritten in the equivalent form dz = −ihωzdt + dQ .
(1.27)
808
V. N. KOLOKOL’TSOV
Equations (1.23) and (1.27) can be easily solved explicitly: ω0 + α tanh(2t/α) , ω0 tanh(2t/α) + α Z t Z τ Z t ω(s)ds} z|t=0 + exp{ih ω(s) ds}dQ(τ ) . z = exp{−ih ω(t) = α
0
0
(1.28) (1.29)
0
In particular, for any initial ω0 with Reω0 > 0, the solution of Eq. (1.23) tends to the same limit (1.30) lim ω(t) = α = h−1/2 (1 − i) , t→∞
and therefore the limits of the dispersions Dxˆ = (2Reω)−1 ,
Dpˆ = h2 |ω|2 (2Reω)−1
of the coordinate and momentum respectively for any solution of form (1.22) are given by formulas √ (1.31) lim Dxˆ = h/2 , lim Dpˆ = h3/2 . t→∞
t→∞
Remark. Approximating the Dirac δ-function by Gaussian functions χ0 and taking limit of the corresponding Gaussian solutions χ , one can obtain Proposition 1.2. At the end of the introduction we collect (for the convenience of the future references) some well-known properties (see, for instance, [19]) of the operator H =−
mΩ2 x2 h2 ∆+ . 2m 2
(1.32)
Note however that in the standard textbooks these properties are given for the case of positive mass m and frequency Ω (when in particular, the operator H is positive and selfadjoint) and we need them in a slightly more general situation, for which these properties are still valid (though H is not more selfadjoint). Proposition 1.4. Let h > 0 and let Ω, m be complex constants such that ReΩ > 0 and Re(Ωm) > 0. Then (i) the operator (1.32) has discrete spectrum {λn = hΩ(n + 12 ), n = 0, 1 . . . , } (ii) the corresponding eigenfunctions are √ 1 ω 1/4 −ωx2 /2 e Hn ( ωx) , ψn (x) = √ n π 2 n!
(1.33)
where ω = mΩ/h and Hn are the Hermite polynomials, defined by the equation n 2 d 2 Hn (y) = (−1)n ey e−y , (1.34) n dy
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
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(iii) the operator exp{−tH/h} is a compact integral operator with the integral kernel s mΩ mΩ exp − ((x2 + y 2 ) cosh(Ωt) − 2xy) . (1.35) 2πh sinh(Ωt) 2h sinh(Ωt) 2. Analytic Properties of the Solutions of Equations (1.13) and (1.16) and a Paley Wiener Type Theorem The aim of this section is to describe natural spaces of analytic functions, which comprise the solutions of the quantum Langevin equation (1.13). In order first to make clear the main ideas, we start with more simple Eq. (1.16) with real positive frequency Ω and mass m. Definition 1. Let A and B be real numbers such that A + B ≥ 0. We denote by SA,B the space of entire analytic functions f such that 1 2 1 2 Ax + By |f (x + iy)| ≤ C exp 2 2 for some positive constant C and all x, y. It is easy to see that the infimum of those C for which the latter inequality holds, defines a norm on SA,B , and that SA,B is a Banach space with respect to this norm. Note also, that the restriction A + B ≥ 0 in the definition is essential, because for A + B < 0 the corresponding space SA,B would comprise only identically vanishing functions (which is not difficult to prove). Our first statement describes the action of the Fourier transform Φ on the family of spaces SA,B . It turns out that though the space of analytical functions of the classical Paley–Wiener theorem (which are the Fourier images of finite smooth functions) are not invariant with respect to Φ, the family of spaces SA,B is invariant with respect to Φ, as is its comprising Schwartz space S. Propositon 2.1. For any B > A > 0, the Fourier transform Φ (as well as its inverse Φ−1 ) is an isomorphism of the Banach spaces S−A,B 7→ S−B −1 ,A−1 with the norm A−1/2 . In particular, S−A,A−1 is invariant under the action of Φ, and Φ is a norm preserving isomorphism of the space S−1,1 . Proof. Let B > A > 0 and f ∈ S−A,B . It is clear that Φf is an entire analytic function. Further, Z 1 e−i(µ+iλ)(x+iy) f (x + iy) dx (Φf )(µ + iλ) = √ 2π for any y ∈ R (due to the Cauchy theorem). It follows that Z A 2 B 2 1 √ dx exp µy + λx − x + y |(Φf )(µ + iλ)| ≤ 2 2 2π 2 B 2 λ −1/2 =A + µy + y . exp 2A 2
810
V. N. KOLOKOL’TSOV
Taking minimum over all y ∈ R we obtain
1 2 1 2 µ + λ . |(Φf )(µ + iλ)| ≤ A−1/2 exp − 2B 2A
We have proved that Φ takes S−A,B in S−B −1 ,A−1 and its norm does not exceed A−1/2 . Let us now consider the function f (z) = (1 − )e−Az
2
/2
+ e−Bz
2
/2
,
z = x + iy ,
which belongs to S−A,B and has the norm equal to one for any ∈ (0, 1). Clearly 2 2 1− Φf (z) = √ e−z /2A + √ e−z /2B A B
belongs to S−B −1 ,A−1 and has the norm
1− √ + √ . A B −1/2
conclude that the norm of Φ is equal to A
Since ∈ (0, 1) is arbitrary, we
, which completes the proof.
ˆ ω denote the integral operator in L2 (R) with the Now, let ω > β ≥ 0 and let K β integral kernel n ω o Kβω (x, ξ) = exp − (x2 + ξ 2 ) + βxξ . (2.1) 2 ˆω In particular, the resolving operator for the Cauchy problem of Eq. (1.16) is K β
(up to a constant coefficient) with ω = coth(Ωt) and β = sinh−1 (Ωt). It turns out ˆ ω almost coincides (up to a “subtle fiber”) with some SA,B . that the image of K β Namely: ˆ ω is a continuous linear Proposition 2.2. For any p ≥ 1, the operator K β 2π 1/2q mapping Lp 7→ S−(ω− β2 ),ω with the norm that does not exceed ( qω ) (respectively, ω
1) for p > 1, where p−1 + q −1 = 1 (respectively, for p = 1). Further, for any real 2 ˆ ω (L∞ ) a, b such that ω − βω ≤ a ≤ b < ω, the space S−a,b belongs to the image of K β and there exists a continuous inverse operator S−a,b 7→ S−(
β2 β2 ω−a −ω), ω−b −ω
.
Proof. The first part of the theorem is a direct consequence of the H¨ older ˆ ω in the product of four inequality and to prove the second part we decompose K β operators ˆ ω = M−ωx2 /2 ◦ Liβ ◦ Φ ◦ M−ωx2 /2 , K β where Mf (x) denotes the multiplication on exp f (x), and Liβ f (x) = f (iβx) is the rotation with dilatation in the complex plane. Then we use Proposition 1 and the trivial remarks that M−ωx2 /2 is a (norm preserving) isomorphism SA,B 7→ SA−ω,B+ω and Liβ is a (also norm preserving) isomorphism SA,B 7→ Sβ 2 B,β 2 A . ˆ ω not only on Lp but also on other It is natural to consider the action of K β functions that do not increase very fast at infinity. Therefore, analogously to the previous result we obtain the following
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Proposition 2.3. For any A, B such that A + B ≥ 0 and A < ω, the operator ˆ ω is an isomorphism K β SA,B 7→ S β2 −ω,ω− β2 , ω−A
B+ω
ˆ ω )−1 is and conversely, for any a, b such that a + b ≥ 0 and b < ω, the operator (K β a continuous mapping Sa,b 7→ Sω− β2 , β2 −ω . ω+a ω−b
We proceed now to the case of the resolvent operator for the Cauchy problem of Eq. (1.16) with complex frequency or mass. More precisely, we shall consider ˆ ω , where constants β, ω should be complex with the only the integral operator K β condition Reω > |Reβ|. To consider that case one should introduce a generalization of the spaces SA,B . Moreover, it is convenient (in order to obtain a simple formula for the linear change of variables) to reparametrize them also. Definition 2. Let µ > 0 and let Γ, a be any complex constants. We denote by Sµ,Γ,a the space of entire analytic functions f (z), z = x + iy, such that µ 2 1 |z| − Re(Γz) + Re(az) (2.2) |f (z)| ≤ exp 2 2 for all z, or equivalently, µ − ReΓ 2 µ + ReΓ 2 x + y + xyImΓ + xRea − yIma . |f (x + iy)| ≤ exp 2 2 Clearly SA,B = S B+A , B−A ,0 , so that the spaces SA,B are the particular cases of 2 2 Sα,Γ,a corresponding to a = ImΓ = 0. We write down first the generalization of Proposition 2.1. Proposition 2.10 . If µ < ReΓ, the Fourier transform Φ is an isomorphism Φ : Sµ,Γ,a 7→ SµD−1 ,ΓD −1 ¯ −1 ,−i(µ¯ ¯ a+Γa)D with the norm a2 ))} , (ReΓ − µ)−1/2 exp{(2D)−1 (|a|2 + Re(Γ¯ where D = |Γ|2 − µ2 . Proof. It is quite similar to Proposition 2.1 and we drop it. We now give the law of the transformation of the spaces Sµ,Γ,a under the linear change of variables. Simple calculations give the following Lemma 2.1. The change of variable Lβ f (z) = f (βz) with a complex β is a norm preserving isomorphism Sµ,Γ,a 7→ Sµ|β|2 ,Γβ 2 ,aβ , and the shift (Tq f )(z) = f (z + q) is an isomorphism Sµ,Γ,a 7→ Sµ,Γ,a+µ¯q −Γq with the norm µ 2 1 |q| − Re(Γq 2 ) + Re(aq) . exp 2 2
812
V. N. KOLOKOL’TSOV
Now we obtain the generalization of Propositions 2.2 and 2.3 on the case of the resolving operator of the Cauchy problem for general Eq. (1.16) and also for (our main object of investigation) Eq. (1.13). Proposition 2.20 . Let Reω > |Reβ| and let a, b, γ be any complex constants. Then the integral operator with the kernel n ω o exp − (x2 + ξ 2 ) + βxξ + ax + bξ (2.3) 2 is a continuous (injective) linear mapping Lp 7→ S |β|2
β2 Reb 2Reω ,ω− 2Reω ,a+ Reω β
whose norm does not exceed
2π qReω
1/2q
exp
(Reb)2 2Reω
and
exp
(Reb)2 2Reω
for p > 1 and p = 1 respectively. Proposition 2.30 . Let the assumptions of the previous Proposition hold. If Re(Γ + ω) > µ, then the integral operator with the integral kernel (2.3) is a bounded linear operator Sµ,Γ,γ 7→ S µ|β|2 ,(Γ+¯ ¯ ω ) β 2 +ω,β(µ(¯ ¯ ω ))D −1 +a , γ +¯ b)+(γ+b)(Γ+¯ D
D
where D = |Γ + ω|2 − µ2 . On the other hand, the inverse operator is defined when Re((ω − Γ)β¯2 ) > µ|β|, as a continuous operator Sµ,Γ,γ 7→ S
¯ ω ¯ ω µ γ ¯ −¯ a µ ¯ Γ− ¯ γ−a ,− Γ− ¯ + β ¯2 D −ω, |β|2 β ¯2 β −b β |β|2 D
,
where D = (|Γ − ω|2 − µ2 )|β|−4 . Proof. The proof of both these Propositions is the same as that of Propositions 2.2 and 2.3: one should represent our integral operator in the product of multiplications operators, change of variable and the Fourier transform, then use Proposition 2.10 and Lemma 2.1. We conclude this section with the following simple Proposition 2.4. For any complex Γ, γ, the space S0,Γ,γ is one-dimensional and is generated by the function exp{γz − Γz 2 /2}. Proof. In fact, if f ∈ S0,Γ,γ , then f (z) exp{−γz + Γz 2 /2} is a bounded entire analytic function and thus it is a constant.
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3. The Convergence of the Solutions of the Quantum Langevin Equation to the Gaussian Form ˜ −γt ) denote a function that is of We now formulate the main theorems. Let O(e −t order O(e ) for any < γ, as t → ∞. We use here (as previously) the notation (1.22) for Gaussian functions and the letter α for the complex constant h−1/2 (1 − i). Moreover, Mf and Tq will denote respectively the operator of the multiplication on the function exp f and the shift Tq f (x) = f (x + q). The symbols k.k and k.kµ,Γ,a will denote respectively the L2 -norm of a function and its norm in the Banach space Sµ,Γ,a introduced above. Theorem 3.1. Let ϕ be the solution of the Cauchy problem for Eq. (1.14) with any initial function ϕ0 ∈ L2 , kϕ0 k = 1. Then for a.a. trajectories of the innovating Wiener process W, √ √ α ˜ − ht ) , (3.1) kϕ − (π h)1/4 ghˆ ˆ ϕ k = O(e xiϕ ,hpi as t → ∞, and moreover, there exist real constants q˜(W ), p˜(W ) such that √ √ Rt ˜ − ht ) hˆ xiϕ = q˜(W ) + p˜(W )t + hW + h 0 W (s) ds + O(e . √ ˜ − ht ) hˆ piϕ = p˜(W ) + hW + O(e
(3.2)
Theorem 3.2. With the assumptions of Theorem 3.1, the asymptotic formula (3.1) for solutions of Eq. (1.14) is still valid if instead of the L2 -norm we put there the norm of the uniform convergence in the space C k of k times continuously differentiable functions (k is arbitrary). Moreover, the same is true for the spaces of entire functions introduced in the previous section, if we centralize the position of the limit Gaussian wave packet. More precisely, for any positive √ √ 1/4 α ˜ − ht ) k ghˆxiϕ ,hpi = O(e (3.10 ) kM−ihpi ,α,0 ˆ ϕ x/h Thˆ xiϕ ϕ − (π h) ˆϕ for a.a. W as t → ∞. This section is devoted to the complete proof of these theorems. In the next section we sketch another proof. The plan of the proof is the following. First of all, due to Proposition 1.2, we represent the resolving operator for the Cauchy problem (1.13) as the composition ˆ ωG ◦ MbG x , CG exp γG MaG x ◦ K βG
(3.3)
where all the coefficients with the index G are given by formulas (1.19)–(1.21) and ˆ ωG denotes the integral operator with the kernel of the form (2.1). the operator K βG ˆ ωG = exp{− t H}, where Therefore, due to Proposition 1.4, CG K βG h H=h
h ∆ + x2 2i
,
(3.30 )
814
V. N. KOLOKOL’TSOV
√ is the operator of form (1.32) with Ω = (1 + i) h = 2/α and m = −i. Equivalently, ˆ ωG is the resolving operator of the Cauchy problem for Eq. (1.16) with the CG K βG same Ω, m. The crucial step in our proof of the theorem will be the statement that the real coefficients a1 , a2 defined by the equation aG = ω G a1 +
i a2 h
(3.4)
have asymptotics (3.2) with some real q˜(W ), p˜(W ). It will imply almost immediately that the coefficient bG has a finite limit b∞ = b∞ (W ) for a.a. W . Next important step is to prove that the projection of the function Mb∞ x ϕ0 on the first eigenfunction ψ0 (x) =
α 1/4 π
n α o exp − x2 2
(3.5)
of the operator (3.30 ) should not vanish for a.a. W . It implies that exp{−tH/h} MbG x ϕ0 tends asymptotically to the Gaussian function (3.5). Using the fact that aG does not tend to infinity very fast, one shows at last that the application of the multiplication operator MaG x reduces asymptotically to the desired shift of (3.5). We proceed now with details. Note first that due to the result of the previous section, we can and will consider ϕ0 to belong to some space SA,B . In particular, |ϕ0 (x)| ≤ C exp{−Ax2 /2}
(3.6)
for all real x and some positive constants A, C. Lemma 3.1. For a.a. W, f (t) ≡ hˆ xiϕ −
√ ReaG = O(e−t h ) . ReωG
(3.7)
=ReaG is the mean coordinate of the Green function Remark. Note that a1Reω G (1.18) for ξ = 0, i.e. the lemma states that the difference between the mean coordinate of each solution of Eq. (1.14) and the mean coordinate of the Green function vanishes rapidly at infinity almost surely with respect to the measure defined by the innovating process W .
R Proof. Let χ(t, x) = G(t, x, ξ)ϕ0 (ξ) dξ be the solution of Eq. (1.13) with initial function χ(0, x) = ϕ0 (x). Let us recall that then ϕ = χ/kχk and (χ, xχ) = hˆ xiϕ (χ, χ). Therefore we obtain by direct calculations that f (t)kχk2 is equal to Z r 2(ReaG )2 βG ξ + β¯G η π dξdηϕ0 (ξ)ϕ¯0 (η) |CG |2 exp 2ReγG + ReωG ReωG 2ReωG 2 2 1 βG 1 β¯G |βG |2 2 2 ωG − ω ¯G − ξ − η + × exp − ξη 2 ReωG 2 ReωG ReωG βG ReaG β¯G ReaG ¯ ξ + bG + 2 η . × exp + bG + 2 ReωG ReωG
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
...
815
The inequality |ξη| ≤ 12 (ξ 2 + η 2 ) implies that |f (t)|kχk2 does not exceed r 2(ReaG )2 π 2 exp 2ReγG + |CG | ReωG ReωG Z |βG |(|ξ| + |η|) g(ξ)g(η) , × dξdη|ϕ0 (ξ)||ϕ0 (η)| 2ReωG where
2ReβG 2ReβG ReaG 2 u + RebG + u . g(u) = exp − ReωG − ReωG ReωG
(3.8)
Estimating now g(ξ)g(η) ≤ 12 (g 2 (ξ) + g 2 (η)) and then using the symmetry between ξ and η we get r 2(ReaG )2 π exp 2ReγG + |f (t)|kχk2 ≤ |CG |2 ReωG ReωG Z |βG |(|ξ| + |η|) 2 g (ξ) . (3.9) × dξdη|ϕ0 (ξ)||ϕ0 (η)| 2ReωG The key point in the proof is the remark that for any ξ Z MarQ (ξ) = |G(t, x, ξ)|2 dx is a positive martingale with respect to the measure defined by the (input) Brownian motion Q (more precisely, it is a martingale for t ≥ t0 for any t0 > 0). It follows from Proposition 1.1. A simple calculation gives r 2(ReaG )2 π g 2 (ξ) exp 2ReγG + MarQ (ξ) = |CG |2 ReωG ReωG and therefore, (3.9) can be rewritten as Z |βG |(|ξ| + |η|) 2 |f (t)|kχk ≤ dξdη|ϕ0 (ξ)||ϕ0 (η)| MarQ (ξ) . 2ReωG
(3.10)
Due to (3.6), we conclude that Z |f (t)|kχk ≤ C|βG | 2
e−ξ
2
/
MarQ (ξ) dξ
(3.11)
for > 0, C > 0. As the integral in the r.h.s. of (3.11) is obviously also some positive martingale, say MarQ , with respect to Q, we have |f (t)| ≤ |βG |
MarQ . kχk2
By the theorem on the transformation of the martingale property by means of the Girsanov transformation (see, for instance [46]), we conclude that MarW = MarQ /kχk2 is a positive martingale with respect to the Wiener measure defined
816
V. N. KOLOKOL’TSOV
by the innovating process W (as MarQ was a martingale with respect to Q), and therefore, due to the Doob convergence theorem, for a.a. W the limit exists of MarW as t → ∞ and this limit is integrable with respect to the Wiener measure of the process W . In particular, MarW is bounded for a.a. W . It implies that |f (t)| = O(|βG |) and (3.7) follows at last from (1.19). Lemma 3.2. The real coefficients a1 , a2 defined by (3.4) satisfy (3.2), i.e. for a.a. W, there exist q˜(W ), p˜(W ) such that √ √ R ˜ − ht ) a1 = q˜(W ) + p˜(W )t + hW + h 0t W (s) ds + O(e . (3.12) √ ˜ − ht ) a2 = p˜(W ) + hW + O(e Proof. Since a1 and a2 are the mean position and momentum respectively of the Gaussian solution G(t, x, 0) (see (1.18) of Eq. (1.13), it follows from Proposition 1.3 that they satisfy Eqs. (1.24) and (1.25) (it can be also directly verified from the explicit formula (1.20) for aG ). Let us rewrite these equations in terms of the innovating process W : ( da1 = (a2 + 2(ReωG )−1 f (t)) dt + (ReωG )−1 dW , (3.13) da2 = −h(ImωG )(ReωG )−1 (2f (t) dt + dW ) where f (t) is defined in (3.7). Due to (3.7) and to the exponentially fast convergence of ωG to α, as t → ∞, we first obtain the second equation in (3.12) from the second equation in (3.13). Then, substituting it in the first Eq. (3.13), we obtain the first Eq. (3.12). Lemma 3.3. The coefficient bG in (1.21) tends to some finite limit b∞ for a.a. W, as t → ∞, and moreover, (also almost surely) ˜ − bG = b∞ + O(e
√ ht
),
˜ kebG x ϕ0 − eb∞ x ϕ0 k = O(e
√
(3.14) ht
).
(3.15)
Proof. Due to (3.12), aG = O(t2 ) for a.a. W , as t → ∞, and therefore (3.14) follows from (1.21). In his turn, (3.15) follows from (3.14) and (3.6). We shall show now that if a function a(t) does not tend to infinity very fast (as t → ∞), then the application of the operator Ta Max does not “spoil” the asymptotic structure of the image of the operator exp{−tH/h} with H of the form (1.32). P Lemma 3.4. Let ψ ∈ L2 (R) and let ψ = µn ψn be its Fourier decomposition with respect to the basis of the eigenfunctions (1.33) of the operator (1.32) such that the assumptions of Proposition 1.4 hold. Let a = a(t) be any real function on time t ≥ 0 such that |a(t)| ≤ Ctκ for some positive C, κ. Then ∞ X 2 t Ta Maωx exp − H ψ = ea ω/2 e−Ωt(k+1/2) µ ˜k ψk , (3.16) h k=0
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
...
817
where ˜ −Ωt )ek/2 µ ˜k = µk + O(e
(3.17)
˜ −Ωt ) depends only on C, κ, kψk. and the function O(e Proof. We have t Maωx exp − H ψ (x) h =
∞ π 1/4 X
ω
√
n=1
√ 2 2 1 e−Ωt(n+1/2) ea ω/2 µn Hn ( ωx)e−ω(x−a) /2 . 2n n!
Due to the well-known formula Hn0 = 2nHn−1 for the Hermite polynomials (1.34), we also have n X n! Hk (y) , 2n−k bn−k Hn (y + b) = k!(n − k)! k=0
and therefore r n ∞ X X √ t n! ψk a2 ω/2 −Ωt(n+1/2) n−k e µn ( 2ωa) Ta Maωx exp − H ψ = e h k! (n − k)! n=0
k=0
=e
=e
a2 ω/2
a2 ω/2
√ ∞ ∞ X ψk X −Ωt(n+1/2) n! √ √ ( 2ωa)n−k e µn (n − k)! k! k=0 n=k ∞ X
e−Ωt(k+1/2) µ ˜k ψk
(3.18)
k=0
with
∞ 1 X −Ωnt e µ ˜k = √ k! n=0
p (n + k)! √ ( 2ωa)n µn+k , n!
or equivalently µ ˜ k = µk + e
−Ωt
∞ X √ √ 1 1 + k 2ωaµk+1 + √ e−Ωt δn k! n=2
p (n + k)! µn+k , n!
(3.19)
where
√ δn = e−Ω(n−1)t ( 2ωa)n . √ We choose now t0 such that e−Ωt/2 2ω|a| < 1 for all t > t0 . Then δn < 1 for all P t > t0 and n ≥ 2. Noting that kψk2 = |µn |2 we obtain consequently that the last term in (3.19) does not exceed in magnitude v u∞ u X (n + k)! 1 √ kψke−Ωt t . k!(n!)2 k! n=2
(3.20)
818
V. N. KOLOKOL’TSOV
Due to the well-known Stirling formula, there exists a constant C1 such that √ √ C1−1 2πnnn e−n < n! < C1 2πnnn e−n for all n > 1. This implies that p 2π(n + k) exp{(n + k) log(n + k) − (n + k)} (n + k)! 4 √ . < C1 k!(n!)2 2πn 2πk exp{k log k − k + 2n log n − 2n}
(3.21)
Estimating here the r.h.s. by means of the trivial inequality (n + k) log(n + k) ≤ n log n + k log k + n + k , we obtain that (3.20) does not exceed in magnitude C2 kψke−Ωt ek/2 for some con˜k follows. stant C2 . Now from (3.19) the desired estimate (3.17) for µ The last important step in the proof of the theorem is the following. Lemma 3.5. For a.a. W, the function eb∞ x ϕ0 (x) has a nonvanishing projection on the Gaussian function ψ0 . Proof. If the assertion of the lemma does not hold, then, using (3.3), (3.15), and applying Lemma 3.4 in the case of operator (3.30 ) we conclude that the following estimate holds for the norm of the solution of (1.13) with the initial function χ0 = ϕ0 : ( √ ) √ h α − ht 2 ˜ a + Reγ − t . (3.22) ) exp kχk = O(e 2 1 2 Now we use the same trick as at the end of the proof of Lemma 3.1. Namely, the squared norm of the solution G(t, x, 0) (see (1.18)) is a positive martingale with respect to the measure of the (input) Wiener process Q. We denote it by MarQ . Then by the martingale transformation theorem we conclude that MarW = MarQ /kχk2 is a martingale with respect to the innovating process W and thus it should be bounded for a.a. W . But the latter property contradicts (3.22) and the √ obvious remark that MarQ is of order exp{αa21 + 2Reγ − ht}. Proof of Theorem 3.1. It follows directly from formula (3.3), Lemmas 3.3 and 3.5, and Lemma 3.4 applied to operator (3.30 ). Proof of Theorem 3.2. We prove only (3.10 ). Let −1 −Reγ−ωG a1 /2 χ(t) ˜ = CG e Ta1 M−ia2 x/h χ(t) . 2
Then
Z χ ˜=
o n ω G (x2 + ξ 2 ) + βG xξ + (βG a1 + bG )ξ χo (ξ) dξ , exp − 2
where, due to Lemmas 3.2 and 3.4, and formula (1.19), ˜ − βG a1 + bG = b∞ + O(e
√ ht
).
(3.23)
(3.24)
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
...
819
Using the trivial inequality |eν2 − eν1 | ≤ |eν1 ||ν2 − ν1 |e|ν2 −ν1 |
(3.25)
(which holds for any complex ν1 , ν2 ), we can estimate (for t > τ ) |(χ(t) ˜ − χ(τ ˜ ))(x + iy)| Z ωG (τ ) ((x + iy)2 + ξ 2 ) + β(τ )(x + iy)ξ + b∞ ξ ≤ exp Re − 2 ˜ − × O(e
√ ht
˜ − ) exp{O(e
√
ht
)(x2 + y 2 + ξ 2 + 1)}|χ0 (ξ)| dξ ,
which implies (due to Proposition 2.20 , or by direct application of H¨ older inequality) that χ(t) ˜ is a Cauchy family, as t → ∞, in the space S,α,0 for any > 0. Therefore, χ(t) ˜ tends in S,α,0 to some (automatically also entire) function F . Further, due ˜ ∈ Sg(t),α,0 with a function g(t) of order to Lemma 2.1 and Proposition 2.20 , χ(t) √ − ht ˜ ), and moreover, the set of norms kχ(t)k ˜ O(e g(t),α,0 is bounded. This obviously implies that the limit F belongs to S0,α,0 and consequently, due to Proposition 2.4, F = A exp{−α(x + iy)2 /2} with some constant A. Noting now that the solution of (1.14) is equal to ˜, ϕ = CG eReγ+ωa1 /2 kχk−1 χ 2
and (as it was shown in the proof of Lemma 3.5) the coefficient before χ ˜ in this formula is bounded, we conclude that A 6= 0 (otherwise, 1 = kϕk would tend to zero), which implies the assertion of the theorem. 4. A Sketch of Another Proof of Theorem 3.2 Due to the results of Sec. 2, we can consider the initial function for the Cauchy problem of Eqs. (1.13) or (1.14) to belong to some space SA,B . In particular, we can present it in the form of “infinite” linear combination of Gaussian functions 2 Z i x (4.1) χ0 = ϕ0 = c(µ) exp − + µx dµ h with some > 0 and some function c(µ) also belonging to some SA,B so that the integral Z 2 (4.2) c(µ)2 eAµ dµ converges for some A > 0. Propositions 1.1 and 1.3 imply that the solution of (1.13) with such initial function can be represented in the form Z i ω(t) 2 (x − q(µ, t)) + p(µ, t)x dµ , (4.3) χ = c(µ, t) exp − 2 h where ω, q, p, c satisfy (1.23)–(1.26) with initial values ω0 = 2−1 ,
c(µ, 0) = c(µ) ,
p(µ, 0) = µ ,
q(µ, 0) = 0 .
(4.4)
820
V. N. KOLOKOL’TSOV
The first step in the second proof of Theorem 3.1 (and just this step was missing in the original exposition of this proof in [52]) is the following. Lemma 4.1. For a.a. W and for any µ0 ˜ − |hˆ xiϕ − q(µ0 , t)| = (|µ0 | + 1)O(e
√ ht
).
Proof. We calculate r Z |zµν (t)|2 π 2 c(µ, t)¯ c(ν, t) exp − kχk = Reω 4Reω(t) i (p(µ, t) − p(ν, t))(q(µ, t) + q(ν, t)) , + 2h where
(4.5)
i zµν (t) = ω(t) q(µ, t) − q(ν, t)) + (p(µ, t) − p(ν, t) . h
Due to (1.29)), zµν (t) =
(4.6)
Z t i (µ − ν) exp −ih ω(s) ds . h 0
(4.7)
By (1.28), we conclude that ˜ e− |zµν (t)| = |µ − ν|O
√ ht
.
(4.8)
Calculating analogously hˆ xiϕ = (χ, xχ) we obtain (hˆ xiϕ − q(µ0 , t))kχk2 Z r 1 π dµdν c(µ, t)¯ c(ν, t)(zµµ0 (t) + z¯νµ0 (t)) = 2 (Reω)3 i |zµν (t)|2 + (p(µ, t) − p(ν, t))(q(µ, t) + q(ν, t)) . × exp − 4Reω 2h Using the obvious inequality 2|c(µ, t)¯ c(ν, t)| ≤ (|c(µ, t)|2 eAµ + |c(ν, t)|2 eAν )e−Aµ 2
2
2
/2 −Aν 2 /2
e
,
the symmetry in µ, ν, and (4.8) we get the estimate Z r √ 2 π 2 − ht ˜ |c(µ, t)|2 eAµ dµ . ) |hˆ xiϕ − q(µ0 , t)|kχk ≤ (|µ0 | + 1)O(e Reω
(4.9)
We can finish the proof of the lemma analogously to the end of the proof of Lemma 3.1, using the fact that the integral in the r.h.s. of (4.9) is a positive martingale with respect to the Wiener measure of the process Qp (which follows from the existence of the integral (4.2), the martingale property of π/Reω|c(µ, t)|2 for any µ, and Fubbini’s theorem).
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
...
Lemma 4.2. For a.a. W, (3.2) holds and also for each µ √ √ R ˜ − ht ) q(µ, t) = q˜(W ) + p˜(W )t + hW + h 0t W (s) ds + (|µ| + 1)O(e
˜ − p(µ, t) = p˜(W ) + hW + (|µ| + 1)O(e
√
ht
821
,
(4.10)
)
˜ − where the random constants q˜(W ) and p˜(W ) and a random function O(e not depend on µ.
√ ht
) do
Proof. It follows from the previous lemma in the same way as Lemma 3.2 follows from Lemma 3.1. Lemma 4.3. For all µ, ν such that c(µ) 6= 0 and c(ν) 6= 0, the limit lim (log c(µ, t) − log c(ν, t)) = −(µ2 − ν 2 )A + (µ − ν)B[W ]
t→∞
(4.11)
exists, where A is some (explicitly calculated) complex constant (depending only on ω0 = 2−1 ) with positive real part and B[W ] is almost surely finite random variable (not depending on µ, ν). Proof. By direct calculations, we get d(log c(µ, t)) − log c(ν, t)) = −(µ2 − ν 2 )f (t) dt + (µ − ν)(g1 (t, [W ]) dt + g2 (t) dW ) ,
(4.12)
where the functions f, g1 , g2 are exponentially small, as t → ∞, and moreover, f does not depend on W and −2
Ref (t) = (hReω)
2 Z t Z t exp{h Imω(s) ds} sin{h Reω(s) ds} > 0 , 0
0
which implies the assertion of the lemma. Lemma 4.4. Let us fix any µ0 such that c(µ0 ) 6= 0. Then for a.a. W, the limit g(µ) = lim (c(µ, t)/c(µ0 , t)) t→∞
exists, the limit function g(µ) is fast decreasing so that Z 2 |g(µ)|eδµ dµ < ∞
(4.13)
for some δ > 0, and also Z g(µ) dµ 6= 0 .
(4.14)
Proof. The first two statements follow directly from the previous lemma. To prove (4.14) we should again use the “martingale” trick of Lemma 3.1. Namely, since
822
V. N. KOLOKOL’TSOV
p MartQ = π/Reω|c(µ0 , t)|2 is a positive martingale with respect to the Wiener process Q, we conclude that MartW = M artQ /kχk2 is a positive martingale with respect to the innovating process W and thus it is bounded almost surely. Using (4.5) and Lemma 4.2, we get Z −2 g(µ) dµ . lim MartW = t→∞
Since it is bounded, we get (4.14). Second proof of Theorem 3.2. Due to (4.14), we get √ 1/4 α ghˆxiϕ ,hpi M−ihpi ˆ ϕ x/h Thˆ xiϕ (ϕ − (π h) ˆ ϕ ) (x + iy) Z c(µ, t) ω α g ≤C δq,δp − g0,0 (x + iy) dµ . c(µ0 , t)
(4.15)
Further, using estimate (3.25), we shall have ω α ˜ − − g0,0 )(x + iy)| ≤ O(e |(gδq,δp
√ ht
)(x2 + y 2 + (|µ| + 1)(|x| + |y|) + (|µ| + 1)2 )
˜ − × exp{O(e
√ ht
)(x2 + y 2 + (|µ| + 1)(|x| + |y|)
1 + (|µ| + 1)2 ) − Re(α(x + iy)2 )} . 2 Making now the integration over µ in (4.15) we conclude that the r.h.s. in (4.15) does not exceed ˜ − O(e
√ ht
1 )O(x2 + y 2 + 1) exp{− Re(α(x + iy)2 )} , 2
which obviously implies (3.10 ). 5. Long Time Asymptotics of the Brownian Motion and its Integral Rt We give here the estimates of growth of the integral V (t) = 0 W (s) ds of the standard d-dimensional Brownian motion W (t), as t → ∞, and as a consequence, obtain the estimates of growth of the mean position and momentum (satisfying (3.2)) of the solutions of Eq. (1.14). In this section the dimension of the problem is essential. Theorem 5.1. Suppose g(t) be an increasing positive function on R+ and d > 1. R∞ Then, if the integral 0 (g(t)t−3/2 )d dt is convergent, then lim inf (|V (t)|/g(t)) = ∞ t→∞
for a.a. W. In particular, for any β
2 and the integral 0 (g(t)/ t)d dt is convergent, then lim inf t→∞ (|W (t)|/g(t)) is almost surely infinite. In particular, lim inf t→∞ |Wtβ(t)| = ∞ for any β < 12 − 1d . From Theorem 5.1, Proposition 5.1, and representation (3.2) for the mean position and momentum of the solution of Eq. (1.14), it follows immediately: Theorem 5.3. Let d > 2 and let ϕ be any solution of the Cauchy problem for Eq. (1.14). Then |hˆ xiϕ | |hˆ piϕ | = ∞ , lim inf β−1 = ∞ lim inf t→∞ t→∞ t tβ almost surely for any β
1 and β < 2 − d . Then by the n can hold. It first Borell–Cantelli lemma only a finite number of the events BA,β means the existence of a constant T such that V (t) ∈ / [−Ag([t]), Ag([t])]d for t > T , where [t] denotes the integer part of t. This obviously implies the statement of Theorem 5.1.
824
V. N. KOLOKOL’TSOV
Proof of Lemma 5.1. Obviously, it is enough to consider the case d = 1. The density pt (x, y) of the joint distribution of W (t) and V (t) is well known (see, [47]): √ 2 2 6 6 2 3 pt (x, y) = 2 exp − x + 2 xy − 3 y . πt t t t In particular,
√ 2 x 3 . pt (x, y) ≤ 2 exp − πt 2t
(5.2)
It is clear that Z t P (BA,g ) = P (V (t) ∈ [−Ag(t), Ag(t)]) + 2
×P
Z
τ
min (y + τ x +
0≤τ ≤1
+∞
dy Ag(t)
Z
∞
−∞
pt (x, y)
W (s) ds) < Ag(t) dx .
(5.3)
0
The first term here is equal to √
y2 exp − 3 dy 2t −Ag(t)
Z
1 2πt3
Ag(t)
and is of order O(g(t)t−3/2 ). The second term can be estimated from above by the integral Z +∞ Z ∞ dy pt (x, y)P min τ x + min W (τ ) < Ag(t) − y dx . (5.4) 2 Ag(t)
0≤τ ≤1
−∞
0≤τ ≤1
We decompose this integral in the sum I1 +I2 +I3 of three integrals, whose domain of integration in the variable x are {x ≥ 0}, {Ag(t) − y ≤ x ≤ 0}, and {x < Ag(t) − y} respectively. We shall show that the integrals I1 and I2 are of order O(t−3/2 ) and the integral I3 is of order O(t−1 ), which will complete the proof of the Lemma. It is clear that Z ∞ Z ∞ dy pt (x, y)P min W (τ ) < Ag(t) − y dx . I1 = 2 Ag(t)
0≤τ ≤1
0
Enlarging the domain of integration over x to the whole line, integrating over x, and using the well-known distribution for the minimum of the Brownian motion we obtain 2 Z ∞ Z ∞ 2 y2 z dz . exp − 3 dy exp − I1 ≤ √ 3 2t 2 π t Ag(t) y−Ag(t) Changing the order of integration we can rewrite the last expression in the form 2 Z Ag(t)+z Z ∞ z y2 2 √ dz exp − exp − 3 dy . 2 2t π t3 0 Ag(t) Consequently, 2 I1 ≤ √ π t3
Z 0
∞
z2 z exp − 2
dz = O(t−3/2 ) .
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
...
825
We proceed with I2 . Making the change of the variable x 7→ −x we obtain Z ∞ Z y−Ag(t) I2 = 2 dy pt (−x, y)P min < Ag(t) − y + x dx . Ag(t)
0≤τ ≤1
0
Making the change of the variable s = y − Ag(t) and using the distribution of the minimum of the Brownian motion we get further that r Z ∞ 2 Z ∞ Z s z 2 dz . ds pt (−x, s + Ag(t)) dx exp − I2 = 2 π s−x 2 0 0 Estimating pt (x, y) by (5.2) and changing the order of integration we get √ Z ∞ 2Z ∞ 2 Z x+z 4 z x 3 I2 ≤ √ dz exp − dx exp − ds . 2 2 2t 2π πt 0 0 x The last integral is obviously of order O(t−3/2 ). It remains to estimate the integral I3 . We have Z ∞ Z ∞ I3 = 2 dy pt (−x, y) dx Ag(t)
Z
Z
∞
Ag(t)+x
pt (−x, y) dx
=2 0
≤
y−Ag(t)
dy Ag(t)
√ Z 2 2 3 ∞ x dx x exp − πt2 0 2t
= O(t−1 ) . The proof is complete. Acknowledgments I am very thankful to Profs. Albeverio, V. P. Belavkin, and A. Ponosov for useful discussions. References [1] S. Albeverio, A. Hilbert and V. Kolokoltsov, “Transience for stochastically perturbed Newton systems”, Ruhr Universit¨ at Bochum, SFB 237, preprint 269 (1995), to appear in Stochastics and Stochastics Reports. [2] S. Albeverio, A. Hilbert and V.Kolokoltsov, “Estimates uniform in time for the transition probability of diffusions with small drift and for stochastically perturbed Newton equations”, Ruhr Universit¨ at Bochum, SFB 237, preprint 291 (1995). [3] S. Albeverio, V. Kolokoltsov and O. Smolyanov, “Continuous quantum measurement: local and global approaches”, to appear in Rev. Math. Phys. [4] A. Barchielli and V. P. Belavkin, “Measurements continuos in time and a posteriori states in quantum mechanics”, J. Phys. A: Math. Gen. 24 (1991) 1495–1514. [5] A. Barchielli, L. Lanz and G. M. Prosperi, “Statistics of continuous trajectories in quantum mechanics: Operator valued stochastic processes”, Found. Phys. 13 (1983) 779–812.
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[6] A. Barchielli, L. Lanz, G. M. Prosperi, “A model for macroscopic description and continuous observation in quantum mechanics”, Nuovo Cimento, 72B (1982) 79– 121. [7] V. P. Belavkin, “Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes”, in: Modelling and Control of Systems, Proc. Bellman Continuous Workshop, Sophia-Antipolis 1988, Lect. Notes in Contr. and Inform. Sci., 121 (1988) 245–265. [8] V. P. Belavkin. “A new wave equation for a continuous nondemolition measurement, Phys. Let. A 140 (1989) 355–358. [9] V. P. Belavkin, “A posterior Schr¨ odinger equation for continuous nondemolition measurement”, J. Math. Phys. 31 (1990) 2930–2934. [10] V. P. Belavkin, “The reconstruction theorem for quantum stochastic process”, Teor. Mat. Fis. 62 (1985) 409–431. English translation in Theor. Math. Phys. [11] V. P. Belavkin, “Quantum continual measurements and a posteriori collapse on CCR, Comm. Math. Phys. 146 (1992) 611–635. [12] V. P. Belavkin, O. Hirota and R. L. Hudson eds, “Quantum communications and measurement”, Proc. Int. Conf. held on July 11-16, 1994, Nottingham, Plenum Press, N. Y., 1995. [13] V. P. Belavkin and V. N. Kolokoltsov, “Quasy-classical asymptotics of quantum stochastic equations”, Teor. i Mat. Fis. 89 (1991) 163–178. English translation in Theor. Math. Phys. [14] V. P. Belavkin and P. Staszewski, “A stochastic solution of Zeno paradox for quantum Brownian motion”, Phys. Rev. A 45 (3) (1992) 1347–1356. [15] D. Bohm and J. Bub, “A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory”, Rev. Mod. Phys. 38 (1966) 453–469. [16] Ph. Blanchard and A. Jadczyk, “Strongly coupled quantum and classical systems and Zeno’s effect”, Phys. Lett. A 183 (1993) 272–276. [17] Ph. Blanchard and A. Jadczyk, “On the interaction between classiacal and quantum systems”, Phys. Lett. A 175 (1993) 157–164. [18] Ph. Blanchard and A. Jadczyk, “Evant-enhanced formalism of quantum theory or Columbus solution to the quantum measurement problem. Univ. Bielefeld, preprint BiBoS 655/7/94 (HEP-TH 9408021), 1994. [19] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon. Schr¨ odinger Operators with Applications to Quantum Mechanics and Global Geometry, Springer-Verlag, 1987. [20] E. B. Davies, “Quantum stochastic processes”, Commun. Math. Phys. 15 (1969) 277– 304. [21] E. B. Davies, Quantum Theory of Open Systems, Academic Press, London, 1976. [22] L. Diosi, “Continuous quantum measurement and Ito formalism”, Phys. Lett. A 129 (1988) 419–423. [23] L. Diosi, “Localized solution of a simple nonlinear quantum Langevin equation”, Phys. Lett. A 132 (1988) 233–236. [24] L. Diosi, “Models for universal reduction of macroscopic quantum fluctuations”, Phys. Rev. A 40 (1989) 1165–1173. [25] L. Diosi, “Quantum measurement and quantum gravity for each other”, in Quantum Chaos, Quantum Measurement; NATO ASI Series C: Math. Phys. Sci. 357 (1992) 299–304. Dordrecht, Kluwer. [26] A. Dvoretski and P. Erd¨ os, “Some problems on random walk in space”, Second Berkeley Simposium in Probability, Univ. of California Press, (1951) 353–367. [27] K. D. Elworthy and A. Truman, “The diffusion equation and classical mechanics: An elementary formula”, in Stochastic Processes in Quantum Physics LNP 173 (1982) 136–146. [28] Ch. N. Friedman, “Semigroup product formulas, compressions, and continuous observation in quantum mechanics”, Indiana Univ. Math. J. 21 (1972) 1001–1011.
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[29] C. W. Gardiner and M. J. Collet, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation”, Phys. Rev. A 31 (1985) 3761–3774. [30] B. M. Garraway and P. L. Knight, “A comparison of quantum state diffusion and quantum jump simulations of two-photon processes in a dissipative environment”, Phys. Rev. A 49 (1994) 1266–1274. [31] D. Gatarek and N. Gisin, “Continuous quantum jumps and infinite dimensional stochastic equations”, J. Math. Phys. 32 (81) (1991) 2152–2156. [32] G. C. Ghirardi, R. Grassi and A. Rimini, “A continuous sponteneous reduction model involving gravity”, Phys. Rev A 42 (1990) 1057–1064. [33] G. C. Ghirardi, R. Grassi and P. Pearle, “Relativistic dynamical reduction models: general framework and examples”, Found. of Phys. 20 (1990) 1271–1316. [34] G. C. Ghirardi, A. Rimini and T. Weber, “A model for unified quantum description of macroscopic and microscopic systems”, in Quantum Probability and Applications II. LNM 1136 (1985) 223–233, Berlin, Springer. [35] N. Gisin, “Quantum measurement and stochastic processes”, Phys. Rev. Lett. 52 (19) (1984) 1657–1660. [36] N. Gisin, “Stochastic quantum dynamics and relativity”, Helvetica Physica Acta 62 (1989) 363–371. [37] N. Gisin, P. L. Knight, I. C. Percival, R. C. Thompson and D. C. Wilson, “Quantum state diffusion theory and a quantum jump experiment”, Lett. J. Modern Optics 40 (9) (1993) 1663–1671. [38] N. Gisin and I. C. Percival, “The quantum state diffusion model applied to open systems”, J. Phys. A 25 (1992) 5677–5691. [39] N. Gisin and I. C. Percival, “Quantum state diffusion, localization and quantum dispersion entropy”, J. Phys. A: Math. Gen. 26 (1993) 2233–2244. [40] N. Gisin and I. C. Percival, “The quantum state diffusion picture of physical processes”, J. Phys. A: Math. Gen. 26 (1993) 2245–2260. [41] A. S. Holevo, “Statistical inference for quantum processes”, in Quantum Aspects of Optical Communications, LNP 378 (1991) 127–137, Berlin, Springer. [42] R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolution”, Commun. Math. Phys. 93 (1998) 301–323. [43] K. Ito and H. P. McKean, Diffusion Processes and Their Sample Paths, SpringerVerlag, 1974. [44] D. Juriev, “Belavkin–Kolokoltsov watch-dog effect in interectively controlled stochastic computer-graphic dynamical systems. A mathematical study”, E-print (LANL Electronic Archive on Chaos Dyn.): chao-dyn/9406013 + 9504008 (1994, 1995). [45] D. Juriev, “Belavkin–Kolokoltsov watch-dog effect in interectively controlled stochastic computer-graphic dynamical systems. A summary of mathematical researches”, E-print (LANL Electronic Archive on Adapt.Self-Org.): adap-org/9410001 (1994). [46] G. Kallianpur, Stochastic Filtering Theory, Springer-Verlag, N. Y., Heidelberg, Berlin, 1980. [47] A. N. Kolmogorov, “Zuf¨ allige Bewegungen (Zur Theorie der Brownschen Bewegung)”, Ann. Math. 35 (1) (1934) 116–117. [48] V. N. Kolokoltsov, “Application of the quasi-classical methods to the investigation of the Belavkin quantum filtering equation”, Mat. Zametki 50 (1991) 153–156. English translation in Math. Notes. [49] V. N. Kolokoltsov, “Long time behavior of continuously observed and controlled quantum systems”, preprint 204 (December 1993), Ruhr Univ. Bochum, SFB 237, to be published in Quantum Probability Communications, ed. R. Hudson, M. Lindsey. [50] V. N. Kolokoltsov, “Long time behavior of the solutions of the Belavkin quantum filtering equation, in [12], 429–438.
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[51] V. N. Kolokoltsov, “Stochastic Hamilton–Jacobi equation and stochastic method WKB”, preprint 236 (November 1994), Ruhr Univ. Bochum, SFB 237, to appear in Proc. Int. Conf. “Idempotency” held on October 1994 in Bristol, Cambridge Univ. Press, 1996. [52] V. N. Kolokoltsov, “Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate”, J. Math. Phys. 36 (6) (1995) 2741– 2760. [53] V. N. Kolokoltsov, “Short deduction and mathematical properties of the main equation of the theory of continuous quantum measurement”, in GROUP21. Physical Applications and Mathematical aspects of Geometry, Groups, and Algebras, Proc. XXI Int. Colloq. on Group Theoretical Methods in Physics 15–20 July 1996 in Goslar, Germany, (eds. H. D. Doebner, P. Nattermann and W. Scherer, World Scientific, 1997, v. 1, 326–330. [54] G. Lindblad, “On the generators of quantum dynamical semigroups”, Commun. Math. Phys. 48 (1976) 119–130. [55] G. J. Milburn, “Intrinsic decoherence in quantum mechanics”, Phys. Rev. A 44 (1991) 5401–5406. [56] B. Misra and E. C. Sudarshan, “Quantum Zeno paradox”, J. Math. Phys. 18 (1977) 756–763. [57] I. C. Percival, “Localization of wide open quantum systems”, J. Phys. A 27 (1994) 1003–1020. [58] P. Pearle, “Reduction of the state vector by a nonlinear Schr¨ odinger equation”, Phys. Rev. D 13 (1976) 857–868. [59] P. Pearle, “Towards explaining why events occur”, Int. J. Theor. Phys. 18 (1979) 489–518. [60] P. Pearle, “Combining stochastic dynamical state-vector reduction with spontaneous localisation”, Phys. Rev. A 39 (1989) 2277–2289. [61] P. Staszewski, “Quantum mechanics of continuously observed systems”, Habilitation thesis, Nicholas Copernicus Univ. Press, Torun, 1993. [62] A. Truman and H. Zhao, “The stochastic Hamilton Jacobi equation, stochastic heat equation and Scr¨ odinger equation”, Swansea Univ. preprint 1994, to appear in Stochastic Analysis and Applications. eds. A. Truman, I. M. Davies and K. D. Elworthy, World Scientific Publ. Co. (1996), 441–464. [63] A. Truman and H. Zhao, “The stochastic Hamilton Jacobi theory and related topics”, in Stochastic Partial Differential equations, London Math. Soc. Lecture Notes Series 276, Cambridge Univ. Press, 1995, 287–303.
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN? H. NEIDHARDT Fachbereich Mathematik Universit¨ at Potsdam Postfach 60 15 53 14 415 Potsdam Germany E-mail : [email protected]
V. A. ZAGREBNOV D´ epartment de Physique Universit´ e de la M´ editerran´ ee (Aix-Marseille II) CPT-Luminy Case 907 13288 Marseille Cedex 9 France E-mail : [email protected] Received 27 September 1996 Revised 25 September 1997 Mathematics Subject Classification: 47A05, 47B25, 81C10, 81C12 We show that any symmetric operator H has a dense maximal b-stability domain Ds (i.e. H|Ds ≥ bI, b ∈ R1 ) if and only if H is unbounded from above. This abstract result allows an application to singular perturbed Schr¨ odinger operators which are not semi-bounded from below, i.e., to the so-called “fall to the center problem”. It turns out that in this case the regularization problem is always ill-posed which implies that there is no unique “right Hamiltonian” for corresponding perturbed system. We give an example of singular perturbed Schr¨ odinger operator for which stability domains are described explicitly. Keywords: singular perturbation, symmetric operator, self-adjoint extension, regularization, Schr¨ odinger operator, fall to the center.
1. Introduction If a self-adjoint operator H on separable Hilbert space H is Hamiltonian of some quantum system of finitely many particles, then for stability of the system it is natural to demand that it has to be semi-bounded from below. Otherwise particles would occupy states with lower and lower energies, which finally leads to collapse. However, having some semi-bounded self-adjoint operator A and some singular perturbation W of A it can happen that perturbed operator H, Hf = Af + W f ,
f ∈ dom(H) = D ⊆ dom(A) ∩ dom(W ) ,
(1.1)
where D is some linear dense subset of dom(A) ∩ dom(W ), is not essentially selfadjoint and, moreover, it is not semi-bounded from below. If H admits self-adjoint 829 Reviews in Mathematical Physics, Vol. 10, No. 6 (1998) 829–850 c World Scientific Publishing Company
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H. NEIDHARDT and V. A. ZAGREBNOV
extensions then none of them can be regarded as the Hamiltonian of the perturbed quantum system because each extension is not semi-bounded from below. Example 1.1. Let A = −d2 /dx2 be the Laplace operator on H = L2 (R1 ). Let W be the multiplication operator corresponding to W (x) = −
1 1 , 4 |x|2
x ∈ R1 ,
(1.2)
and H κ1 f = Af +
1 Wf , κ
f ∈ dom(H κ1 ) = D = C0∞ (R1 \ {0}) ,
0 < κ < ∞ . (1.3)
If κ ≥ 1, then the operator H κ1 is semi-bounded from below (see [?, ?]) while for κ ∈ (0, 1) not: “fall to the center of attraction”. An instructive discussion of this phenomenon can be found in [?, ?, ?]. In the latter paper it is shown how the “fall to the center” is related to an infinite family of self-adjoint extensions of H1/κ , see also Sec. 4. Since the operator H κ1 commutes with the complex conjugation, it has equal deficiency indices (von Neumann’s theorem, see e.g. [?, Th. X3]). Hence H κ1 has several self-adjoint extensions and, moreover, if κ ∈ (0, 1), then each of these extensions is not semi-bounded from below. Therefore, in the last case there is no self-adjoint extension which would be a candidate for a stable Hamiltonian. To find a way out of this situation one can take the point of view that the domain dom(H) is too large and, hence, even the symmetric operator H is badly defined. In other words, it might be possible that in H there is a dense domain Ds ⊆ dom(H) such that H|Ds is semi-bounded from below. Definition 1.2. Let H be a (closed or not closed) symmetric operator. A dense linear subset Ds ⊆ dom(H) is called a stability domain of H if H|Ds is semi-bounded from below. Stability domain Ds of H is called a b-stability domain, b ∈ R1 , if inf
f ∈B1 (Ds )
(Hf, f ) ≥ b ,
(1.4)
where B1 (Ds ) = {f ∈ Ds : kf k ≤ 1} .
(1.5)
A b-stability domain Ds is called maximal if for each b-stability domain Ds0 obeying Ds ⊆ Ds0 one gets Ds = Ds0 . Therefore, the problem of constructing a semi-bounded from below Hamiltonian is reduced to existence of stability domain of H. However, this leads to a general question: whether each symmetric operator has a stability domain? The general answer is, of course, no.
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DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
Definition 1.3. Let H be a densely defined unbounded symmetric operator. We say H is unbounded from above if def
λ+ 1 (H) =
sup
(Hf, f ) = +∞ .
(1.6)
f ∈B1 (dom(H))
The operator is called unbounded from below if (−H) is unbounded from above. If H is unbounded from below but semi-bounded from above, then evidently H has no stability domain. Otherwise the existence of a stability domain Ds of H would imply that H is bounded. So the problem reduces to the question: whether each symmetric operator, which is unbounded from above, has a stability domain? The answer to this question is yes. The aim of this paper is to prove this as well as to describe them and corresponding stable Hamiltonians. Of course, the question is trivial if the operator H is unbounded from above but semi-bounded from below. So we face a real problem if H is unbounded from below and above. A partial solution of this problem is obtained in [?]. There it is proven that two (closed or unclosed) densely defined unbounded symmetric operators H1 and H2 which satisfy ∞ \ dom(Hs ) = dom(Hsn ) , s = 1, 2 , (1.7) n=1
possess unitarily equivalent densely defined restrictions if and only if both operators are strongly unbounded from below or from above. Definition 1.4. Let H be a densely defined unbounded symmetric operator such that the condition H(dom(H)) ⊆ dom(H) is satisfied and let k, n ∈ N, where N = {1, 2, . . .}. We set def
Bn (dom(H)) = {f ∈ dom(H) : |(H j f, f )| ≤ 1
for
j = 0, . . . , n − 1}
(1.8)
and def
2k−1 λ+ f, f ) 2k−1 (H) = supf ∈B2k−1 (dom(H)) (H def λ− 2k−1 (H) =
inf f ∈B2k−1 (dom(H)) (H
2k−1
(1.9)
f, f ) .
The operator H is called strongly unbounded from above (below) if λ+ 2k−1 (H) = +∞ (H) = −∞) for all k ∈ N. (λ− 2k−1 In application to our situation this means that if the symmetric operator H satisfies conditions: H(dom(H)) ⊆ dom(H) and dom(H) =
∞ \
dom(H n ) ,
(1.10)
n=1
and it is strongly unbounded from above, then H possesses a stability domain. To see this we set H1 = H and choose for H2 a restriction of an arbitrary unbounded def T∞ non-negative self-adjoint operator K to the domain D∞ = n=1 dom(K n ), i.e. H2 = K|D∞ which is essentially self-adjoint. One can easily prove that H1 and H2
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H. NEIDHARDT and V. A. ZAGREBNOV
obey (??) and, moreover, that H2 is strongly unbounded from above. Hence H1 and H2 have densely defined restrictions which are unitarily equivalent. In particular, this means that H has densely defined non-negative restriction, i.e., H has a stability domain with b = 0. However, the condition (??) and the strong unboundedness from above are too limited for our purpose. The paper is organized as follows. In the next section we give necessary and sufficient conditions (Theorem ??) that a symmetric operator has a maximal b-stability domain. In Sec. 3 we discuss applications of the main Theorem ?? to the problem of the unique, stable “right Hamiltonian” for non-positive singular perturbations. Our principal observation is a kind of “no-go” Theorem ?? which says that if a naturally defined perturbed operator is not semi-bounded from below, then there is no unique “right Hamiltonian”. We demonstrate these above abstract statements by an instructive quantum mechanical example in Sec. 4. Concluding remarks and discussions are postponed to the last Sec. 5. 2. Stability The aim of this section is to prove the assertion made above about the existence of a stability domain for an unbounded from above symmetric operator. Lemma 2.1. Let H be a densely defined unbounded symmetric operator which is unbounded from above. Let F ⊆ dom(H) be a finite dimensional subspace. Then sup
(Hf, f ) = +∞
(2.1)
f ∈B1 (dom(H) F )
Proof. Let PFH be the orthogonal projection from H onto F . Since F ⊆ dom(H) and dim(F) < ∞ there is a constant C > 0 such that ||HPFH f || ≤ C||f ||, f ∈ H. Hence |(H(I − PFH )f, (I − PFH )f ) − (Hf, f )| ≤ 2C||f ||2 ,
f ∈ dom(H) .
(2.2)
Since H is unbounded from above, for each n ∈ N there is a fn ∈ dom(H), ||fn || ≤ 1, such that (Hfn , fn ) ≥ n. We set gn = (I − PFH )fn ,
n ∈ N.
(2.3)
Then gn ∈ dom(H) F, ||gn || ≤ 1 and by (??) one gets (Hgn , gn ) = (H(I − PFH )fn , (I − PFH )fn ) ≥ n − 2C for n ∈ N which proves (??).
(2.4)
Next we need some facts from linear algebra. Lemma 2.2. Let H be a symmetric operator and {gl }m l=1 be a sequence of linearly independent elements gl ∈ dom(H). Let Gm be the subspace spanned by
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
833
{gl }m l=1 . Then one has (Hf, f ) > 0 for any non-trivial element f ∈ Gm if and only if the determinants (Hg1 , g1 ) · · · (Hgk , g1 ) (Hg1 , g2 ) · · · (Hgk , g2 ) (2.5) D(g1 , . . . , gk ) = det ...................... (Hg1 , gk ) · · · (Hgk , gk ) obey D(g1 , . . . , gk ) > 0, for each k = 1, 2, . . . , m. Proof. We set def
Let f =
1 ≤ l, n ≤ m .
aln = (Hgn , gl ) ,
Pm
(2.6)
n=1 cn gn .
Obviously one has ! m m m m m X m X X X X X H cn g n , cl g l = cn cl (Hgn , gl ) = aln cn cl . n=1
l=1
n=1 l=1
Let Am = kaln km l,n=1
(2.7)
n=1 l=1
(Hg1 , g1 ) · · · (Hgm , g1 )
(Hg1 , g2 ) · · · (Hgm , g2 )
=
........................
(Hg , g ) · · · (Hg , g ) 1
m
m
(2.8)
m
Note that the matrix Am is symmetric, i.e. anl = aln , 1 ≤ l, n ≤ m. Furthermore, the kernel of Am would be non-trivial if {gl }m l=1 are not linearly independent. Introducing the vector c1 · (2.9) ~c = · · cm one gets (see (??)) (Hf, f ) = hAm~c, ~c i ,
(2.10)
where h· , ·i is the usual scalar-product in Cm . Therefore one has (Hf, f ) > 0 for each f ∈ Gm , f 6= 0, if and only if the matrix Am is positive. However, by a wellknown criterion (Theorem 27, [?]) this is equivalent to the fact the determinants D(g1 , . . . , gk ) are positive for each k = 1, 2, . . . , m. In the following we use decomposition of determinants into minors. One gets D(g1 , . . . , gm+1 ) = am+1,1 (−1)m+2 M (g2 , . . . , gm+1 ) + · · · + am+1,m+1 (−1)2m+2 M (g1 , . . . , gm ) ,
(2.11)
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H. NEIDHARDT and V. A. ZAGREBNOV
where am+1,j = (Hgj , gm+1 ) ,
j = 1, 2, . . . , m + 1 .
(2.12)
The minors M (g1 , g2 , . . . , gj−1 , gj+1 , . . . , gm+1 ) are given by M (g1 , . . . , gj−1 , gj+1 , . . . , gm+1 ) (Hg1 , g1 ) · · · (Hgj−1 , g1 ) (Hgj+1 , g1 ) · · · (Hgm+1 , g1 ) (Hg1 , g2 ) · · · (Hgj−1 , g2 ) (Hgj+1 , g2 ) · · · (Hgm+1 , g2 ) = det . (2.13) ........................................................ (Hg1 , gm ) · · · (Hgj−1 , gm ) (Hgj+1 , gm ) · · · (Hgm+1 , gm ) Note that D(g1 , . . . , gm ) = M (g1 , . . . , gm ). If g1 , . . . , gm are fixed elements, then there are constants Cj (g1 , . . . , gm ) such that |M (g1 , . . . , gj−1 , gj+1 , . . . , gm+1 )| ≤ Cj (g1 , . . . , gm )kgm+1 k ,
j = 1, 2, . . . , m . (2.14)
This comes from the estimate |(Hgm+1 , gn )| ≤ kHgn k kgm+1k, 1 ≤ n ≤ m. Theorem 2.3. Let H be a densely defined symmetric operator. Then for any b ∈ R1 this operator has a maximal b-stability domain if and only if H is unbounded from above. Proof. Let b = 0 and {ξn }∞ n=1 be an orthonormal basis in H such that ξn ∈ dom(H) for each n = 1, 2, . . . . Such basis always exists. On the other hand, by Lemma ?? there is a sequence {fm }∞ m=1 , fm ∈ dom(H), kfm k ≤ 1, such that (i) fm ⊥ f1 , . . . , fm−1 , m = 2, 3, . . . , (ii) fm ⊥ ξ1 , . . . , ξ[m]+1 , m = 1, 2, . . . , (iii) (Hfm , fm ) > γm > 0 , m = 1, 2, . . . , where {γm }∞ m=1 is a sequence of positive numbers such that limm→∞ γm = +∞, which will be defined in the following, and n(n + 1) def < m , m ∈ N, (2.15) [m] = max n ∈ N0 : 2 where N0 = {0, 1, 2, . . .} and N = {1, 2, . . .}. We introduce the function p(·) : N → N, [m]([m] + 1) def (2.16) p(m) = m − 2 and the elements def
gm = fm + q(m)ξp(m) ,
m ∈ N,
def
q(m) = ([m] − p(m) + 2) .
(2.17)
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
835
Few first terms of the sequence {gm }∞ m=1 are given by g1 = f1 + 1 · ξ1 g2 = f2 + 2 · ξ1 g3 = f3 + 1 · ξ2 g4 = f4 + 3 · ξ1
(2.18)
g5 = f5 + 2 · ξ2 g6 = f6 + 1 · ξ3 g7 = f7 + 4 · ξ1 ..............
Note that the sequence {gm }∞ m=1 consists of linearly independent elements. To see this let us assume that L X cl g l = 0 . (2.19) l=1
By the properties (i) and (ii) one has fL ⊥ f1 , . . . , fL−1 and fL ⊥ ξ1 , . . . , ξ[L]+1 . Therefore ! L X 0= cl gl , fL = cL kfL k2 . (2.20) l=1
PL−1 Hence l=1 cl gl = 0. Repeating this reasoning one gets c1 = c2 = · · · = cL = 0. Let us prove that there is a suitable sequence {γm }∞ m=1 such that the sequence obeys D(g , . . . , g ) > 0 for each m = 1, 2, . . . . The existence of {γm }∞ {gm }∞ 1 m m=1 m=1 we prove by induction. We choose γ1 ≥ 3kHξ1 k .
(2.21)
Then we obtain D(g1 ) = (Hg1 , g1 ) = (H(f1 + ξ1 ), (f1 + ξ1 )) ≥ (Hf1 , f1 ) − 3kHξ1 k > 0 .
(2.22)
Let us assume that there are γ1 , . . . , γm such that the corresponding g1 , . . . , gm satisfy the condition D(g1 , . . . , gl ) > 0 for l = 1, 2, . . . , m. We have to show that there is a γm+1 such that D(g1 , . . . , gm , gm+1 ) > 0. Let γm+1 be an arbitrary positive number which satisfies the condition γm+1 ≥ ([m + 1] + 1)([m + 1] + 3)
sup
kHξk k
1≤k≤[m+1]+1
+
m 1 + ([m + 1] + 1)2 X kHgj kCj (g1 , . . . , gm ) , D(g1 , . . . , gm ) j=1
(2.23)
836
H. NEIDHARDT and V. A. ZAGREBNOV
where Cj (g1 , . . . , gm ) is given by (??). Taking into account (??) and (??) one gets D(g1 , . . . , gm+1 ) ≥ (Hgm+1 , gm+1 )D(g1 , . . . , gm ) −
m X
kHgj k kgm+1k |M (g1 , . . . , gj−1 , gj+1 , . . . , gm+1 )| . (2.24)
j=1
Using (??) we obtain D(g1 , . . . , gm+1 ) ≥ (Hgm+1 , gm+1 )D(g1 , . . . , gm ) − kgm+1 k2
m X
kHgj kCj (g1 , . . . , gm ) .
(2.25)
j=1
Since D(g1 , . . . , gm ) > 0 by assumption, one gets D(g1 , . . . , gm+1 ) > 0 if kgm+1 k2 X kHgj kCj (g1 , . . . , gm ) > 0 D(g1 , . . . , gm ) j=1 m
def
∆ = (Hgm+1 , gm+1 ) −
(2.26)
From (??) and (??) we find kgm+1 k2 ≤ 1 + q(m + 1)2 ≤ 1 + ([m + 1] + 1)2
(2.27)
and (Hgm+1 , gm+1 ) ≥ (Hfm+1 , fm+1 ) − 2q(m + 1)kHξp(m+1) k − q(m + 1)2 kHξp(m+1) k .
(2.28)
Hence (Hgm+1 , gm+1 ) ≥ (Hfm+1 , fm+1 ) − ([m + 1] + 1)([m + 1] + 3)
sup
kHξk k .
1≤k≤[m+1]+1
(2.29) Therefore, the estimate (??) gets the form: ∆ ≥ (Hfm+1 , fm+1 ) − ([m + 1] + 1)([m + 1] + 3)
sup
kHξk k
1≤k≤[m+1]+1
−
m 1 + ([m + 1] + 1)2 X kHgj kCj (g1 , . . . , gm ) . D(g1 , . . . , gm ) j=1
(2.30)
Using condition (??) one finally obtains ∆ ≥ (Hfm+1 , fm+1 ) − γm+1 .
(2.31)
Applying (iii) we immediately obtain ∆ > 0 which yields D(g1 , . . . , gm+1 ) > 0. So, by Lemma ?? the proof of stability follows by induction. It remains to show that the linear span G of {gm }∞ m=1 is dense in H. To this end we note that by (??) def
ηm =
1 fm + ξp(m) ∈ G , q(m)
m = 1, 2, . . . .
(2.32)
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
837
If m = n(n+1) + k, 1 ≤ k ≤ n + 1. Then by definitions (??) and (??) we have 2 [m] = n and p(m) = k. Hence q(m) = n − k + 2. Therefore, (??) transforms into η n(n+1) +k = 2
1 f n(n+1) +k + ξk . 2 n−k+2
(2.33)
Fixing k and tending n → ∞ one gets limn→∞ η n(n+1) +k = ξk . However, by con2 struction {ξk }∞ k=1 forms an orthonormal basis in H. Hence G is dense in H. def
If b 6= 0, then we consider instead of H the operator Hb = H − bI. Applying the first part of the proof we obtain a stability domain Ds such that Hb |Ds ≥ 0. Hence H|Ds ≥ bI. It remains to show that there is always a maximal stability domain. To prove this we introduce the set Xb of all dense domains Ds ⊆ dom(H) such that H|Ds ≥ bI. The set Xb is partially ordered with respect to the set-inclusion relation. Moreover, for every linearly ordered subset Yb of Xb there is a upper bound Dp . The upper S bound is given by Dp = Ds ∈Yb Ds . Obviously, Dp is a linear dense subset of dom(H) such that H|Dp ≥ bI. Then by the Zorn’s lemma (see e.g. [?, Th. I.2]) Xb contains at least one maximal element Dm , i.e., such that from Ds0 ⊇ Dm , Ds0 ∈ Xb , it follows Ds0 = Dm . Note that the Zorn lemma does not say that the maximal element Dm is unique. We shall illustrate this in Sec. 4 by Example. 3. Singular Perturbations Let us relate the above result to our previous papers [?, ?]. There we associated with two self-adjoint operators A ≥ 0 and W ≤ 0 having a common dense domain D ⊆ dom(A) ∩ dom(W ) the symmetric operator Hα f = Af + αW f ,
dom(Hα ) = D ,
α > 0.
(3.1)
A dense linear subset Ds ⊆ D was called a stability domain of the pair {A, W } if there are constants 0 < a < 1, b ≥ 0, such that the estimate (−W f, f ) ≤ a(Af, f ) + b(f, f ) ,
f ∈ Ds ,
(3.2)
takes place. In general, however, such a stability domain Ds ⊆ D might not exist for given constants 0 < a < 1, b ≥ 0. So, a natural question arises whether there is always a stability domain of the pair {A, W } with respect to D ? Moreover, does such stability domain exist for any constants 0 < a < 1, b ≥ 0 ? Finally, is there always a maximal stability domain for given constants a, b ? This means a domain Dm such that for any other stability domain Ds , Dm ⊂ Ds ⊆ D, for which (??) holds, one gets Dm = Ds . Theorem 3.1. Let {A ≥ 0, W ≤ 0} be a pair of self-adjoint operators with common dense domain D ⊆ dom(A) ∩ dom(W ). For each 0 < a < 1, and b ≥ 0 there is a maximal stability domain Da,b ⊆ D of the pair {A, W } if and only if for α = 1/a the symmetric operator H1/a given by (??) is unbounded from above.
838
H. NEIDHARDT and V. A. ZAGREBNOV
Proof. Let H1/a be unbounded from above. Applying Theorem ?? to the operator H1/a we find (− ab )-stability domain Da,b ⊆ D of H1/a . This means that 1 b − (f, f ) ≤ (Af, f ) + (W f, f ) , a a
f ∈ Da,b ,
(3.3)
which immediately yields that Da,b is a stability domain of {A, W }. Moreover, by Theorem ?? the domain Da,b can be chosen maximal for H1/a which implies that Da,b is a maximal stability domain of {A, W } for constants a, b. The converse is obvious. Consequently, constants a, b obeying 0 < a < 1, b ≥ 0 exist if and only if at least for one α > 1 the operator Hα (??) is unbounded from above. However, this seems to be a natural assumption from the physical point of view. To proceed further we have to recall briefly main results of [?, ?]. Assume that D itself is a stability domain of {A, W }, i.e. Ds = D, and that symmetric def
operators A0 = A|D and Hα , 0 < α ≤ 1, are not essentially self-adjoint on this stability domain. Hence the problem arises to find out a right Hamiltonian for the perturbed system which is expected to be a semi-bounded from below self-adjoint extension of Hα . To obtain this extension one associates with Hα an approximating sequence of ˜ α,n }∞ self-adjoint operators {H n=1 of the form ˜ + αWn f , ˜ α,n f = Af H
˜ α,n ) = dom(A) ˜ , f ∈ dom(H
(3.4)
where {Wn }∞ n=1 is a regularizing sequence of bounded operators, which converges in the strong resolvent sense to W , and A˜ is a self-adjoint extension of A0 . If the ˜ α,n }∞ sequence {H n=1 tends in the strong resolvent sense to some self-adjoint operator ˜ α and H sup kWn f k < ∞ , f ∈ D , (3.5) n
˜ α is a self-adjoint extension of Hα and it is usually adopted as the then indeed H right Hamiltonian of the perturbed system. An essential ingredient of our approach is a maximality of the Friedrichs extension Aˆ of A0 with respect to the perturbation W . This means that if for some semi-bounded self-adjoint extension A˜ we have √ (3.6) dom(ˆ ν ) ⊆ dom(˜ ν ) ⊆ dom( −W ) , ˜ respectively, then where νˆ and ν˜ are quadratic forms associated with Aˆ and A, ˜ ˆ A coincides with √ the Friedrichs extension A. Note that by (??) one always has dom(ˆ ν ) ⊆ dom( −W ). Under the assumption that Aˆ is maximal with respect to the perturbation W , that the regularizing sequence {Wn }∞ n=1 satisfies the condition p √ √ |Wn | + If = −W + If , f ∈ dom( −W ) , lim n→∞
(3.7)
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
839
˜ α,n }∞ obeys and that the approximating sequence {H n=1 p p ˜ α,n − z)−1 |Wn |k < ∞ , =m(z) 6= 0 , sup k |Wn |(H
(3.8)
n
we obtain that if convergence of the sequence (??) takes place in the strong resolvent sense to some semi-bounded self-adjoint operator, then the limit is an extension of Hα , which is canonical in the sense that it necessarily coincides with the form-sum . ˆ α = Aˆ + αW . Conversely, for each semi-bounded self-adjoint extension A˜ of A0 H there is a regularizing sequence {Wn }∞ n=1 such that the corresponding approximatˆ α and the condition (??) is satisfied. So, ing sequence (??) tends to the form-sum H ˆ α or we have no either we have convergence to the unique canonical form-sum H convergence at all. That is why the case, when the Friedrichs extension is maximal with respect to the perturbation, was called in [?] a well-posed regularization problem. We have shown (see e.g. [?, ?]) that the well-posed regularization problem takes place if and only if √ (3.9) dom( −W ) ∩ ker(A∗0 − ηI) = {0} for some η < 0. √ The opposite case when dom( −W ) ∩ ker(A∗0 − ηI) 6= {0} for some η < 0 and, hence for each η < 0, was called in [?] an ill-posed regularization problem. In this √ case ν ) ⊆ dom( −W ) for each semi-bounded self-adjoint extension A˜ of A0 obeying dom(˜ the approximating sequence (??) converges in the strong resolvent sense to the form . ˜ α = A˜ + αW at least for sufficiently small coupling constants α. Moreover, sum H the condition (??) is satisfied. Consequently, in the case of the ill-posed problem we have as a disadvantage the lost of the uniqueness of the approximating procedure. ˜ α for physical Hamiltonian of the perturbed Hence, there are a lot of candidates H system. In other words, the approximating procedure is not helpful in finding a right Hamiltonian because it gives a variety of possible Hamiltonians. To single out in this variety a right Hamiltonian one has to apply some additional physical or mathematical arguments, for instance, to restrict the class of allowed semi-bounded self-adjoint extensions A˜ of A0 . Let us show that the ill-posed case realizes if D itself is not a stability domain of the pair {A, W }. Lemma 3.2. Let ν ≥ 0 and γ ≥ 0 be two closed quadratic forms such that dom(ν) ∩ dom(γ) is dense. Then we have a dense domain D ⊆ dom(ν) ∩ dom(γ) and constants α, a, b, 0 < α ≤ 1, 0 < a < 1, b ≥ 0, such that αγ(f, f ) ≤ aν(f, f ) + b(f, f ) ,
f ∈ D,
(3.10)
if and only if there is a densely defined closed restriction νˆ of ν such that dom(ˆ ν) ⊆ dom(γ). Proof. If (??) is satisfied, then one obviously has that the closure νˆ of ν|D obeys dom(ˆ ν ) ⊆ dom(ν) and dom(ˆ ν ) ⊆ dom(γ). Conversely, let us assume that
840
H. NEIDHARDT and V. A. ZAGREBNOV
there is a closed restriction νˆ of ν such that dom(ˆ ν ) ⊆ dom(γ). Let us introduce a scalar product (3.11) (f, g)ν = ν(f, g) + (f, g) , f, g ∈ dom(ν) . Since ν is a closed form, the domain dom(ν) endowed with the scalar product (??) def
ν ) is a closed subspace forms a Hilbert space Hν = {dom(ν), (· , ·)ν }. The set dom(ˆ ν ), then dom(γ) contains the closed subspace of dom(ˆ ν ). of Hν . If dom(γ) ⊇ dom(ˆ Applying the closed graph principle (see e.g. [?, III Sec. 5.4]) one gets a constant C > 0 such that γ(f, f ) ≤ C(ˆ ν (f, f ) + (f, f )) ,
f ∈ dom(ˆ ν) .
Setting D = dom(ˆ ν ) we find constants α, a, b such that (??) holds.
(3.12)
Theorem 3.3. Let {A ≥ 0, W ≤ 0} be a pair of self-adjoint operators with common dense domain D ⊆ dom(A) ∩ dom(W ). If D is not a stability domain of {A, W }, then for any stability domain Ds ⊆ D of {A, W } the Friedrichs extension Aˆ of A|Ds is not maximal with respect to W . Proof. Let us denote closed quadratic forms associated with A and −W and the Friedrichs extension Aˆ of A0 = A|Ds by ν, γ and νˆ, respectively. Since Ds ⊆ D is a stability domain of {A, W } there are constants 0 < a < 1, b ≥ 0, such that (??) is satisfied. By Lemma ?? this ν ) is a closed subspace of Hν obeying √ yields that dom(ˆ ν ) ⊆ dom(ν) ∩ dom(γ). dom(ˆ ν ) ⊆ dom(γ) = dom( −W ). Hence one gets dom(ˆ Note that D ⊆ dom(ˆ ν ) is impossible. Otherwise D would be a stability domain of {A, W }. Hence there is an element f ∈ D such that f 6∈ dom(ˆ ν ). Since f ∈ dom(ν) the element f admits an orthogonal decomposition in Hν of the form f = g +h,
g ∈ dom(ˆ ν) ,
h ⊥ dom(ˆ ν) .
(3.13)
Since f , g ∈ dom(γ), one obviously has h ∈ dom(γ). Hence h ∈ dom(ν) ∩ dom(γ) and h ⊥ dom(ˆ ν ) in Hν , i.e. ν(h, f ) + (h, f ) = 0 ,
∀f ∈ Ds .
(3.14)
Since Ds ⊆ dom(A), one gets (h, (A + I)f ) = (h, (A0 + I)f ) = 0 , ∀f ∈ Ds . (3.15) √ √ Therefore, h ∈ ker(A∗0 + I). Since h ∈ dom( −W ), one gets dom( −W )∩ker(A∗0 + I) 6= {0}. Hence Aˆ is not maximal with respect to W . If D is not a stability domain, by Theorem ?? there is always a smaller stability domain Ds ⊆ D of {A, W } provided Hα is unbounded from above at least for one α > 1. Hence results of [?, ?] can be generalized. However, this has the price that the ill-posed case inevitably emerges even if the stability domain Ds of {A, W } is maximal for given constants 0 < a < 1, b ≥ 0. In particular this means that approximating method fails in definition of the right Hamiltonian.
841
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
Note that the D is not a stability domain of {A, W } if and only if Hα is unbounded from below for any α > 1. Hence conclusions of Theorem ?? are meaningful if for any α > 1 the operator Hα is unbounded from below (D is no stability domain) and at least for one α > 1 the operator Hα is unbounded from above (existence of stability domain Ds ⊆ D). 4. Example Let us illustrate Theorem ?? by Example ?? when κ ∈ (0, 1): “fall to the center problem”. Since we fix κ ∈ (0, 1), in the following we omit the κ indices for def simplicity, i.e., we set H = H κ1 . As far as the operator H is direct sum of two symmetric operators H ± which corresponds to the positive and negative half axis, i.e. (H ± f )(x) = −
d2 1 1 f (x) − f (x) , dx2 4κ x2
f ∈ dom(H ± ) = C0∞ (R1± ) ,
x ∈ R1± , (4.1)
we can concentrate our considerations only on H + . Since H + is unbounded from above, by Theorem ?? one gets that for each b ∈ R1 there is a maximal b-stability domain Db ⊆ C0∞ (R1+ ) for H + . Below we describe this domain in explicit form. Let us introduce an auxiliary operator: (H(a)f )(x) = −
1 1 d2 f (x) − f (x) , dx2 4κ (x + a)2
dom(H(a)) = {f ∈
W22 ([0, ∞))
: f (0) = 0} ,
(4.2) a > 0.
The operator H(a) is self-adjoint and semi-bounded from below for each a > 0. Let def
λ(a) = inf σ(H(a)) ,
a > 0,
(4.3)
where by σ(X) we denote the spectrum of an operator X. Then the function λ(·) is continuous. Since the family of operators {H(a)}a>0 is non-decreasing, one gets that λ(·) is non-decreasing too. A straightforward computation shows that lima→∞ λ(a) = 0. Since H + is unbounded from below one has that inf a>0 λ(a) = −∞. Let b < 0. Then we set def
N (b) = {a > 0 : λ(a) ≥ b} ,
b < 0.
(4.4)
def
Let a0 = inf N (b). Then by continuity of λ(·) one gets λ(a0 ) = b. Furthermore, a < a0 yields λ(a) < b. Further let us introduce a self-adjoint operator (K(r)f )(x) = −
1 1 d2 f (x) − f (x) , 2 dx 4κ (x + r)2
dom(K(r)) = {f ∈
W22 ([0, 1])
: f (0) = f (1) = 0} ,
(4.5) r > 0.
and def
µ(r) = inf σ(K(r)) .
(4.6)
842
H. NEIDHARDT and V. A. ZAGREBNOV
First of all we note that µ(r) is continuous and non-decreasing function of r > 0 and one has (4.7) lim µ(r) = inf σ(K(∞)) , r→∞
where (K(∞)f )(x) = −
d2 f (x) , dx2
dom(K(∞)) = {f ∈
W22 ([0, 1])
(4.8) : f (0) = f (1) = 0} .
Since inf σ(K(∞)) = π 2 , one gets lim µ(r) = π 2 > 0 .
(4.9)
r→∞
As above one can prove that inf r>0 µ(r) = −∞. Next we introduce the set a2 b ≥0 , M(a) = r ∈ (0, ∞) : µ(r) − (1 + r)2 2
a > 0.
(4.10)
2
a b a b 2 Since limr→0 (1+r) 2 = a b < 0 and limr→∞ (1+r)2 = 0, the set M(a) is always not empty. Let a0 be chosen as above. For a = a0 we set def
r1 = inf M(a0 ) .
(4.11)
By the continuity of µ(·) one always has a20 b . (1 + r1 )2
(4.12)
r1 < a0 . 1 + r1
(4.13)
µ(r1 ) = Let a1 = a 0
def
Now we consider the set M(a1 ). Then we define r2 = inf M(a1 ) and set a2 = a1
r2 < a1 < a0 . 1 + r2
(4.14)
Moreover, one has
r2 r1 . (4.15) 1 + r2 1 + r1 By this manner we work out the sequence of numbers {rn }∞ n=1 which defines the sequence {an }∞ n=1 , where n Y rk an = a 0 . (4.16) 1 + rk a2 = a0
k=1
Let us show that the sequence {an }∞ n=0 tends to zero as n → ∞. To this end we def
note that rj ≤ r0 , j = 1, 2, . . . , where r0 = inf M(0). Since r0 rj ≤ < 1, 1 + rj 1 + r0
j = 1, 2, . . . ,
(4.17)
843
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
one gets an = a0
n Y k=1
rk ≤ 1 + rk
r0 1 + r0
n a0 ,
n = 1, 2, . . . ,
(4.18)
which immediately implies limn→∞ an = 0. We set ∞ [ {an } . Γ=
(4.19)
n=0
and define domain Db = {f ∈ C0∞ (R1+ ) : f (an ) = 0 , an ∈ Γ} which is obviously dense in L2 (R1+ ). To prove that Db is a stability domain for H + we use the following lemma. Lemma 4.1. The inequality Z +∞ Z +∞ Z +∞ 1 1 2 |f 0 (x)|2 dx − |f (x)| dx ≥ b |f (x)|2 dx , 2 4κ x a a a
(4.20)
holds for f ∈ C0∞ (R1+ ), f (a) = 0, 0 < a < +∞ and b < 0 if and only if λ(a) ≥ b. Furthermore, one has Z
β
|f 0 (x)|2 dx −
α
1 4κ
Z
β
α
1 |f (x)|2 dx ≥ b x2
Z
β
|f (x)|2 dx ,
(4.21)
α
for f ∈ C0∞ (R1+ ), f (α) = f (β) = 0, 0 < α < β < +∞ and b < 0 if and only if µ(r) ≥
β2 (1+r)2 b
def
where r =
α β−α .
Proof. Setting x = t + a and g(t) = f (t + a) one gets that (??) is equivalent to Z
+∞ 0
|g 0 (t)|2 dt −
1 4κ
Z 0
+∞
1 |g(t)|2 dt ≥ b (t + a)2
Z
+∞
|g(t)|2 dt ,
(4.22)
0
g ∈ C ∞ (R1+ ), g(0) = 0. Since (??) is equivalent to H(a) ≥ bI one gets that (??) is valid if and only if λ(a) ≥ b. By x = αr (y+r) and h(y) = f ( αr (y +r)), y ∈ [0, 1], we find that (??) is equivalent to Z 1 Z 1 Z 1 1 α2 1 2 |h0 (y)|2 dy − |h(y)| dy ≥ b |h(y)|2 dy , (4.23) 4κ 0 (y + r)2 r2 0 0 α2 r 2 bI.
h ∈ C ∞ ([0, 1]), h(0) = h(1) = 0. However, (??) is equivalent to K(r) ≥ Therefore, the inequality (??) is satisfied if and only if µ(r) ≥
α2 r2 b
=
2
β (1+r)2 b.
Let f ∈ C0∞ (R1+ ) and f (a0 ) = 0. Then we have to verify that Z ∞ Z ∞ Z ∞ 1 1 2 |f 0 (x)|2 dx − |f (x)| dx ≥ b |f (x)|2 dx . 4κ a0 x2 a0 a0
(4.24)
By Lemma ?? this condition is satisfied if λ(a0 ) ≥ b which is right due to (??).
844
H. NEIDHARDT and V. A. ZAGREBNOV
Let f ∈ C0∞ (R1+ ) and f (an ) = f (an−1 ) = 0, n = 1, 2, . . . . Then one has to prove that Z an−1 Z an−1 Z an−1 1 1 0 2 2 |f (x)| dx − |f (x)| dx ≥ b |f (x)|2 dx , n = 1, 2, . . . 4κ an x2 an an (4.25) a2
n−1 Applying Lemma ?? we obtain that (??) is valid if µ(rn ) ≥ (1+r 2 b, n = 1, 2, . . . . n) Since rn = inf M(an−1 ), this immediately follows from (??). Next let us prove that the domain Db is maximal. In fact the reasoning below is the proof of a constrained variational principle. To this end we assume that Db is not maximal. Therefore, there is at least one function f0 ∈ C0∞ (R1+ ) such that / Db but f0 ∈ (4.26) (H + f0 , f0 ) ≥ b(f0 , f0 )
and f0 (ap ) 6= 0 for at least one ap ∈ Γ. Otherwise f0 ∈ Db . We set Γ0 = {an ∈ Γ : f0 (an ) 6= 0} ⊆ Γ .
(4.27)
Denoting by Db0 the linear span of the linear set Db and the element f0 we find that ) ( ) = 0 a ∈ Γ \ Γ g(a n n 0 (4.28) Db0 = g ∈ C0∞ (R1+ ) : f0 (am )g(an ) = f0 (an )g(am ) am , an ∈ Γ0 In other words, if for some am ∈ Γ0 the value g(am ), g ∈ Db0 , is given, then g(an ) is given for all an ∈ Γ0 . Since f0 ∈ C0∞ (R1+ ) there is an m such that inf Γ0 = am , i.e. f0 (an ) = 0 for n = m + 1, m + 2, . . . . For a given a00 ∈ (a1 , a0 ) we define the sequence {a0n }∞ n=1 by a0n = a00
n Y k=1
rk , 1 + rk
n = 1, 2, . . . .
(4.29)
Choosing a suitable a00 ∈ (a1 , a0 ) we satisfy conditions an+1 < a0n < an ,
n = 1, 2, . . . , m .
(4.30)
Since a00 < a0 one has λ(a00 ) < λ(a0 ) = b. Otherwise, we have λ(a00 ) = λ(a0 ) which implies a00 = a0 . Hence by Lemma ?? there is a b00 < b, λ(a00 ) < b00 < b, and a non-trivial element g0 ∈ C0∞ (R1+ ), g0 (a00 ) = 0, such that Z ∞ Z ∞ Z ∞ 1 1 2 0 |g00 (x)|2 dx − |g (x)| dx ≤ b |g0 (x)|2 dx . (4.31) 0 0 4κ a00 x2 a00 a00 Since µ(rn ) ≥ Z
a0n−1
a0n
a2n−1 (1+rn )2 b
|g 0 (x)|2 dx −
1 4κ
0
2 an−1 (1+rn )2
= Z
a0n−1
a0n
an−1 a0n−1
2 b, by Lemma ?? we get that
1 |g(x)|2 dx ≥ x2
an−1 a0n−1
2 Z b
a0n−1
a0n
|g(x)|2 dx ,
(4.32)
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
845
for g ∈ C0∞ (R1+ ), g(a0n ) = g(a0n−1 ) = 0, n = 1, 2, . . . , m. Fixing a sequence {b0n }m n=1 , which satisfies the condition 2 an−1 b < b0n < b < 0 , n = 1, 2, . . . , m , (4.33) a0n−1 ∞ 1 0 by Lemma ?? we find a sequence of elements {gn }m n=1 , gn ∈ C0 (R+ ), gn (an ) = gn (a0n−1 ) = 0, n = 1, 2, . . . , m, such that
Z
a0n−1
a0n
|gn0 (x)|2 dx
1 − 4κ
Z
a0n−1
1 |gn (x)|2 dx ≤ b0n x2
a0n
Z
a0n−1
a0n
|gn (x)|2 dx
(4.34)
Let us show that gn (an ) 6= 0 ,
an ∈ Γ 0 .
(4.35)
If gn (an ) = 0 for some an ∈ Γ0 , then Z
a0n−1
|gn0 (x)|2 dx
an
and
Z
an
a0n
1 − 4κ
Z
|gn (x)|2 dx −
a0n−1
an
1 4κ
Z
1 |gn (x)|2 dx ≥ b x2
an
a0n
which contradicts (??). We set g=
Z
|gn (x)|2 dx
(4.36)
|gn (x)|2 dx
(4.37)
an
1 |gn (x)|2 dx ≥ b x2
m X
a0n−1
Z
an
a0n
αn gn ,
(4.38)
n=0
where
0 1 αn = f0 (an ) gm (am ) f0 (am ) gn (an )
an ∈ Γ \ Γ0 n=m
(4.39)
n 6= m, an ∈ Γ0 .
Then g ∈ C0∞ (R1 ) and, moreover, it satisfies f0 (am )g(an ) = f0 (an )g(am ) for an ∈ Γ0 . Hence g ∈ Db0 . Moreover, by (??) one gets that Z ∞ Z ∞ 1 1 |g 0 (x)|2 dx − |g(x)|2 dx 4κ 0 x2 0 Z ∞ Z ∞ ≤ sup b0n |g(x)|2 dx < b |g(x)|2 dx . (4.40) n=0,1,...,m
0
0
which shows that Db0 is not a b-stability domain. The proof of the maximality gives an idea how to find other maximal domains. To this end one chooses a number a = a000 > a0 (see (??)). After that we find numbers a001 , a002 , . . . following the formula (??). Assume that a001 < a0 < a000 which can be arranged by the choice of a000 . Then we set Db00 = {f ∈ C0∞ (R1+ ) : f (a0m ) = 0 ,
m = 0, 1, 2, . . .} .
(4.41)
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H. NEIDHARDT and V. A. ZAGREBNOV
Repeating the above line of reasoning for a = a000 we obtain that the domain Db00 is also maximal but Db00 6= Db . Note that by construction the symmetric b-stable restriction H + |Db has (as one should anticipate) infinite deficiency indices {∞, ∞} which corresponds to infinite cardinality of the set Γ. Furthermore, Example ?? allows to describe explicitly the Friedrichs extension Aˆ+ of A+ 0 = A|Db . Denoting by ∆n the intervals ∆0 = [a0 , ∞) and ∆n = [an , an−1 ], n = 1, 2, . . ., one can decompose the Hilbert space H = L2 (R1+ ) into H=
∞ M
Hn ,
Hn = L2 (∆n ) .
(4.42)
n=0
ˆ n , n = 0, 1, 2, . . ., the operators Denoting by L ˆ 0 f )(x) = − (L
d2 f (x) , dx2
ˆ 0 ) = {f ∈ W22 (∆0 ) : f (a0 ) = 0} f ∈ dom(L
(4.43)
and d2 ˆ n ) = {dom(W22 (∆n ) : f (an ) = f (an−1 ) = 0} , f (x) , f ∈ dom(L dx2 (4.44) n = 1, 2, . . ., one gets ∞ M ˆn . (4.45) L Aˆ+ = ˆ n f )(x) = − (L
n=0
To illustrate Theorem ?? we show that the Friedrichs extension Aˆ+ is not maximal with respect to W . The element h belongs to the deficiency subspace Nη = ker(A+∗ 0 − ηI) if and only if h is continuous and admits the representation h(x) = b0 e−
√ −ηx
,
x ∈ ∆0 ,
b0 ∈ C ,
(4.46)
and h(x) = bn e−
√ −ηx
+ cn e
√ −ηx
,
x ∈ ∆n ,
bn ∈ C ,
n = 1, 2, . . . .
(4.47)
The continuity of h imposes some relations between the coefficients bn and cn , n = 0, 1, 2, . . .. Obviously there are elements h ∈ Nη such that h(x) = 0 for x ∈ ∆n for n greater than some natural number N > 1. For instance, if N = 2 the function √ − −ηx b e x ∈ ∆0 0 √ √ sinh( −η(x − a )) 1 √ x ∈ ∆1 h(x) = b0 e− −ηa0 (4.48) sinh( −η(a − a 0 1 )) 0 x ∈ ∆n , n = 2, 3, . . . belongs to Nη and has the property that √ h ∈ dom( −W ) ,
(4.49)
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
which means that
Z
∞
0
1 |h(x)|2 dx = x2
Z
∞ a1
1 |h(x)|2 dx < +∞ . x2
847
(4.50)
√
On the other hand, the element h(x) =√be− −ηx , x ∈ R1+ , also belongs to Nη , √ but h ∈ / dom( −W ). Therefore, dom( −W ) ∩ ker(A+∗ 0 − ηI) 6= {0}, i.e., the regularization problem is ill-posed. . Note that by construction (??) of Aˆ+ , the spectrum of form sum H + = A+ + 1 κ W , 0 < κ < 1, for b < 0 has in addition to a discrete part an absolutely continuous part which results from the infinite interval ∆0 . The situation changes, if the b is chosen non-negative, i.e. b ≥ 0. Since the set N (b) (??) is empty for b ≥ 0 (see e.g. [?]), we proceed as follows. First we choose an arbitrary a = a0 > 0. Then as above we define with this a0 the sequence {an }∞ n=1 and we can prove that limn→∞ an = 0. After that we introduce the set a2 b def (4.51) M− (a) = r ∈ (0, ∞) : µ(r) − 2 ≥ 0 , a > 0 . r 2
2
Since limr→0 ar2b = ∞ (or zero if b = 0) and limr→∞ ar2b = 0, (??) implies that the set M− (a) is always not empty. Furthermore, one has inf M− (a) > 0. So we can define (4.52) r−1 = inf M− (a0 ) and a−1 = a0
1 + r−1 > a0 . r−1
(4.53)
Then we pass to r−2 = inf M− (a1 ) and a−2 = a1
1 + r−2 1 + r−2 1 + r−1 = a0 > a−1 > a0 . r−2 r−2 r−1
(4.54)
∞ Finally, we obtain a sequence {r−n }∞ n=1 which defines the sequence {a−n }n=1 , where
a−n = a0
n Y 1 + r−k . r−k
(4.55)
k=1
Let us show that limn→∞ a−n = ∞. By monotonicity limn→∞ a−n = a−∞ exist. Assume that a−∞ < +∞. We set r−∞ = inf M− (a−∞ ). Obviously we have r−j ≤ r−∞ which yields 1< Since a−n = a0
1 + r−j 1 + r−∞ ≤ , r−∞ r−j
j = 1, 2, . . . .
n n Y 1 + r−k 1 + r−∞ ≥ a0 , r−k r−∞
n = 1, 2, . . . ,
(4.56)
(4.57)
k=1
one gets limn→∞ a−n = a−∞ = ∞. Therefore, the sequence {a−n }∞ n=0 tends to infinity as n → ∞.
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Setting −
Γ =
∞ [
{an }
(4.58)
n=−∞
one defines as above Db = {f ∈ C0∞ (R1+ ) : f (an ) = 0 , an ∈ Γ− }. Similarly to the case b < 0 one can show that Db is a maximal b-stability domain for H + . Moreover, starting with a = a00 6= a0 one can construct another maximal b-stability domain Db0 for b ≥ 0. In contrast to the case b < 0, when limn→∞ an = 0, it turns out that for . b ≥ 0, when limn→∞ a−n = +∞, the spectrum of the form sum H + = Aˆ + κ1 W , 0 < κ < 1, is purely point one, i.e., absolutely continuous part of the spectrum is absent. 5. Conclusions This paper is initiated by a general problem of definition and construction of the right Hamiltonian for non-positive singular perturbations, see [?, ?, ?, ?]. The theory developed there for a pair of self-adjoint operators {A ≥ 0, W ≤ 0} includes as an essential ingredient a stability domain of the pair {A, W }, i.e., a dense domain Ds ⊆ dom(A) ∩ dom(W ) such that (??) is satisfied. A priori the existence of Ds is not evident. Our abstract Theorem ?? gives a simple criterion for that: symmetric operator Hα = A + αW with domain dom(Hα ) = dom(A) ∩ dom(W ) should be unbounded from above at least for one α > 1. From the physical point of view this is a very reasonable condition which is, of course, very often satisfied in physical systems. Naturally this leads to a simple idea to handle non-positive singular perturbations as follows: we choose a stability domain of the pair {A, W } and a regularizing sequence for the non-positive singular perturbation W , with this we can construct the approximating Hamiltonian sequence and determine the limit of this sequence which we call the “right Hamiltonian” of the perturbed system, i.e., unique, self-adjoint, semi-bounded from below operator. However, this simple idea is false. The main reason for that is that in general the “right Hamiltonian” is not uniquely determined in this way. In order to guarantee uniqueness of the “right Hamiltonian” one has to verify maximality of Friedrichs extension Aˆ of A|Ds with respect to the singular perturbation W [?, ?]. If one has maximality, then the “right Hamiltonian” for {A, W } is unique and coincides with the form-sum of Friedrichs extension Aˆ and the perturbation W . Since this case is very satisfactory it was called in [?, ?] the “well-posed regularization problem”. In the opposite case, i.e., if there is no maximality, one gets a variety of “right Hamiltonians”. So, the pleasant and important from the physical point of view uniqueness is lost. Consequently, the regularization problem was called ill-posed. In [?] it was mentioned that in some sense an ill-posed regularization problem is an incomplete well-posed one and that it can be reduced to the latter but not in a unique way. Consequently, in order to guarantee the uniqueness of the “right Hamiltonian” it is necessary to introduce besides the regularization method some supplementary physical principles to select it.
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
849
Unfortunately the ill-posed case always realizes if the set dom(A) ∩ dom(W ) is not a stability domain of {A, W }. This means, that the Theorem ??, which guarantees the existence of a stability domain Ds ⊆ dom(A) ∩ dom(W ) of {A, W }, does not save the situation: the ill-posed case necessarily appears due to Theorem ??. Hence the regularization method of [?, ?, ?, ?] is unsuitable to find a “right Hamiltonian” if we have no canonical stability domain from the beginning. In other words, the regularization problem is well-posed for the pair {A, W } only if dom(A) ∩ dom(W ) is a stability domain of {A, W } and, moreover, the Friedrichs extension Aˆ of A|dom(A) ∩ dom(W ) is maximal with respect to the singular perturbation W . This is the physical quintessence of Sec. 3. Besides this we hope that Theorem ?? has an independent meaning since it gives a criterion when a symmetric operator can be restricted to semi-bounded one. In particular it implies that well-known quantum momentum (or coordinate) operator on the real axis can be restricted to a non-negative one.a Finally, in Sec. 4 we present a solution of the quantum mechanics “fall to the center” problem for the Hamiltonian H κ1 , κ ∈ (0, 1), of Example ??. Theorem ?? suggests the idea that H κ1 is defined on a domain which is too large. Hence one has to restrict H κ1 in a suitable manner such that it becomes semi-bounded. In Sec. 4 it is shown how this can be done. From the construction of (maximal) stability domains it follows that the physical interpretation of the way to prevent the fall to the center is to introduce a suitable sequence of point barriers of infinite height which come closer and closer to the center. How to choose the sequence of point barriers is indicated in by Γ (??) and Γ− (??). However, it is easy to see that there are several possibilities to do this. Finally, to illustrate our “no-go” Theorem ?? we show that the Friedrichs extension Aˆ+ is not maximal with respect to the perturbation W . Acknowledgments We would like to thank the referee for numerous useful remarks and especially for his help in finding out of a flaw in the first proof of maximality of the domain Db in our Example (Sec. 4). This flaw is corrected in the present version of the paper. We are grateful to Charles Radin who attracted our attention to the papers [?, ?, ?]. The first author (H. N.) thanks the Center de Physique Th´eorique, Universit´e de Toulon et du Var and Universit´e de Provence for hospitality during his visits to Marseille–Luminy and for financial support. Note added in Proof We would like to thank Werner Timmermann who attracts our attention to the von Neumann’s remark in Math. Ann. 102 (1929) 49–131 (“Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren”) about “patalogies” of unbounded symmetric operators. There von Neumann discovered the same phenomenon as a N. Neidhardt and V. A. Zagrebnov, “On semi-bounded restrictions of self-adjoint operators”, preprint CPT-97/P. 3512, Marseille-Luming, 1997.
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we discuss in Sec. 2 of the present paper. References [1] K. M. Case, “Singular potentials”, Phys. Rev. 80 (5) (1950) 797–806. [2] H. van Haeringen, “Bound states for r −2 -like potentials in one and more dimensions”, J. Math. Phys. 19 (10) (1978) 2171–2179. [3] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966. [4] H. Neidhardt and V. A. Zagrebnov, “Regularization and convergence for singular perturbations”, Commun. Math. Phys. 149 (1992) 573–586. [5] H. Neidhardt and V. A. Zagrebnov, “Singular perturbations, regularization and extension theory”, in Operator Theory: Advances and Applications 70, 299–305, Birkh¨ auser Verlag, Basel, 1994. [6] H. Neidhardt and V. A. Zagrebnov, “On the right Hamiltonian for singular perturbations: General Theory”, Rev. Math. Phys. 9 (5) (1997). [7] H. Neidhardt and V. A. Zagrebnov, “Towards the right Hamiltonian for singular perturbations via regularization and extension theory”, Rev. Math. Phys. 8 (5) (1996) 715–740. [8] E. Nelson, “Feynman integrals and Schr¨ odinger equation”, J. Math. Phys. 5 (3) (1964) 332–343. [9] C. Radin, “Some remarks on the evolution of a Schr¨ odinger particle in an attractive 1/r2 potential”, J. Math. Phys. 16 (3) (1975) 544–547. [10] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I: Functional Analysis, Academic Press, 1972. [11] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press, 1975. [12] K. Schm¨ udgen, “On restrictions of unbounded symmetric operators”, J. Operator Theory. 11 (1984) 379–393. [13] G. E. Shilov, An Introduction to the Theory of Linear Spaces, Prentice-Hall, Inc., N. J., 1961.
CONSTRUCTION OF KINK SECTORS FOR TWO-DIMENSIONAL QUANTUM FIELD THEORY MODELS AN ALGEBRAIC APPROACH DIRK SCHLINGEMANN II. Institut f¨ ur Theoretische Physik Universit¨ at Hamburg Germany and Erwin Schr¨ odinger International Institute for Mathematical Physics (ESI) Boltzmanngasse 9, A-1090 Wien, Austria Several two-dimensional quantum field theory models have more than one vacuum state. Familiar examples are the Sine-Gordon and the φ42 -model. It is known that in these models there are also states, called kink states, which interpolate different vacua. A general construction scheme for kink states in the framework of algebraic quantum field theory is developed in a previous paper. However, for the application of this method, the crucial condition is the split property for wedge algebras in the vacuum representations of the considered models. It is believed that the vacuum representations of P (φ)2 -models fulfill this condition, but a rigorous proof is only known for the massive free scalar field. Therefore, we investigate in a construction of kink states which can directly be applied to a large class of quantum field theory models, by making use of the properties of the dynamics of a P (φ)2 and Yukawa2 models.
1. Introduction Studying 1 + 1-dimensional quantum field theories from an axiomatic point of view shows that kink sectors naturally appear in the theory of superselection sectors [21, 22, 58]. This paper is concerned with the construction of kink sectors for concrete quantum field theory models, like P (φ)2 and Yukawa2 models.a Our subsequent analysis is placed into the framework of algebraic quantum field theory which has turned out to be a successful formalism to describe physical concepts like observables, states, superselection sectors (charges) and statistics. These notions can be appropriately described by mathematical concepts like C ∗ -algebras, positive linear functionals and equivalence classes representations. For the convenience of the reader, we shall state the relevant definitions and assumptions here. Let O ⊂ R1,s be a region in space-time. We denote by A(O) the algebra generated by all observables which can be measured within O. For technical reasons we always suppose that A(O) is a C ∗ -algebra and O is a double cone, i.e. a bounded and causally complete region. Motivated by physical principles, we make the following assumptions: a Parts
are extracted from the PhD thesis [61]. 851
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D. SCHLINGEMANN
(1) The assignment A : O 7→ A(O) is an isotonous net of C ∗ -algebras, i.e. if O1 is contained in O2 , then A(O1 ) is a C ∗ -sub-algebra of A(O2 ). The isotony encodes the fact that each observable which can be measured within O can also be measured in every larger region. Furthermore, the C ∗ -inductive limit C ∗ (A) of the net A can be constructed since the set of double cones is directed. We refer to [57] for this notion. (2) Two local operations which take place in space-like separated regions should not influence each other. The principle of locality is formulated as follows: If the regions O1 and O2 are space-like separated, then the elements of A(O1 ) commute with those of A(O2 ). (3) Each operator a which is localized in a region O should have an equivalent counterpart which is localized in the translated region O + x. The principle of translation symmetry is encoded by the existence of an automorphism group {αx ; x ∈ R1,s } which acts on the C ∗ -algebra C ∗ (A) such that αx maps A(O) onto A(O + x). A net of C ∗ -algebras which fulfills conditions (1) to (3) is called a translationally covariant Haag–Kastler net . In order to discuss particle-like concepts, we select an appropriate class S of normalized positive linear functionals, called states, of C ∗ (A). We require the states ω ∈ S to fulfill the conditions: (1) There exists a strongly continuous unitary representation of the translation group U : x 7→ U (x) on the GNSb -Hilbert space H which implements the translations in the GNS-representation π, i.e. π(αx a) = U (x)π(a)U (−x) for each a ∈ C ∗ (A). (2) The stability of a physical system is encoded in the spectrum condition (positivity of the energy), i.e. the spectrum (of the generator) of U (x) is contained in the closed forward light cone. These conditions are also known as the Borchers criterion. States which satisfy the Borchers criterion and which are, in addition, translationally invariant are called vacuum states. a state ω ∈ S, we obtain via GNS-construction a Hilbert space H, a ∗ -representation π of C ∗ (A) on H and a vector Ω ∈ H such that hΩ, π(a)Ωi = ω(a) for each a ∈ C ∗ (A). The triple (H, π, Ω) is called the GNS-triple of ω.
b Given
CONSTRUCTION OF KINK SECTORS FOR
...
853
Kinks already appear in classical field theories and the typical systems in which they occur are 1 + 1-dimensional. Familiar examples are the Sine-Gordon and the φ42 -model. We briefly describe the latter: The Lagrangian density of the model is given by L(φ, x) =
1 ∂µ φ(x)∂ µ φ(x) − U (φ(x)) , 2
where the potential U is given by U (z) := λ/2 (z 2 − a)2 . The energy of a classical field configuration φ is Z 1 1 (∂0 φ(0, x))2 + (∂1 φ(0, x))2 + U (φ(0, x)) . E(φ) = dx 2 2 With the choice of U , given above, the absolute minimum value of U is zero and thus the energy functional E : φ 7→ E(φ) is positive. There are two configurations φ± with zero energy E(φ± ) = 0: φ± : (t, x) 7→ ± a . These configurations are invariant under space-time translations and represent the vacua of the classical system. There are two further configurations φs , φs¯ which are stationary points of the energy functional E. They are given by √ √ φs : (t, x) 7→ a tanh( λax) and φs¯ : (t, x) 7→ −a tanh( λax) . These configurations represent the kinks of the classical system which interpolate the vacua φ± . Indeed, we have for the kink φs lim φs (t, x) = φ± (t, x) = ±a .
x→±∞
(1)
The configuration φs¯, which interpolates the vacua φ± in the opposite direction, represents the anti-kink of φs . Both of them have the same energy, namely E(φs ) = E(φs¯) =
4√ 3 λa . 3
From the classical example above, we see that the crucial properties of a kink are to interpolate vacuum configurations as well as to be a configuration of finite energy. Motivated by these properties, in quantum field theory a kink state ω is defined as follows: The interpolation property: For each observable a, the limits lim ω(α(t,x) (a)) = ω± (a)
x→±∞
(2)
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D. SCHLINGEMANN
exist and ω± are vacuum states. Note that Eq. (2) is the quantum version of the interpolation property (1). Positivity of the energy: ω fulfills the Borchers criterion. In the literature the concept of kink as described above is often called soliton (see [27, 28]) or more seldom lump (see [12]). In the following, we shall use the word kink. In [59], a construction scheme for kink states has been developed which is based on general principles. In order to make the comprehension of the subsequent sections easier we shall state the main ideas here. The construction of an interpolating kink state is based on a simple physical idea: Let A be a Haag–Kastler net of W ∗ -algebras in 1 + 1-dimensions. Each double cone O splits our system into two infinitely extended laboratories, namely the laboratory which belongs to the left space-like complement OLL , and the laboratory ORR which belongs the right spacelike complement ORR . In order to prepare an interpolating kink state, we wish to prepare one vacuum state ω1 in the left laboratory OLL , and another vacuum state ω2 in the right laboratory ORR . This can only be done if the preparation of ω1 does not disturb the preparation procedure of ω2 . In other words, the physical operations which take place in the laboratory on the left side OLL should be statistically independent of those which take place in ORR . Therefore, we require that there exists a vacuum representation π0 such that the W ∗ -tensor product Aπ0 (OLL ) ⊗ Aπ0 (ORR ) is unitarily isomorphic to the von Neumann algebra Aπ0 (OLL ) ∨ Aπ0 (ORR ) , where Aπ0 is the net in the vacuum representation π0 .c This condition is equivalent to the existence of a type I factor N which sits between Aπ0 (ORR ) and Aπ0 (OR ): Aπ0 (ORR ) ⊂ N ⊂ Aπ0 (OR ) . Here OR is the space-like complement of OLL . In other words, the inclusion Aπ0 (ORR ) ⊂ Aπ0 (OR )
(3)
is split . A detailed investigation of standard split inclusions of W ∗ -algebras has been carried out by S. Doplicher and R. Longo [19]. We also refer to the results of D. Buchholz [9], C. D’Antoni and R. Longo [14] and C. D’Antoni and K. Fredenhagen [13]. an unbounded region U , Aπ0 (U ) denotes the von Neumann algebra which is generated by all local algebras Aπ0 (O) with O ⊂ U .
c For
CONSTRUCTION OF KINK SECTORS FOR
...
855
Let ω1 and ω2 be two inequivalent vacuum states whose restrictions to each local algebra A(O) are normal. Using the isomorphy Aπ0 (OLL ) ⊗ Aπ0 (ORR ) ∼ = Aπ0 (OLL ) ∨ Aπ0 (ORR ) we conclude that the map ab 7→ ω1 (a)ω2 (b), a is localized in OLL and b is localized in ORR , defines a state of the algebra C ∗ (A, OLL ∪ ORR ) which, by the Hahn–Banach theorem, can be extended to a state ω of the C ∗ -algebra of all observables. The state ω interpolates the vacua ω1 and ω2 correctly, but for an explicit construction of an interpolating state which satisfies the Borchers criterion, some technical difficulties have to be overcome. The condition that the inclusion (3) is split is sufficient to develop a general construction scheme for interpolating kink states. We shall give a brief description of it here. Step 1 : We consider the W ∗ -tensor product of the net A with itself: A ⊗ A : O 7→ A(O) ⊗ A(O) . The map αF which is given by interchanging the tensor factors, αF : a1 ⊗ a2 7→ a2 ⊗ a1 is called the flip automorphism. Since the inclusion (3) is split, the flip automorphism is unitarily implemented on Aπ0 ⊗ Aπ0 (ORR ) by a unitary operator θ which is contained in Aπ0 ⊗ Aπ0 (OR ) [14]. The adjoint action of θ induces an automorphism β := (π0 ⊗ π0 )−1 ◦ Ad(θ) ◦ π0 ⊗ π0 which maps local algebras into local algebras. Here we have assumed that the representation π0 is faithful in order to build the inverse π0−1 . For each observable a which is localized in the left space-like complement of O we have β(a) = a, and for each observable b which is localized in the right space-like complement of O we have β(b) = αF (b). Note that β may depend on the choice of the vacuum representation π0 . Step 2 : It is obvious that the state ω := ω1 ⊗ ω2 ◦ β|C ∗ (A)⊗1 interpolates ω1 and ω2 . Let π1 and π2 be the GNS-representations of ω1 and ω2 respectively. Then the GNS-representation π = π1 ⊗ π2 ◦ β|C ∗ (A⊗1) of ω is translationally covariant because the automorphism αx ◦ β ◦ α−x ◦ β
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is implemented by a cocycle γ(x) of local operators in C ∗ (A). The positivity of the energy can be proven by showing the additivity of the energy-momentum spectrum for automorphisms like β. This together implies that ω is an interpolating kink state. In comparison to previous constructions, in particular the work of J. Fr¨ ohlich in which the existence of kink states for the φ42 and the Sine-Gordon model is proven [27, 29, 28], our construction scheme has the following advantages: ⊕ It is independent of specific details of the considered model because the split property (3), which is the crucial condition for applying the construction scheme, can be motivated by general principles. ⊕ It can be applied to pairs of vacuum sectors which are not related by a symmetry transformation, whereas the techniques of J. Fr¨ohlich rely on the existence of a symmetry transformation connecting different vacua. Indeed, according to J. Z. Imbrie [44], there are examples for P (φ)2 models possessing more than one vacuum state, but where the different vacua are not related by a symmetry. We also mention here the papers of K. Gawedzki [35] and S. J. Summers [65]. Unfortunately, there is one disadvantage which is the price we have to pay for using a model independent analysis. The split property for wedge algebras (3) has to be proven for the vacuum states of the model under consideration if we want to apply our construction scheme to it. It is believed that the vacuum states of the P (φ)2 - and Yukawa2 models fulfill this condition, but a rigorous proof is only known for the massive free Bose and Fermi field [13, 9, 64]. In the present paper, we investigate an alternative construction of kink states which can directly be applied to models. It is convenient to formulate our setup in the time slice formulation of a quantum field theory. The time slice-formulation has two main aspects. First, the Cauchy data with respect to a given space-like plane Σ which describes the boundary conditions at time t = 0. Second, the dynamics which describes the time evolution of the quantum fields. The Cauchy data of a quantum field theory are given by a net of v Neumannalgebras M := {M(I) ⊂ B(H0 ) ; I is open and bounded interval in Σ} represented on a Hilbert-space H0 . This net has to satisfy the following conditions: (1) The net is isotonous, i.e. if I1 ⊂ I2 , then M(I1 ) ⊂ M(I2 ). (2) The net is local, i.e. if I1 ∩ I2 = ∅, then M(I1 ) ⊂ M(I2 )0 . (3) There exists a unitary and strongly continuous representation U : x ∈ R 7→ U (x) ∈ U(H0 ) of the spatial translations in Σ ∼ = R, such that αx := Ad(U (x)) maps M(I) onto M(I + x).
CONSTRUCTION OF KINK SECTORS FOR
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857
A one-parameter group of automorphisms α = {αt ∈ Aut(M); t ∈ R} (Aut(M) denotes the automorphisms of C ∗ (M)) is called a dynamics of the net M if the following conditions are fulfilled: (a) The automorphism group α has propagation speed ps(α) ≤ 1, where ps(α) is defined by ps(α) := inf{β 0 |αt M(I) ⊂ M(Iβ 0 |t| ) ; ∀ t, I} . Here Is := I + (−s, s) denotes the interval, enlarged by s > 0. (b) The automorphisms {αt ∈ Aut(M); t ∈ R} commute with the automorphism group of spatial translations {αx ∈ Aut(M); x ∈ R}, i.e. αt ◦ αx = αx ◦ αt ; ∀ x, t . The set of all dynamics of M is denoted by dyn(M). For our purposes it is crucial to distinguish carefully the C ∗ -inductive limit C ∗ (M) of the net M and the corresponding C ∗ - and W ∗ -algebras, which belong to an unbounded region J ⊂ Σ. They are denoted by C ∗ (M, J ) :=
[
k·k
M(I)
and M(J ) :=
I⊂J
_
M(I) respectively .
I⊂J
We claim here that the Cauchy data of the P (φ)2 - and the Yukawa2 model are given by the nets of the corresponding free fields at time t = 0. Before we continue to discuss our methods for constructing kink states, we briefly give here a review of methods and techniques which has been applied in previous papers. During the 70s, examples for interacting quantum field theory models were constructed. It was proven by J. Glimm, A. Jaffe and T. Spencer that two-dimensional models with P (φ)2 -interaction exist, and their vacuum states satisfy the Wightman axioms [36, 40]. Interactions between fermions and bosons have also been studied, in particular the Yukawa2 interactions [36, 37, 62, 63]. Furthermore, an investigation of the Sine-Gordon model has been carried out by J. Fr¨ ohlich an E. Seiler [34]. A few years later, a great deal of attention has been paid to the construction of new superselection sectors which are different from vacuum sectors. In 1976, the existence of kink sectors for the (Φ · Φ)22 -and the Sine-Gordon model was established by J. Fr¨ ohlich [27, 29] (compare also [30, 31]). To illustrate the ideas and techniques which have been used in [27], we give a short review of the construction of the kink sectors of the (Φ · Φ)22 -model. The basic ingredients for the construction of kink sectors of the (Φ · Φ)22 -model have been taken from the work of J. Glimm, A. Jaffe and T. Spencer [36]. They have proven that there are two inequivalent vacuum states ω± for the (Φ·Φ)22 model which are related by a symmetry χ ∈ Aut(M) αt ◦ χ = χ ◦ αt
858
D. SCHLINGEMANN
in the following way: ω+ ◦ χ = ω− . The construction proceeds in several steps: Step 1 : Let s be a smooth test function with the property: There exists a bounded interval I ⊂ Σ such that π if x ∈ IRR , s(x) = 0 if x ∈ ILL where IRR is the right and ILL is the left complement of I. The O(2)-valued function cos(s(x)) sin(s(x)) ∈ O(2) gs : x 7→ gs (x) = − sin(s(x)) cos(s(x)) induces a Bogoliubov automorphism ρs which is defined on the Weyl operators by ρs : exp(iΦ(f1 ) + iΠ(f2 )) 7→ exp(iΦ(gs f1 ) + iΠ(gs f2 )) where Φ = (Φ1 , Φ2 ) is a massive free two-component Bose field and Π its canonically conjugate, acting as operator valued distributions on the Fock space H0 . Since −12 if x ∈ IRR , gs (x) = 12 if x ∈ ILL the automorphism ρs acts trivially on operators which are localized in ILL and as the symmetry χ on those which are localized in IRR . ¯ s := ω+ ◦ ρ−1 Obviously, the states ωs := ω− ◦ ρs and ω s fulfill the interpolation condition for kink states. Step 2 : The explicit knowledge of the dynamics α can be used to prove the existence of a strongly continuous function γ : (t, x) 7→ γ(t, x) , where γ(t, x) is a unitary operator, localized in a sufficiently large interval I(t,x) . It implements the automorphism α(t,x) ◦ ρs ◦ α(−t,−x) ◦ ρ−1 s = Ad(γ(t, x)) and satisfies the cocycle condition: γ(t1 + t, x1 + x) = α(t,x) (γ(t1 , x1 ))γ(t, x) .
(4)
The operators γ(t, x) describe the translation by (t, x) of the kink charge [ω− ◦ ρs ]. It follows from the properties of γ that ωs is translationally covariant and satisfies the spectrum condition. The same holds for the state ω ¯ s := ω+ ◦ ρ−1 s . This implies ¯ s are kink states. that ωs and ω In 1977 J. Fr¨ohlich proved the existence of the kink states of the one-component φ42 -model [28] by using, in comparison to [27], an alternative method. The technical
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difficulties which arise here are due to the fact that one has to deal with a onecomponent Bose field. Therefore, there is no a priori choice for a Bogoliubov transformation ρs . We shall give a brief summary of the results of [28] to illustrate the main differences to the construction of the (Φ · Φ)22 -kinks. The construction of the vacuum sectors of the φ42 -model, which is presented in [28], uses the methods of field theory. The vacuum states of the φ42 -model can be obtained from two measures µ± on S 0 (R2 ) which satisfy the Osterwalder–Schrader axioms. We briefly explain how the measures µ± are constructed as limits of perturbations of the Gaussian measure µ0 . Step 1 : Let µ0 be the Gaussian measure on the space of tampered distributions S 0 (R2 ) with mean zero and covariance C where the integral kernel of C is Z C(x − y) = d2 p (p2 + m2 )−1 eip(x−y) . The regularized interaction part of the Euclidean action is Z S1 (g, φ) = d2 x g(x) (λ : φ(x)4 :µ0 −σ : φ(x)2 :µ0 ) , where : · :µ0 is the normal ordering with respect to the Gaussian measure µ0 and g is a smooth test function. The action S1 (g, φ) is invariant under the substitution φ 7→ −φ. To approximate one of the measures µ± the Z2 symmetry has to be broken explicitly by introducing appropriate boundary terms. The test function g can be chosen in such a way that it is one in the region IT × IL and zero outside a slightly larger region. Here the interval Is is defined by Is := (−s/2, s/2). For L1 < L the region IL \IL1 has two connected components I± and there are two possibilities (corresponding to µ+ or µ− ) to choose boundary conditions with respect to each of the regions IT × I± This gives four different boundary terms {δSj,± (φ) = φ(gj,± ) + cj,± ; j = ±} , where gj,± are suitable test functions which have support in a neighborhood of IT × I± and cj,± are appropriate constants. The regularized interaction part of the Euclidean action with boundary terms is Sij (g, φ) = S1 (g, φ) + δSi,+ (φ) + δSj,− (φ) . Step 2 : To approximate the measure µ± , we perturb µ0 by a positive L1 -function dµT,L,± (φ) := Z(T, L, ±) dµ0 (φ) exp(−S±± (g, φ)) , where the constant Z(T, L, ±) is for normalization. According to J. Glimm, A. Jaffe and T. Spencer [40], the limits Z Z dµT,L,± (φ) exp(φ(f )) , dµ± (φ) exp(φ(f )) = lim lim L→∞ T →∞
860
D. SCHLINGEMANN
which determine the measures µ± , exist for each test function f . Since the different choices for the boundary terms are related by the the map φ 7→ −φ, i.e. φ(g+,± ) = −φ(g−,± ) , the measures µ+ and µ− satisfy the relation dµ+ (−φ) = dµ− (φ) . Step 3 : There are four Hamilton operators {Hij (L); i, j = ±} acting on the Fock space H0 of the massive free scalar field. They are related to the unnormalized measures dµT,L,ij (φ) := dµ0 (φ) exp(−Sij (g, φ)) by Nelson’s Feynman–Kac formula: Z dµT,L,ij (φ) = hΩ0 , exp(−T Hij (L))Ω0 i . Here Ω0 is the bare vacuum vector in H0 . Let M : I 7→ M(I) be the net of Cauchy data for the massive free scalar field. The dynamics of the φ42 -model can be obtained by the prescription αt (a) := lim eitHij (L) a e−itHij (L) , L→∞
where the limit is independent of the choice of the boundary conditions. Finally, by using the Osterwalder–Schrader reconstruction theorem, two vacuum states ω± with respect to the dynamics α can be constructed from the measures µ± . The crucial property which allows us to carry through the analysis of [28] is the following: Let I be a bounded interval, then the observables which are localized in the left complement ILL of I statistically independent of those which are localized in the right complement IRR . This means, formulated in the language of operator algebras, that the W ∗ -tensor product M(ILL ) ⊗ M(IRR ) is unitarily isomorphic to the W ∗ -algebra M(ILL ) ∨ M(IRR ) . The statistical independence for half-line algebras has been proven according to [9, 64, 59]. We now describe the main steps of the construction of the kink sectors of the φ42 -model. Step 1 : According to [14, 19], the statistical independence of M(ILL ) and M(IRR ) implies the existence of a unitary operator uI which has the following properties: Let a and b be operators which are localized in ILL and IRR respectively. Then the relations uI a u∗I = a and uI b u∗I = χ(b) hold .
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Here χ is the Bogoliubov automorphism which is induced by the map φ 7→ −φ. Step 2 : According to the results of [27], it can be shown that for each t the limit γ 0 (t) := lim exp(itH++ (L))uI exp(−itH−+ (L))u∗I L→∞
exists and that the operator γ 0 (t) is localized in a sufficiently large interval It . Note that the Hamiltonian H−+ (L) belongs to the following interpolating boundary conditions: The left boundary term is chosen with respect to the boundary conditions for the vacuum ω− and the right boundary term is chosen with respect to the boundary conditions for the vacuum ω+ . Finally, the charge transporters are given by γ(t, x) := αx (γ 0 (t)uI )u∗I and the corresponding interpolating automorphism ρ can be obtained from γ by the uniform limit ρ(a) = lim γ(t, x) a γ(t, x)∗ . x→−∞
It follows from its construction that ρ acts trivially on the observables which are localized in ILL and acts as the symmetry χ on those which are localized in IRR . The kink sector and its anti-kink sector are θ = [ω+ ◦ ρ] and θ¯ = [ω+ ◦ ρ−1 ] respectively . This result is in complete analogy to the result for the (Φ · Φ)22 -model, i.e. in both models the same four irreducible sectors appear. At this point, we shall mention here some further treatments of kink sectors: (i) In [27, Chap. 5], the existence of kink states in general P (φ)2 -models is discussed. However, this construction leads only to kink states which interpolate vacua which are connected by the internal symmetry transformation φ 7→ −φ. We shall see later that we achieve a generalization of this result. (ii) In the late 80s, J. Fr¨ ohlich and P. A Marchetti developed a quantization of kinks in terms of Euclidean functional integrals which has been applied to several lattice field theories [32, 51, 33]. (iii) Recently, a construction of kink sectors for a lattice version of the XY model has been carried out by H. Araki [1] and for XXZ models by T. Matsui [52, 53]. (iv) Moreover, by using euclidean techniques (compare [28]), an estimate for the mass of the (λφ4 )2 soliton has been established in [3]. In the following, we briefly explain how the construction of kink states can be generalized to a larger class of quantum field theory models for which the conditions, listed below, hold: (i) The dynamics of the model satisfies an appropriate extendibility condition which we shall explain later. (ii) The vacuum states are local Fock states which is automatically satisfied for P (φ)2 and Yukawa2 models [36, 63].
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D. SCHLINGEMANN
Step 1 0 : We consider the two-fold net M ⊗ M : I 7→ M(I) ⊗ M(I) . Like in Step 1 of our previous construction scheme, the split property implies that on M(IRR ) ⊗ M(IRR ), the flip automorphism is implemented by a unitary operator θI [14]. The adjoint action of θI is an automorphism β I which has the following properties: (i) The automorphism β I acts trivially on observables which are localized in the left complement of I and it acts like the flip on observables which are localized in the right complement of I. (ii) The automorphism β I maps local algebras into local algebras. Note that the automorphism β I does not depend on the dynamics α. Step 2 0 : Let ω1 , ω2 be two vacuum states with respect to a given dynamics α. The state ω := ω1 ⊗ ω2 ◦ β I |C ∗ (M)⊗C1 interpolates the vacua ω1 and ω2 . Moreover, it is covariant under spatial translations since for each x the operator γ(0, x) = (αx ⊗ αx )(θI )θI is localized in a sufficiently large bounded interval. Indeed, the unitary operators U (0, x) := (U1 (0, x) ⊗ U2 (0, x)) (π1 ⊗ π2 )(γ(0, −x)) implement the spatial translations in the GNS-representation of ω where U1 and U2 implement the translations in the GNS-representations π1 , π2 of ω1 and ω2 respectively. Step 3 0 : It remains to be proven that ω is translationally covariant with respect to the dynamics α. For this purpose, we wish to construct a cocycle γ(0, t) such that the operators U (t, 0) := (U1 (t, 0) ⊗ U2 (t, 0)) (π1 ⊗ π2 )(γ(−t, 0)) implement the dynamics α in the GNS-representation of ω. The operator γ(t, 0) := (αt ⊗ αt )(θI )θI is a formal solution. Unfortunately, the flip implementer θI is not contained in any local algebra and the term (αt ⊗ αt )(θI ) has no mathematical meaning unless α is the free dynamics. However, it can be given a meaning in some cases. We shall see that for an interacting dynamics there exists a suitable cocycle of the operators γ(t, 0) such that γ(t, 0) is localized in a bounded interval whose size depends linearly on |t|.
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In order to formulate a sufficient condition for the existence of γ(t, 0), we conˆ to be the von Neumann struct an extension of the net M ⊗ M. We define M(I) algebra which is generated by M(I) ⊗ M(I) and the operator θI . The net ˆ : I 7→ M(I) ˆ M is an extension of M ⊗ M which does not fulfill locality. This is due to the nontrivial implementation properties of θI . We shall call a dynamics α extendible if ˆ which is an extension of α ⊗ α. Indeed, there exists a dynamics α ˆ of M t 7→ γ(t, 0) := α ˆ t (θI )θI is a cocycle which has the desired properties. Finally, we conclude like in Step 3 of our previous construction scheme that the state ω := ω1 ⊗ ω2 ◦ β I |C ∗ (M)⊗C1 is a kink state where ω1 , ω2 are vacuum states with respect to the dynamics α. Since the extendibility condition is rather technical one might worry that it is only fulfilled for few exceptional cases. Fortunately, this is not true. There is a large class of quantum field theory models whose dynamics are extendible. We shall prove that the extendibility holds for the following models: (i) P (φ)2 -models. (ii) Yukawa2 models. (iii) Special types of Wess–Zumino models. Note that a Dirac spinor field contributes to the field content of the Yukawa2 and Wess–Zumino models, and the nets of Cauchy data fulfill twisted duality instead of Haag duality [64]. According to recent results which have been established by M. M¨ uger [54], our results remain true for these cases also. Wess–Zumino models have been studied in several papers. We refer to the work of A. Jaffe, A. Lesniewski, J. Weitsman and S. Janowsky [45, 48, 49, 46, 47]. It has been proven in [46] that some Wess–Zumino models possess more than one vacuum sector. An application of our construction scheme proves the existence of kink states for these models. 2. Preliminaries In the first part (Sec. 2.1) of this preliminary section, we briefly describe how to construct a Haag–Kastler net from a given net of Cauchy data and a given dynamics. Examples for physical states with respect to an interacting dynamics are given in the second part (Sec. 2.2). 2.1. From Cauchy data to Haag Kastler nets We denote by U (M) the group of unitary operators in C ∗ (M). Let G(R, M) be the group which is generated by the set {(t, u)|t ∈ R modulo the following relations:
and u ∈ U (M)}
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D. SCHLINGEMANN
(1) For each u1 , u2 ∈ U (M) and for each t1 , t2 , t ∈ R, we require (t, u1 )(t, u2 ) = (t, u1 u2 ) and (t, 1) = 1 (2) For u1 ∈ M(I1 ) and u2 ∈ M(I2 ) with I1 ⊂ (I2 + [−|t|, |t|])c we require for each t1 ∈ R: (t1 + t, u1 )(t1 , u2 ) = (t1 , u2 )(t1 + t, u1 ) . We conclude from relation (1) that (t, u) is the inverse of (t, u∗ ). Furthermore, a localization region in R×Σ can be assigned to each element in G(R, M). A generator (t, u), u ∈ M(I) is localized in O ⊂ R × Σ if {t} × I ⊂ O. The subgroup of G(R, M) which is generated by elements which are localized in the double cone O, is denoted by G(O). We easily observe that relation (2) implies that group elements commute if they are localized in space-like separated regions. The translation group in R2 is naturally represented by group-automorphisms of G(R, M). They are defined by the prescription β(t,x) (t1 , u) := (t + t1 , αx u) . Thus the subgroup G(O) is mapped onto G(O + (t, x)) by β(t,x) . To construct the universal Haag–Kastler net, we build the group C ∗ -algebra B(O) with respect to G(O). For convenience, we shall describe the construction of B(O) briefly. In the first step we build the ∗ -algebra B0 (O) which is generated by all complex valued functions a on G(O), such that a(u) = 0 for almost each u ∈ G(O) . We write such a function symbolically as a formal sum, i.e. X a(u) u a= u ∗
The product and the -relation is given as follows: X X XX 0 0 −1 0 ab = a(u) u · b(u ) u = a(u)b(u u ) u0 u
u0
a∗ =
u0
X
u
a ¯(u−1 ) u
u
It is well known, that the algebra B0 (O) has a C ∗ -norm which is given by kak := sup kπ(a)kπ , π
where the supremum is taken over each Hilbert space representation π of B0 (O). Finally, we define B(O) as the closure of B0 (O) with respect to the norm above. The C ∗ -algebra which is generated by all local algebras B(O) is denoted by ∗ C (B). By construction, the group isomorphisms β(t,x) induce a representation of the translation group by automorphisms of C ∗ (B).
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Observation. The net of C ∗ -algebras B := {B(O)|O is a bounded double cone in R2 } is a translationally covariant Haag–Kastler net. We have to mention that the universal net B is not Lorentz covariant. The universal properties of the net B are stated in the following proposition: Proposition 2.1. Each dynamics α ∈ dyn(M) induces a C ∗ -homomorphism ια : C ∗ (B) → C ∗ (M) such that ια ◦ β(t,x) = α(t,x) ◦ ια , for each (t, x) ∈ R2 . In particular, Aα : O 7→ Aα (O) := ια (B(O))00 is a translationally covariant Haag–Kastler net. Proof. Given a dynamics α of M. We conclude from ps(α) ≤ 1 that the prescription (t, u) 7→ αt u defines a C ∗ -homomorphism ια : C ∗ (B) → C ∗ (M) . In particular, ια is a representation of C ∗ (B) on the Hilbert space H0 . This statement can be obtained by using the relations, listed below. (a) ια ((t, u1 )(t, u2 )) = αt u1 αt u2 = αt (u1 u2 ) = ια (t, u1 u2 ) (b) If (t1 , u1 ) and (t1 + t, u2 ) are localized in space-like separated regions, then we obtain from ps(α) ≤ 1: [ια (t1 , u1 ), ια (t1 + t, u2 )] = αt1 [u1 , αt u2 ] = 0 (c) ια (β(t,x) (t1 , u)) = ια (t + t1 , αx u) = α(t,x) αt1 u
In general we expect that for a given dynamics α the representation ια is not faithful. Hence each dynamics defines a two-sided ideal ∗ J(α) := ι−1 α (0) ∈ C (B)
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D. SCHLINGEMANN
in C ∗ (B) which we call the dynamical ideal with respect to α and the quotient C ∗ -algebras B(O)/J(α) ∼ = Aα (O) may depend on the dynamics α. Indeed, if O is a double cone whose base is not contained in Σ, then for different dynamics α1 , α2 the algebras Aα1 (O) and Aα2 (O) are different. On the other hand, if the base of O is contained in Σ, then we conclude from the fact that the dynamics α has finite propagation speed and from Proposition 2.1. Corollary 2.2. If I ⊂ Σ is the base of the double cone O, then the algebra Aα (O) is independent of α. In particular, the C ∗ -algebra C ∗ (M) =
[
k·k
M(I)
I
=
[
k·k
Aα (O)
O
is the C ∗ -inductive limit of the net Aα . From the discussion above, we see that two dynamics with the same dynamical ideal induces the same quantum field theory. 2.2. Examples for physical states Let us consider the set S of all locally normal states on C ∗ (M), i.e. for each state ω ∈ S and for each bounded interval I, the restriction ω|M(I) is a normal state on M(I). As mentioned in the introduction, we are interested in states with vacuum and particle-like properties, i.e. states satisfying the Borchers criterion (See the Introduction for this notion). Notation. Given a dynamics α ∈ dyn(M). We denote the corresponding set of all locally normal states which satisfies the Borchers criterion by S(α) and analogously the set of all vacuum states by S0 (α). Moreover, we write for the set of vacuum sectors sec0 (α) := {[ω]|ω ∈ S0 (α)} ,
(5)
where [ω] denotes the unitary equivalence class of the the GNS-representation of ω. Examples. Examples for vacuum states are the vacua of the P (φ)2 -models [36, 37]. The interacting part of the cutoff Hamiltonian is given by a Wick polynomial of the time zero field φ0 , i.e. H1 (I) = H1 (χI ) =: P (φ0 ) : (χI ) ,
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where χI is a test function with χI (x) = 1 for x ∈ I and χI (y) = 0 on the complement of a slightly larger region Iˆ ⊃ I. It is well known that H1 (I) is a self-adjoint operator, which has a joint core with the free Hamiltonian h0 , and is ˆ The operator h1 (I) induces a automorphism group αI which affiliated with M(I). is given by αI,t (a) := eiH1 (I)t a e−iH1 (I)t . Consider the inclusion of intervals I0 ⊂ I1 ⊂ I2 . Then we have for each a ∈ M(I0 ): αI1 ,t (a) = αI2 ,t (a) Hence, there exists a one-parameter automorphism group {α1,t ∈ Aut(M); t ∈ R} such that α1,t acts on a ∈ M(I) as follows: α1,t (a) = αI,t (a) ; ∀ t ∈ R . The automorphism group {α1,t ∈ Aut(M); t ∈ R} is a dynamics of M with zero propagation speed, i.e. ps(α1 ) = 0. Since H1 (I) has a joint core with the free Hamiltonian H0 , we are able to define the Trotter product of the automorphism groups α0 and α1 which is given for each local operator a ∈ M(I) by αt (a) := (α0 × α1 )t (a) = s − lim (α0,t/n ◦ α1,t/n )n (a) . n→∞
The limit is taken in the strong operator topology. Furthermore, the propagation speed is sub-additive with respect to the Trotter product [36], i.e. ps(α0 × α1 ) ≤ ps(α0 ) + ps(α1 ) and we conclude that α ∈ dyn(M) is a dynamics of M. We call the dynamics α interacting. It is shown by Glimm and Jaffe [36] that there exist vacuum states ω with respect to the interacting dynamics α. We have to mention, that there is no vector ψ in Fock space H0 , such that the state a 7→ hψ, aψi is a vacuum state with respect to an interacting dynamics α, but there is a sequence of vectors (Ωn ) in H0 such that the weak∗ limit ω = w∗ − limhΩn , ·Ωn i n
is a vacuum state with respect to the dynamics α.
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3. On the Existence of Kink States The main theorem (Theorem 3.2) of this paper is formulated in the first part (Sec. 3.1) of the present section. In order to prepare the proof of Theorem 3.2, we need some technical preliminaries which are given in Sec. 3.2. In the last part (Sec. 3.3), we prove a criterion for the existence of kink states (the extendibility of the dynamics) which turns out to be satisfied by the P (φ)2 and Yukawa2 models. 3.1. The main result We now reformulate the definition (see Introduction) of a kink state within the time-slice formulation. Definition 3.1. Let α ∈ dyn(M) be a dynamics of M. A state ω of M is called a kink state, interpolating vacuum states ω1 , ω2 ∈ S0 (α) if (a) ω satisfies the Borchers criterion (b) and there exists a bounded interval I, such that ω fulfills the relations: π|C ∗ (M,ILL ) ∼ = π1 |C ∗ (M,ILL )
and π|C ∗ (M,IRR ) ∼ = π2 |C ∗ (M,IRR )
where the symbol ∼ = means unitarily equivalent and (H, π, Ω), (Hj , πj , Ωj ) are the GNS-triples of the states ω ∈ S(α) and ωj ∈ S0 (α); j = 1, 2 respectively. The set of all kink states which interpolate ω1 and ω2 is denoted by S(α|ω1 , ω2 ). As already mentioned in the Introduction, a criterion for the existence of an interpolating kink state, can be obtained by looking at the construction method of [59]. In our context, we have to select a class of dynamics which are equipped with good properties. Such a selection criterion is developed in Sec. 3. We shall show that each dynamics of a P (φ)2 -model satisfies this criterion which leads to the following result: Theorem 3.2. If α ∈ dyn(M) is a dynamics of a model with P (φ)2 plus Yukawa2 interaction, then for each pair of vacuum states ω1 , ω2 ∈ S0 (α) there exists an interpolating kink state ω ∈ S(α|ω1 , ω2 ). We shall prepare the proof of Theorem 3.2 during the subsequent sections. 3.2. Technical preliminaries Definition 3.3. Let M be a net of Cauchy data. We denote by G(M) the group of unitary operators u ∈ B(H0 ) whose adjoint actions χu := Ad(u) commute with the spatial translations, i.e. χu ◦ αx = αx ◦ χu .
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Let α ∈ dyn(M) be a dynamics of the net M. Then we define the following subgroup of G(M): G(α, M) := {u ∈ G(M)|χu ◦ αt = αt ◦ χu for each t ∈ R} . Remark. Each operator u ∈ G(α, M) induces a symmetry of the Haag–Kastler net Aα . We make the following assumptions for the net of Cauchy data M: Assumption. (a) The net M fulfills duality, i.e. M(I)0 = M(ILL ) ∨ M(IRR )
(6)
(b) There exists a dynamics α0 and a normalized vector Ω0 in H0 , such that ω0 = hΩ0 , (·)Ω0 i is a vacuum state with respect to the dynamics α0 . (c) For each bounded interval I, the inclusion (M(IRR ), M(IR )) is split. (d) The net fulfills weak additivity. According to our assumption, we conclude from the Theorem of Reeh and Schlieder that Ω0 is a standard vector for the inclusion (M(IRR ), M(IR )) which implies that Λ(I) := (M(IRR ), M(IR ), Ω0 )
(7)
is a standard split inclusion for each interval I, and hence (see [19]) there exists a unitary operator wI : H0 ⊗ H0 → H0 such that for a ∈ M(ILL ) and b ∈ M(IRR ) we have wI (a ⊗ b)wI∗ = ab . Thus there is an interpolating type I factor N ∼ = B(H0 ), i.e. M(IRR ) ⊂ N ⊂ M(IR ) which is given by N := wI (1 ⊗ B(H0 ))wI∗ . Hence we obtain an embedding of B(H0 ) into the algebra M(IR ): ΨI : F ∈ B(H0 ) 7→ wI (1 ⊗ F )wI∗ ∈ M(x, ∞) This embedding is called the universal localizing map.
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Remark. We shall make a few remarks on the assumptions given above. (i) The results, which we shall establish in the following, remain correct if the net of Cauchy data fulfills twisted duality instead of duality [54, 64]. (ii) For the application of our analysis to quantum field theory models, like P (φ)2 - or Yukawa2 models, we can choose as Cauchy data tensor products of the time-zero algebras of the massive free Bose or Fermi field. The timezero algebras of the massive free Bose field fulfill the assumptions (a) and (b) and it has been shown [60, Appendix] (compare also [9]) that (c) is also fulfilled. Replacing duality by twisted duality, the assumptions (a) to (c) hold for the massive free Fermi field, too [64]. In addition to that, we claim that the weak additivity (d) is also fulfilled in these cases. (iii) The state ω0 plays the role of a free massive vacuum state, called the bare vacuum. Proposition 3.4. Let u ∈ G(M) be an operator and let I be a bounded interval. Then there exists a canonical automorphism χIu with the properties: (1) The relations χIu |C ∗ (M,ILL ) = idC ∗ (M,ILL )
and χIu |C ∗ (M,IRR ) = χu |C ∗ (M,IRR )
(8)
hold. 1 : Σ → C ∗ (M) such that : (2) There exists a strongly continuous map γ(u,I) (i) 1 (x)) = αx ◦ χIu ◦ α−x ◦ (χIu )−1 . Ad(γ(u,I)
(ii) The cocycle condition is fulfilled : 1 1 1 (x + y) = αx (γ(u,I) (y))γ(u,I) (x) . γ(u,I)
Proof. (1) In the same manner as in [59], we show that ˆ ⊂ M(I) ˆ Ad(ΨI (1 ⊗ u))(M(I)) if the interval Iˆ contains I. This implies that χIu := Ad(ΨI (1 ⊗ u)) is a well defined automorphism of C ∗ (M). By using the properties of the universal localizing map ΨI , we conclude that χIu fulfills Eq. (8). (2) By a straight forward generalization of the proof of [59, Proposition 4.2], we 1 (x) is given by conclude that the statement (2) holds where γ(u,I) 1 γ(u,I) (x) = ΨI+x (1 ⊗ u)ΨI (1 ⊗ u∗ ) .
Let ω be a vacuum state with respect to the dynamics α and let u ∈ G(α, M), then the state ωuI := ω ◦ χIu
CONSTRUCTION OF KINK SECTORS FOR
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seem to be a good candidate for an interpolating kink state. Indeed, it follows from the construction of χIu that ωuI |C ∗ (M,IRR ) = ω ◦ χu |C ∗ (M,IRR ) ωuI |C ∗ (M,ILL ) = ω|C ∗ (M,ILL ) . Hence ωuI interpolates ω and ω ◦ χu . To decide whether ωuI is a positive energy state, we investigate in the subsequent section, how χIu is transformed under the action of a dynamics α. 3.3. When does a theory possess kink states? Let α be a dynamics and G ⊂ G(α, M) be a finite subgroup. By using the universal localizing map ΨI , we obtain for each bounded interval I a unitary representation of G UI : G 3 g 7→ UI (g) := ΨI (1 ⊗ g) ∈ M(IR ) . In the previous section it has been shown that UI (g) implements an automorphism χIg which is covariant under spatial translations (Proposition 3.4). For a dynamics α ∈ dyn(M), we wish to construct a cocycle γ(g,I) in order to show that χIg is an interpolating automorphism. The formal operator γ(g,I) (t, x) := α(t,x) (UI (g))UI (g)∗ seems to be a useful Ansatz since it formally implements the automorphism α(t,x) ◦ χIg ◦ α(−t,−x) ◦ (χIg )−1 . Unfortunately, the operators UI (g) are not contained in C ∗ (M) and the term α(t,x) (UI (g)) has no well-defined mathematical meaning. To get a well-defined solution for γ(g,I) , we construct an extension of the net M which contains the operators UI (g) (compare also [54]). Definition 3.5. Let G ⊂ G(M) be a compact sub-group. The net M o G is defined by the assignment M o G : I 7→ (M o G)(I) := M(I) ∨ UI (G)00 . Proposition 3.6. Let I be a bounded interval, then the map π I : M(I) o G 3 a · g 7→ a UI (g) ∈ M(I) ∨ UI (G)00 is a faithful representation of the crossed product M(I) o G. Proof. First, we easily observe that π I is a well-defined representation of M(I) o G. According to [43, Theorem 2.2, Corollary 2.3], we conclude that the crossed product M(I)oG is isomorphic to the von Neumann algebra M(I)∨UI (G)00 and π I is a W ∗ -isomorphism.
872
D. SCHLINGEMANN
Definition 3.7. A one parameter automorphism group α, which satisfies the conditions, listed below, is called a G-dynamics of the extended net M o G. (a) α is a dynamics of the net M o G (See Introduction). (b) The automorphisms αt commute with the automorphisms χg , i.e. αt ◦ χg = χg ◦ αt ; for each t ∈ R and for each g ∈ G . The set of all G-dynamics of M o G is denoted by dynG (M o G). Proposition 3.8. Let α ∈ dynG (M o G) be a G-dynamics and I be a bounded interval. Then the operator 0 (t) := αt (UI (g))UI (g)∗ γ(g,I)
is contained in M(I|t| ) where I|t| denotes the enlarged interval I + (−|t|, |t|) and the operator γ(g,I) (t, x) := α(t,x) (UI (g))UI (g)∗ fulfills the cocycle condition: γ(g,I) (t + t0 , x + x0 ) = α(t,x) (γ(g,I) (t0 , x0 ))γ(g,I) (t, x) . Proof. For a ∈ C ∗ (M, I|t|,RR ), the operator α−t (a) is contained in C ∗ (M, IRR ) which implies a αt (UI (g))UI (g)∗ = αt (α−t (a)UI (g))UI (g)∗ = αt (UI (g)χg α−t (a))UI (g)∗ = αt (UI (g)α−t χg (a))UI (g)∗ = αt (UI (g))χg (a)UI (g)∗ = αt (UI (g))UI (g)∗ a and we conclude αt (UI (g))UI (g)∗ ∈ C ∗ (M, I|t|,RR )0 = M(I|t|,L ) . By a similar argument, αt (UI (g))UI (g)∗ is contained in M(I|t|,R ) and we conclude from duality that it is contained in M(I|t| ). The cocycle condition for γ(g,I) is obviously fulfilled and the proposition follows. Definition 3.9. Let α ∈ dyn(M) be a dynamics and G ⊂ G(M) be a compact subgroup. We shall call α G-extendible if there exists a G-dynamics α ˆ of the extended net M o G, such that α ˆ t |C ∗ (M) = αt for each t ∈ R.
CONSTRUCTION OF KINK SECTORS FOR
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We are now prepared to prove one of our key results: Theorem 3.10. Let α ∈ dyn(M) be a G-extendible dynamics and let χIg be the automorphism which can be constructed by Proposition 3.4. Then for each vacuum state ω with respect to α the state ωgI := ω ◦ χIg is a kink state which interpolates ω and ω ◦ χg . Proof. As postulated, there exists an extension α ˆ ∈ dynG (M o G) of α. We show that for each g ∈ G the operator 0 (t) := α ˆt (UI (g))UI (g)∗ γ(g,I)
implements the automorphism αt ◦ χIg ◦ α−t ◦ (χIg )−1 on C ∗ (M). Indeed, we have for each a ∈ C ∗ (M): 0 Ad(γ(g,I) (t))a = α ˆ t (UI (g))UI (g)∗ a UI (g)ˆ αt (UI (g))∗
=α ˆ t (UI (g)) (χIg )−1 (a) α ˆ t (UI (g))∗ −1 =α ˆ t UI (g)α−t χIg (a) UI (g)∗ −1 = αt UI (g)α−t χIg (a) UI (g)∗ = αt ◦ χIg ◦ α−t ◦ (χIg )−1 (a) . Finally we conclude from Proposition 3.8 that γ(g,I) (t, x) := α(t,x) (UI (g))UI (g)∗ is a cocycle where γ(g,I) (t, x) is localized in a sufficiently large bounded interval. By a straight forward generalization of the proof of [59, Proposition 5.5] we conclude that ωuI is a positive energy state which implies the result. The dynamics α of P (φ)2 - and Yukawa2 models are locally implementable by unitary operators. More precisely, for each bounded interval I and for each positive number τ > 0, there exists a unitary operator u(I, τ |t) with the properties: (1) If |t1 |, |t2 |, |t1 + t2 | < τ , then we have u(I, τ |t1 + t2 ) = u(I, τ |t1 )u(I, τ |t2 ) . (2) For |t| < τ , the operator u(I, τ |t) implements αt on M(I), i.e. αt (a) = u(I, τ |t) a u(I, τ |t)∗ ; for each a ∈ M(I) .
(9)
874
D. SCHLINGEMANN
Let G ⊂ G(α, M) be a compact sub-group. In order to show that α is G-extendible, it is sufficient to prove that the operators u(I1 , τ |t)UI (g)u(I1 , τ |t)∗ , which are the obvious candidates for α(U ˆ I (g)), are independent of I1 for I1 ⊃ I and |t| ≤ τ . Lemma 3.11. If for each I ⊂ I1 , for each τ < τ1 and for each g ∈ G the equation u(I, τ |t)UI (g)u(I, τ |t)∗ = u(I1 , τ1 |t)UI (g)u(I1 , τ1 |t)∗
(10)
holds, then the dynamics α is G-extendible. Here u(I, τ |t) are unitary operators which fulfill Eq. (9). Proof. Let (In , τn )n∈N be a sequence, such that limn In = R and limn τn = ∞. We conclude from our assumption (Eq. (10)) that the uniform limit α ˆ t (a) := lim Ad(u(In , τn |t))(a) n→∞
exists. Thus α ˆ : t 7→ α ˆ t is a well-defined one-parameter automorphism group, extending the dynamics α. It remains to be proven that α ˆ has propagation speed ps(ˆ α) ≤ 1. Since α ˆ is an extension of α and ps(α) ≤ 1, we conclude for each a ∈ C ∗ (M, I|t|,RR ) and for each b ∈ C ∗ (M, I|t|,LL ): ab α ˆ t (UI (g)) = α ˆ t (α−t (a)α−t (b)UI (g)) =α ˆ t (UI (g)α−t χg (a)α−t (b)) =α ˆ t (UI (g)) χg (a)b Thus the operator α ˆt (UI (g)) is contained in M(I|t|,R ) and implements χg on M(I|t|,RR ). This finally implies α ˆt (UI (g))UI|t| (g)∗ ∈ M(I|t| )
and the lemma follows.
Let us consider the two-fold W ∗ -tensor product of the net of Cauchy data, i.e. M ⊗ M : I 7→ M(I) ⊗ M(I) . Observation. (i) If the net M fulfills the conditions (a) to (c) of the previous section, then the net M ⊗ M fulfills them, too. (ii) Let α ∈ dyn(M) be a dynamics of M, then α⊗2 is a dynamics of M ⊗ M. Note that the flip operator uF , which is given by uF : H0 ⊗ H0 → H0 ⊗ H0 ; ψ1 ⊗ ψ2 7→ ψ2 ⊗ ψ1
CONSTRUCTION OF KINK SECTORS FOR
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is contained in G(α⊗2 , M ⊗ M). Hence uF induces an embedding of Z2 into G(α⊗2 , M ⊗ M). (iii) According to Definition 3.5, we can construct a non-local extension ˆ := (M ⊗ M) o Z2 M of the two-fold net M ⊗ M. Let ΨI be the universal localizing map of the standard split inclusion Λ(I) ⊗ Λ(I) = (M(IRR )
⊗ 2
, M(IR )
⊗ 2
, Ω0 ⊗ Ω0 )
ˆ is simply given by and define θI := ΨI (1 ⊗ uF ). Then the algebra M(I) ˆ M(I) = ((M ⊗ M) o Z2 )(I) = (M(I) ⊗ M(I)) ∨ {θI }00 . (iv) By Proposition 3.4, there exists a canonical automorphism β I := Ad(θI ) ,
(11)
associated with the pair (uF , I). Notation. Let α be a dynamics of M. In the sequel, we shall call α extendible if α⊗2 is Z2 -extendible. We conclude this section by the following corollary which can be derived by a direct application of Theorem 3.10: Corollary 3.12. Let α ∈ dyn(M) be an extendible dynamics, then for each pair of vacuum states ω1 , ω2 ∈ S0 (Aα ), the state ω = µβ I (ω1 ⊗ ω2 ) is a kink state. 4. Application to Quantum Field Theory Models We show that a sufficient condition for the existence of interpolating automorphisms, i.e. the extendibility of the dynamics, is satisfied for the P (φ)2 , the Yukawa2 and special types of Wess–Zumino models. 4.1. Kink states in P (φ)2 -models We shall show that the dynamics of P (φ)2 -models are extendible. As described in Sec. 2.2 the dynamics of a P (φ)2 -model consists of two parts. (1) The first part is given by the free dynamics α0 , with propagation speed ps(α0 ) = 1, α0,t (a) = eiH0 t a e−iH0 t , where (H0 , D(H0 )) is the free Hamiltonian which is a self-adjoint operator on the domain D(H0 ) ⊂ H0 .
876
D. SCHLINGEMANN
(2) The second part is a dynamics α1 with propagation speed ps(α1 ) = 0, i.e. α1,t maps each local algebra M(I) onto itself. The interaction part of the full Hamiltonian is given by a Wick polynomial of the time-zero field φ: Z H1 (I) = H1 (χI ) =
dx : P (φ(x)) : χI (x)
where χI is a smooth test function which is one on I and zero on the complement of a slightly lager region Iˆ ⊃ I. The unitary operator exp(itH1 (I)) implements the dynamics α1 locally, i.e. for each a ∈ M(I) we have α1,t (a) := eiH1 (I)t a e−iH1 (I)t . Definition 4.1. An operator valued distribution v : S(R) → L(H0 ) is called an ultra local interaction, if the following conditions are fulfilled: (1) For each real valued test function f ∈ S(R), v(f ) is self-adjoint and has a common core with H0 . (2) Let f ∈ S(R) be a real valued test function with support in a bounded interval I, then the spectral projections of v(f ) are contained in M(I). (3) For each pair of test functions f1 , f2 ∈ S(R), the spectral projections of v(f1 ) commute with the spectral projections of v(f2 ). Remark. It has been proven in [36], that the Wick polynomials of the time zero fields are ultra local interactions. Furthermore, each ultra local interaction v induces a dynamics αv ∈ dyn(M) with propagation speed ps(αv ) = 0. Let I be a bounded interval and let χI ∈ S(R) be a positive test function with χI (x) = 1 for each x ∈ I. Indeed, by an application of J. Glimm’s and A. Jaffe’s analysis [36], we conclude that the automorphisms αvt : M(I) → M(I) ; a 7→ Ad(exp(itv(χI ))) a define a dynamics with zero propagation speed. In the sequel, we shall call a dynamics αv ultra local if it is induced by an ultra local interaction v. In order to prove that a dynamics α, which is given by the Trotter product α = α0 × αv of a free and an ultra local dynamics, is extendible, we show that each part of the dynamics can be extended separately. Since the free part of the dynamics can be extended to the algebra B(H0 ) of all bounded operators on the Fock space H0 , it is obvious that α0 is extendible. Thus it remains to be proven the following: Lemma 4.2. Each ultra local dynamics αv ∈ dyn(M) is extendible.
CONSTRUCTION OF KINK SECTORS FOR
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Proof. Let us consider any ultra local interaction v. For each test function f ∈ S(R), we introduce the unitary operator u(f |t) := eitv(f ) ⊗ eitv(f ) . Let I be a bounded interval and denote by I , > 0, the enlarged interval I+(−, ). We choose test functions χ(I,) ∈ S(R) such that 1 x∈I . χ(I,) (x) = 0 x ∈ Ic = I \R For an interval Iˆ ⊃ I , the region Iˆ \I consists of two connected components (Iˆ \I )± and there exist test functions χ± ∈ S(R) with supp(χ− ) ⊂ (Iˆ \I )− ⊂ ILL supp(χ+ ) ⊂ (Iˆ \I )+ ⊂ IRR ˆ
χ(I,) − χ(I,) = χ+ + χ− . Let us write u(I, |t) := u(χ(I,) |t) and u± (t) := u(χ± |t) . Since we have [u(f1 |t), u(f2 |t)] = 0 for any pair of test functions f1 , f2 ∈ S(R), we ˆ obtain for each > 0 and for I ⊂ I: ˆ |t) = u(I, |t)u− (t)u+ (t) u(I,
(12)
If we make use of the fact that u+ (t) is αF -invariant and localized in IRR , we conclude that θI and u± (t) commute. Thus we obtain ˆ |t))θI = Ad(u(I, |t))θI Ad(u(I,
(13)
which depends only of the localization interval I since > 0 can be chosen arbitrarily small. According to Lemma 3.11, the automorphisms ˆ ˆ α ˆ vt : M(I) 3 a 7→ Ad(u(I, |t))a ∈ M(I) ˆ whose restriction to M ⊗ M is αv ⊗ αv . Thus αv is define a dynamics of M extendible. ˆ ˆv If α ˆ 0 denotes the natural extension of the free dynamics α⊗2 0 to M and let α v v be the extension of the ultra local dynamics α ⊗ α then, by using the Trotter product, we conclude that the dynamics α ˆ := α ˆ0 × α ˆv ˆ This leads to the following result: is an extension of the dynamics (α0 ×αv )⊗2 to M. Proposition 4.3. Each dynamics of a P (φ)2 -model is extendible.
878
D. SCHLINGEMANN
Proof. The statement follows from Lemma 4.2 and from the fact that each dynamics of a P (φ)2 -model is a Trotter product of the free dynamics α0 and an ultra local dynamics α1 . The existence of interpolating kink states in P (φ)2 -models is an immediate consequence of Proposition 4.3. Corollary 4.4. Let α ∈ dyn(M) be a dynamics of a P (φ)2 -model. Then for each pair of vacuum states ω1 , ω2 ∈ S0 (α, M) there exists an interpolating kink state ω ∈ S(ω1 , ω2 ). Proof. By Proposition 4.3 each dynamics of a P (φ)2 -model is extendible and we can apply Corollary 3.12 which implies the result. 4.2. The dynamics of the Yukawa2 model Since the dynamics of a Yukawa2 -like model cannot be written as a Trotter product which consists of a free and an ultra local dynamics, it is a bit more complicated to show that these dynamics are extendible. We briefly summarize here the construction of the Yukawa2 dynamics which has been carried out by J. Glimm and A. Jaffe [36]. We also refer to the work of R. Schrader [62, 63]. Let Ms and Ma be the nets of Cauchy data for the free Bose and Fermi field, represented on the Fock spaces Hs and Ha respectively. The Cauchy data of the Yukawa2 model are given by the W ∗ -tensor product M := Ms ⊗ Ma of the nets Ms and Ma . Moreover, we set H0 := Hs ⊗ Ha . Step 1 : In the first step, a Hamiltonian, which is regularized by an UV-cutoff c0 > 0 and an IR-cutoff c1 > 1, c0 c1 , is constructed. For this purpose, one chooses test functions δc0 , χc1 ∈ S(R) with the properties: (a) Z dx δc0 (x) = 1 supp(δc0 ) ⊂ (−c0 , c0 ) and (b) supp(χc1 ) ⊂ (−c1 − 1, c1 + 1) and χc1 (x) = 1 for each x ∈ (−c1 , c1 ) . The UV-regularized fields are given by φ(c0 , x) := (φ ∗ δc0 )(x)
and ψ(c0 , x) := (ψ ∗ δc0 )(x) ,
(14)
where φ is a massive free Bose field and ψ a free Dirac spinor field at t = 0. The fields, defined by Eq. (14), act on H0 via the operators Φ(c0 , x) := φ(c0 , x) ⊗ 1Ha
and Ψ(c0 , x) := 1Hs ⊗ ψ(c0 , x) .
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The regularized Hamiltonian H(c0 , c1 ) can be written as a sum of three parts: (1) The free Hamiltonian H0 which is given by H0 = H0,s ⊗ 1Ha + 1Hs ⊗ H0,a , where H0,s and H0,a are the free Hamilton operators of the Bose and the Fermi field respectively. (2) The regularized Yukawa interaction term: Z ¯ 0 , x)Ψ(c0 , x) . HY (c0 , c1 ) = dx χc1 (x) Φ(c0 , x) : Ψ(c (3) The counterterms: HC (c0 , c1 ) =
N X
Z zn (c0 )
dx χc1 (x) : Φ(x)n ,
n=0
where zn (c0 ) are suitable renormalization constants. The following statement has been established by J. Glimm and A. Jaffe [36, 38]: Theorem 4.5. The counterterms HC (c0 , c1 ) can be chosen in such a way that (1) the cutoff Hamiltonian H(c0 , c1 ) = (H0 + HY (c0 , c1 ) + HC (c0 , c1 ))∗∗ is a positive and self adjoint operator with domain D(H0 ). (2) The uniform limit R(c1 , ζ) = lim (H(c0 , c1 ) − ζ)−1 c0 →0
is the resolvent of a self adjoint operator H(c1 ). (3) H(c1 ) is the limit of H(c0 , c1 ) in the strong graph topology. Notation. In the sequel, we shall use the following notation: u(c0 , c1 , t) := exp(itH(c0 , c1 )) and u(c1 , t) := exp(itH(c1 )) . Remark. The aim is to show that H(c1 ) induces a dynamics of M, given locally by the equation αt |M(I) = Ad(u(c1 , t)) for I|t| := I + (−|t|, |t|) ⊂ (−c1 , c1 ) . However, in comparison to the P (φ)2 -models, there are some more technical difficulties which have to be overcome. (i) The Hamiltonian H(c1 ) is only defined as a limit of the Hamiltonians H(c0 , c1 ) and it has no mathematical meaning when written as a sum H0 + HY (c1 ) + HC (c1 ) . Thus the construction scheme for a dynamics, as it has been used for P (φ)2 models, does not apply.
880
D. SCHLINGEMANN
(ii) On the other hand, one might try to apply P (φ)2 -like methods to the Hamiltonian H(c0 , c1 ), for which the UV-cutoff is not removed. For this purpose, one wishes to write H(c0 , c1 ) as a sum H(c0 , c1 ) = H1 (c0 , c1 ) + H2 (c0 , c1 ) where H1 (c0 , c1 ) induces a dynamics α1 with propagation speed ps(α1 ) ≤ 1 and H2 (c0 , c1 ) induces a dynamics α2 with propagation speed ps(α2 ) = 0. The difficulty with writing such a decomposition for H(c0 , c1 ) arises from the fact that the Yukawa interaction term HY (c0 , c1 ) induces an automorphism group with infinite propagation speed. Step 2 : In the next step, one introduces test functions χ(I,s,c0 ) (see Fig. 1), depending on a bounded interval I, a real number s > 0 and the UV-cutoff c0 , fulfilling the conditions supp(χ(I,s,c0 ) ) ⊂ I2c0 +|s|+ \I|s|−
and (15)
χ(I,s,c0 ) (x) = 1 if x ∈ I2c0 +|s| \I|s| .
Here c0 is any sufficiently small positive number. The Hamiltonian H(c0 , c1 ) is replaced by the operator H(I, s, c0 , c1 ) := H0 + HC (c0 , c1 ) + HY (I, s, c0 , c1 )
(16)
depending additionally on I and s, where HY (I, s, c0 , c1 ) is given by Z ¯ 0 , x)Ψ(c0 , x) : (χc1 (x) − χ(I,s,c ) (x)) . HY (I, s, c0 , c1 ) := dx Φ(c0 , x) : Ψ(c 0 In order to construct from these data a c1 -independent approximation of the dynamics which maps M(I) onto M(I|t| ), one defines the unitary operators n Y t w(I, c0 , c1 , t) := exp i H(I, (n − j)n−1 t, c0 , c1 ) , n j=1 where n is equal to the integral part of |c−1 0 t|. The lemma, given below, has been established in [36].
1
2c0
|s|
I
|s|
Fig. 1. The figure shows the graph of the function χ(I,s,c0 ) .
2c0
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Lemma 4.6 [36, Lemma 9.1.2]. The adjoint action of w(I, c0 , c1 , t) induces an automorphism (I,c0 )
αt
:= Ad(w(I, c0 , c1 , t)) : M(I) → M(I|t| )
which is independent of c1 . Step 3 : For technical reasons, to control convergence as c0 tends to zero, the length of time propagation is scaled, and one defines for λ ∈ [0, 1] the c1 -independent automorphism (I,c0 ,λ)
αt
:= Ad(w(I, c0 , c1 , λ, t)) : M(I) → M(I|t| )
where w(I, c0 , c1 , λ, t) is given by w(I, c0 , c1 , λ, t) :=
λ·t H(I, (n − j)n−1 t, c0 , c1 ) . exp i n j=1 n Y
The final approximation is given by averaging over λ: Z (I,c0 ,`) (I,c ,λ) (a) := dλ f` (λ) αt 0 (a) , αt where f` is a positive continuous function such that Z dλ f` (λ) = 1 and supp(f` ) ⊂ [1 − `, 1], ` ≤ 1 . Finally, J. Glimm and A. Jaffe have established the result: Theorem 4.7 [36, Theorem 9.1.3]. There exists a function c : ` 7→ c` with lim`→0 c` = 0 such that (I,c` ,`)
αYt (a) := w − lim αt `→0
(a) = u(c1 , t) a u(c1 , t)∗
(17)
for each a ∈ M(I) and for each sufficiently large c1 . 4.3. Kink states in models with Yukawa2 interaction We shall use an analogous strategy as above (Steps 1–3) in order to show that the dynamics αY , which is given due to Theorem 4.7 is extendible. Theorem 4.8. The dynamics αY of the Yukawa2 model is extendible. Let us prepare the proof. First, we give a few comments on the notation to be used.
882
D. SCHLINGEMANN
Notation. (a) In the sequel, we write w(· ˆ · · ) = w(· · · )⊗2 and uˆ(· · · ) = u(· · · )⊗2 for the corresponding quantities of the two-fold theory. As in Step 3 above, we also define the automorphism (I,c0 ,λ)
α ˆt
:= Ad(w(I, ˆ c0 , c1 , λ, t))
and the average
Z (I,c0 ,`)
α ˆt
(I,c0 ,λ)
(a) =
dλ f` (λ) α ˆt
(a) .
(b) Let ω0 be the vacuum state with respect to the free dynamics which is induced by H0 . We denote by ΨI the universal localizing map of the standard split inclusion Λ(I) ⊗ Λ(I) and we define θI := ΨI (1 ⊗ uF ). Lemma 4.9. The adjoint action of w(I, ˆ c0 , c1 , t) induces an automorphism (I,c0 )
ˆ ˆ |t| ) : M(I) → M(I
α ˆt which is independent of c1 .
Proof. By Lemma 4.6, it is sufficient to prove that Ad(w(I, ˆ c0 , c1 , t))θI is c1 -independent. Indeed, following the arguments in the proof of Proposition 4.3, we conclude that θI0 := exp(iτ H(I, s, c0 , c1 ))⊗2 θI exp(−iτ H(I, s, c0 , c1 ))⊗2 ˆ |s|+|τ | ). Composing is c1 -independent for |τ | ≤ c0 and that θI0 is contained in M(I n such maps, we obtain the lemma. In complete analogy to Theorem 4.7 we have Lemma 4.10. (I,c` ,`)
ˆt α ˆ Y (a) := w − lim α `→0
(a) = u ˆ(c1 , t) a u ˆ(c1 , t)∗ .
ˆ For each a ∈ M(I) and for each sufficiently large c1 . Proof. By Theorem 4.7, we conclude that the lemma holds for each a ∈ M(I) ⊗ M(I). Hence it remains to be proven that (I,c` ,`)
w − lim α ˆt `→0
(θI ) = uˆ(c1 , t) θI u ˆ(c1 , t)∗ .
The Corollary 9.1.9 of [36] states: Z w − lim dλ (w(I, ˆ c` , c1 , λ, t) − u ˆ(c` , c1 , λt))f` (λ) = 0 . `→0
CONSTRUCTION OF KINK SECTORS FOR
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We define Z (I,c` ,`)
θI (`, t) := α ˆt
(θI )
and θ¯I (`, t) :=
dλ f` (λ) Ad(ˆ u(c` , c1 , λt))θI .
The Schwarz’s inequality implies for each ψ ∈ H0 ⊗ H0 : |hψ, θI (`, t) − θ¯I (`, t)ψi| 1/2 Z 2 ˆ c` , c1 , λ, t) − u ˆ(c` , c1 , λt))ψk ≤ 2kψk · dλ f` (λ) k(w(I, Since k(v − u)ψk2 = 2 · Re(h(v − u)ψ, uψi), we obtain |hψ, θI (`, t) − θ¯I (`, t)ψi| Z ˆ c` , c1 , λ, t) ≤ 4kψk · dλ f` (λ)Re (w(I, 1/2 −ˆ u(c` , c1 , λt))ψ, u ˆ(c` , c1 , λt)ψ which proves the lemma.
Proof of Theorem 4.8. We conclude from Lemmas 4.10 and 3.11 that the ˆ whose restriction to automorphism group α ˆ Y is a dynamics of the extended net M Y Y Y M ⊗ M is α ⊗ α . Thus α is extendible. Remark. According to [63], each dynamics αY +P of a quantum field theory model with Yukawa2 plus P (φ)2 boson self-interaction is extendible. Finally, we conclude from Theorem 4.8: Corollary 4.11. Let αY +P be a dynamics of a quantum field theory model with Yukawa2 plus P (φ)2 boson self-interaction. For each pair ω1 , ω2 of vacuum states with respect to αY +P , there exists a kink state ω in S(αY +P |ω1 , ω2 ). 4.4. Wess Zumino models One interesting class of quantum field theory models which possess more than one vacuum sector are the N = 2 Wess–Zumino models in two-dimensional spacetime. Their properties have been studied in several papers [45, 48, 49, 46, 47] and we summarize the main results which are established there in order to setup our subsequent analysis. The field content of these models with a finite volume cutoff c > 0 consists of one complex Bose field φc and one Dirac spinor field ψc , acting as operator valued distributions on the Fock spaces
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Ha (c) :=
∞ M
L2 (Tc , C2 )⊗a ,
n=0
Hs (c) :=
∞ M
L2 (Tc , C)⊗s ,
n=0
where a, s stands for symmetrization or anti-symmetrization of the tensor product and L2 (Tc , Ck ) (k = 1, 2) denotes the Hilbert space of Ck -valued and square integrable functions, living on the circle Tc of length c. The net of Cauchy data for the finite volume theory is given by Mc : (−c, c) ⊃ I 7→ Mc (I) = Mc,s (I) ⊗ Mc,a (I) , where the nets Mc,s and Mc,a are defined by the assignments: 00 Mc,s : (−c, c) ⊃ I 7→ Mc,s (I) := ei(φc (f1 )+πc (f2 )) supp(fj ) ⊂ I , Mc,a
00 ¯ : (−c, c) ⊃ I → 7 Mc,a (I) := ψc (f1 ), ψc (f2 ) supp(fj ) ⊂ I ,
where πc is the canonically conjugate of φc . Let M := Mc=∞ be the net of Cauchy data in the infinite volume limit, then the map φc (f11 ) πc (f12 ) φ(f11 ) π(f12 ) → 7 ; supp(fij ) ⊂ (−c, c) ιc : ¯ 22 ) ψ(f21 ) ψ(f ψc (f21 ) ψ¯c (f22 ) is a W ∗ -isomorphism which identifies the nets M and Mc for those regions I which are contained in (−c, c). The interaction part of the formal Hamiltonian consists of two parts. (a) A P (φ)2 -like part: Z HP (v, c) = dx : |v 0 (Φc )|2 : − : |Φc |2 Tc
(b) A Yukawa2 -like part:
Z ¯c dx : Ψ
HY (v, c) := Tc
v 00 (Φc ) − 1 0
0 00 v (Φc )∗ − 1
Ψc
where v is a polynomial of degree deg(v) = n, called superpotential , and the fields Φc and Ψc are given by Φc := φc ⊗ 1Ha (c)
and Ψc := 1Hs (c) ⊗ ψc .
According to the results of [45, 47, 48, 49], it has been shown that, there is a self-adjoint Fredholm operator Q(v, c), called supersymmetry generator , on H0 (c) := Hs (c) ⊗ Ha (c). The Fredholm index of Q(v, c) ind(Q(v, c)) = dim ker(Q(v, c)) − dim coker(Q(v, c))
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has been computed in [48]. The result is |ind(Q(v, c))| = deg(v) − 1 . The space H0 (c) may be decomposed H0 (c) = H+ (c)⊕H− (c) into the eigenspaces of the fermion parity operator Γ := (−1)Na , where Na is the fermion number operator. With respect to this decomposition, the operator Q(v, c) has the form Q(v, c) =
0 Q− (v, c)
Q+ (v, c) 0
.
The full Hamiltonian of the finite volume model is given by H(v, c) = Q(v, c)2 which implies dim ker(H(v, c)) = |dim ker(Q+ (v, c)) − dim ker(Q− (v, c))| = deg(v) − 1 . The Hamiltonian H(v, c) induces a dynamics α(v,c) of the finite volume model and we conclude from the results of [45]: Theorem 4.12 [45, Theorem 1]. There exists at least deg(v) − 1 vacuum sectors with respect to the dynamics αv := α(v,c=∞) of the model in the infinite volume limit. 4.5. Kink states in Wess Zumino models In order to prove the existence of kink sectors, we now apply the results which have been established in Secs. 4.1 and 4.3 to N = 2 Wess–Zumino Models. The case deg(v) = 3: Let us have a closer look at the simplest non-trivial case deg(v) = 3. We let v 0 (z) = λ2 z 2 + λ1 z + λ0 . As in the previous sections (Eq. (14)), we introduce the UV-regularized fields: Φ(c0 , x) := (Φ ∗ δc0 )(x)
and Ψ(c0 , x) := (Ψ ∗ δc0 )(x) ,
where δc0 is a smooth test function with support in (−c0 , c0 ). We obtain for the P (φ)2 -like part of the regularized interaction Hamiltonian Z HP (v; c0 , c1 ) =
2 2 2 dxχc1 (x) : |λ2 Φ(c0 , x) + λ1 Φ(c0 , x) + λ0 | : − : |Φ(c0 , x)| :
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and for the Yukawa2 -like part: HY (v; c0 , c1 ) Z = dxχc1 (x) ¯ 0 , x) × : Ψ(c
0 2λ2 Φ(c0 , x) + λ1 − 1 ¯ 2 Φ(c0 , x)∗ + λ ¯1 − 1 0 2λ
Ψ(c0 , x) :
Using the same techniques as in Secs. 4.1 and 4.3, we obtain the corollary (see also Corollary 4.11): Corollary 4.13. Let v be a superpotential of degree deg(v) = 3. Then the following statements are true: (1) The dynamics αv ∈ dyn(M) of the model in the infinite volume limit is extendible. (2) There exists two different vacuum sectors e1 , e2 ∈ sec0 (αv , M) and two different kink sectors θ ∈ sec(e1 , e2 ), θ¯ ∈ sec(e2 , e1 ). The case deg(v) > 3: We close this section by discussing the remaining case. In order to show the extendibility of αv ∈ dyn(M), we can try to proceed in the same manner as for the case deg(v) = 3. According to Sec. 4.3 (Steps 2 and 3), we construct an approximation Z (v;I,c0 ,`) (v;I,c0 ,λ) ˆt (a) := dλ f` (λ) α ˆt (a) M(I) ⊗ M(I) 3 a 7→ α of the dynamics αv ⊗ αv of the two-fold theory. Provided that the corresponding (v;I,c0 ,`) result of Lemma 4.9 is true for the case deg(v) > 3 also, the linear maps α ˆt ˆ can be extended to the algebra M(I). For the generalization of Theorem 4.8, it seems that the most difficult part is to show that there exists a function c : ` 7→ c` with lim`→0 c` = 0 such that (v;I,c` ,`)
αvt (a) := w − lim αt `→0
(a) .
(18)
The regularized Yukawa-like part of the Hamilton density contains terms of the form : Ψ(i) (c0 , x)Ψ(j) (c0 , x) : : Φ(c0 , x)k : i, j ∈ {0, 1}, i 6= j
and k ≤ deg(v) − 2 ,
where Ψ(j) denotes the j-component of the Dirac spinor field Ψ. Since there are contributions with k > 1, the proof of Theorem 4.7 does not directly apply. Provided that for each superpotential v the dynamics αv is extendible, we conclude that for each pair of vacuum sectors e1 , e2 ∈ sec0 (αv , M) there exists a kink state ω ∈ S(e1 , e2 ). Then the model possesses at least deg(v)(deg(v) − 1) different non-trivial kink sectors.
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5. Conclusion and Outlook In the present paper, a construction scheme for kink sectors has been developed which can be applied to a large class of quantum field theory models. Most of the techniques which are used, except those in the proof of the extendibility of the dynamics, concern operator algebraic methods. They are model independent in the sense that they can be derived from first principles. There are still some interesting open problems and we shall make a few remarks on them here. Some further remarks on kink states: Let us consider a quantum field theory model (P (φ)2 , Y2 ), possessing vacua ω1 , ω2 which are related by a symmetry χ. According to Theorem 3.10, there exists an automorphism χI which induces a kink state ω = ω1 ◦ χI . Note that ω is a pure state in this case. Alternatively, we obtain a kink state ω ˆ by passing to the two-fold tensor product of the theory with itself first and then by restricting the αF -interpolating automorphism β I , whose existence follows also from Theorem 3.10, to the first tensor factor, i.e. ω ˆ = ω1 ⊗ ω2 ◦ βC ∗ (A)⊗1 . Provided the split property for wedge algebras holds for the interacting vacua [59], then, by applying a recent result of M. M¨ uger [55], we conclude that [ˆ ω ] is nothing else but the infinite multiple of [ω]. The problem of reducibility: The problem of reducibility arises if the vacua under consideration are not related by a symmetry since then our construction scheme leads to kink representations of the form π = π1 ⊗ π2 ◦ β|C ∗ (A)⊗1 , where β is an automorphism and π1 , 4π2 are vacuum representations. The representation π is not irreducible and whether π can be decomposed into irreducible sub-representations is still an open problem. Some of our results [60, Theorem 4.4.3] suggest that π is, in non exceptional cases, an infinite multiple of one irreducible component. Kink sectors in d > 1+1 dimensions: It would be desirable to apply our program to quantum field theories in higher dimensions. Let us suppose a theory, given by a net of W ∗ -algebras A, possesses two locally normal vacuum states ω1 , ω2 . As a sensible generalization of a kink states to d > 1 + 1, we propose to consider locally normal states ω which fulfill the interpolation condition: ω|C ∗ (A,S1 ) = ω1 |C ∗ (A,S1 )
and ω|C ∗ (A,S20 ) = ω2 |C ∗ (A,S20 ) ,
(19)
where S1 , S2 , S1 ⊂ S2 , are space-like cones. The state ω describes the coexistence of two phases which are separated by the phase boundary ∂S := S10 ∩ S2 . Let us assume duality for space-like cones in the vacuum representations under consideration. Furthermore, we presume that the inclusion
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Λ = (Aπ1 (S1 ), Aπ1 (S2 ), Ω1 ) is standard split. Here (H1 , π1 , Ω1 ) is a GNS-triple with respect to ω1 . Unfortunately, for d > 1+1 the phase boundary ∂S is not compact and therefore our construction scheme can not directly be generalized to higher dimensions. In order to overcome this difficulties, we consider a sequence of standard split inclusions Λn := (Aπ1 (O1n ), Aπ1 (O2n ), Ω1 ) , where O1n ⊂⊂ O2n are bounded double cones such that Ojn tends to Sj for n → ∞. As in the 1 + 1-dimensional case we pass now to the two-fold tensor product of the theory with itself. Denote by ΨΛn ⊗Λn the universal localizing map with respect to the inclusion Λn ⊗ Λn . Since the operators θn := ΨΛn ⊗Λn (1 ⊗ uF ) are localized in a bounded region, we may define the following automorphisms of C ∗ (A): βn := (π1 ⊗ π1 )−1 ◦ βn ◦ (π1 ⊗ π1 ) . We obtain a sequence of states {ωn , n ∈ N} where ωn is given by ωn := ω1 ⊗ ω2 ◦ βn |C ∗ (A)⊗1 . For large n the states ωn have almost the correct interpolation property, namely for each pair of local observables a, b where a is localized S20 and b is localized in S1 , there exists a sufficiently large N such that ωn (a) = ω1 (a) and ωn (b) = ω2 (b) for each n > N . Note that each state ωn fulfills the Borchers criterion since ωn belongs to the vacuum sector [ω1 ]. In order to obtain generalized kink states, we propose to investigate weak∗ -limit points of the sequence {ωn , n ∈ N}. Note that each weak*-limit ωι point of the sequence {ωn , n ∈ N} fulfills the interpolation condition (19). It remains to be proven that the weak∗ -limit points are locally normal. Acknowledgment I am grateful to Prof. K. Fredenhagen for supporting this investigation with many ideas. I am also grateful to Dr. K. H. Rehren for many hints and discussion. This investigation is financially supported by the Deutsche Forschungsgemeinschaft which is also gratefully acknowledged. References [1] H. Araki, “Soliton sectors of the XY-model”, preprint from Research Inst. for Mathematical Sciences, Kyoto Univ., Japan 1995. [2] H. Araki and R. Haag, “Collision cross sections in terms of local observables”, Commun. Math. Phys. 4 (1967) 77–91.
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[3] J. Bellissard, J. Fr¨ ohlich and B. Gidas, “Soliton mass and surface tension in the (Λ|φ|4 )2 quantum field model”, Commun. Math. Phys. 60 (1978) 37–72. [4] H.-J. Borchers, “Energy and momentum as observables in quantum field theory”, Commun. Math. Phys. 2 (1966) 49. [5] H.-J. Borchers, “CPT-Theorem in the theory of local observables”, Commun. Math. Phys. 143 (1992) 315–332. [6] H.-J. Borchers, “On the converse of the Reeh–Schlieder theorem”, Commun. Math. Phys. 10 (1968) 269–273. [7] H.-J. Borchers, Commun. Math. Phys. 4 (1967) 315–323. [8] O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics I, Berlin, Heidelberg, New York, Springer, 1979. [9] D. Buchholz, “Product states for local algebras”, Commun. Math. Phys. 36 (1974) 287–304. [10] D. Buchholz, S. Doplicher and R. Longo, “On Noether’s theorem in quantum field theory”, Ann. Phys. 170 (1) (1986). [11] D. Buchholz and K. Fredenhagen, “Locality and the structure of particle states”, Commun. Math. Phys. 84 (1982) 1–54. [12] S. Coleman, Aspects of Symmetry, Cambridge Univ. Press, 1985. [13] C. D’Antoni and K. Fredenhagen, “Charges in spacelike cones”, Commun. Math. Phys. 94 (1984) 537–544. [14] C. D’Antoni and R. Longo, “Interpolation by Type I factors and the flip automorphism”, J. Funct. Anal. 51 (1983) 361–371. [15] S. Doplicher, R. Haag and J. E. Roberts, “Fields, observables and gauge transformations I”, Commun. Math. Phys. 13 (1969) 1–23. [16] S. Doplicher, R. Haag and J. E. Roberts, “Fields, observables and gauge transformations II”, Commun. Math. Phys. 15 (1969) 173–200. [17] S. Doplicher, R. Haag and J. E. Roberts, “Local observables and particle statistics I”, Commun. Math. Phys. 23 (1971) 199–230. [18] S. Doplicher, R. Haag and J. E. Roberts, “Local observables and particle statistics II”, Commun. Math. Phys. 35 (1971) 49–58. [19] S. Doplicher and R. Longo, “Standard split inclusions of von Neumann algebras”, Invent. math. 75 (1984) 493–536. [20] W. Driessler, “On the type of local algebras in quantum field theory”, Commun. Math. Phys. 53 (1977) 295. [21] K. Fredenhagen, “Generalization of the theory of superselection sectors”, The Algebraic Theory of Superselection Sectors, World Scientific, 1989. [22] K. Fredenhagen, “Superselection sectors in low dimensional quantum field theory”, J. Geom. Phys. 11 (1993) 337–348. [23] K. Fredenhagen, “On the existence of antiparticles”, Commun. Math. Phys. 79 (1981) 141–151. [24] K. Fredenhagen, “A Remark on the Cluster Theorem”, Commun. Math. Phys. 97 (1985) 461. [25] K. Fredenhagen, K. H. Rehren and B. Schroer, “Superselection sectors with braid group statistics and exchange algebras I”, Commun. Math. Phys. 125 (1989) 201. [26] K. Fredenhagen, K. H. Rehren and B. Schroer, “Superselection sectors with braid group statistics and exchange algebras II”, Rev. Math. Phys. Special Issue, (1992) 111–154. [27] J. Fr¨ ohlich, “New superselection sectors (soliton states) in two-dimensional bose quantum field models”, Commun. Math. Phys. 47 (1976) 269–310. [28] J. Fr¨ ohlich, Quantum Theory of Nonlinear Invariant Wave (Field) Equations, Erice, Sicily, Summer 1977.
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[52] T. Matsui, “On ground states of the one-dimensional ferromagnetic XXZ model”, Lett. Math. Phys. 37 (1996) 397–403. [53] T. Matsui, “On the spectra of the kink for ferromagnetic XXZ models”, to appear in Lett. Math. Phys. (1997). [54] M. M¨ uger, “Quantum double actions on operator algebras and orbifold quantum field theories”, DESY-96-117, June 1996, to appear in Commun. Math. Phys. [55] M. M¨ uger, “Superselection structure of massive quantum field theories in 1 + 1 dimensions”, to be published. [56] H. Roos, “Independence of local algebras in quantum field theory”, Commun. Math. Phys. 16 (1970) 238–246. [57] S. Sakai, C ∗ -Algebras and W ∗ -Algebras, Berlin, Heidelberg, New York, Springer, 1971. [58] D. Schlingemann, “On the algebraic theory of soliton and anti-soliton sectors”, Rev. Math. Phys. 8 (2) (1996) 301–326. [59] D. Schlingemann, “On the existence of kink (soliton) states”, Rev. Math. Phys. 8 (8) (1996) 1187–1203. [60] D. Schlingemann, Kink States in P (φ)2 -Models, DESY 96-051, Apr. 1996. pp. 37. [61] D. Schlingemann, “On the algebraic theory of kink sectors: application to quantum field theory models and collision theory”, PhD thesis, DESY 96-228, Oct. 1996. pp. 163. [62] R. Schrader, “A remark on Yukawa Plus boson self-interaction in two space-time dimensions”, Commun. Math. Phys. 21 (1971) 164–170. [63] R. Schrader, “A Yukawa quantum field theory in two space-time dimensions without cutoffs”, Ann. Phys. 70 (1972) 412–457. [64] S. J. Summers, “Normal product states for fermions and twisted duality for CCR and CAR type algebras with application to Yukawa2 quantum field model”, Commun. Math. Phys. 86 (1982) 111–141. [65] S. J. Summers, “On the phase diagram of a P (φ)2 quantum field model”, Ann. Inst. Henri Poincar´e 34 (1981) 173–229.
SYMMETRIES OF THE QUANTUM STATE SPACE AND GROUP REPRESENTATIONS∗ GIANNI CASSINELLI Dipartimento di Fisica Universit` a di Genova, I.N.F.N. Sezione di Genova, Via Dodecaneso 33 16146 Genova, Italy E-mail : [email protected]
ERNESTO DE VITO Dipartimento di Matematica Universit` a di Modena via Campi 213/B, 41100 Modena Italy and I.N.F.N., Via Dodecaneso 33 16146 Genova, Italy E-mail : [email protected]
PEKKA LAHTI Department of Physics University of Turku FIN-20014 Turku, Finland E-mail : [email protected]
ALBERTO LEVRERO Dipartimento di Fisica Universit` a di Genova, I.N.F.N. Sezione di Genova, Via Dodecaneso 33 16146 Genova, Italy E-mail : [email protected] Received 17 November 1997 The homomorphisms of a connected Lie group G into the symmetry group of a quantum system are classified in terms of unitary representations of a simply connected Lie group associated with G. Moreover, an explicit description of the T-multipliers of G is obtained in terms of the R-multipliers of the universal covering G∗ of G and the characters of G∗ . As an application, the Poincar´e group and the Galilei group, both in 3 + 1 and 2 + 1 dimensions, are considered.
1. Introduction In the standard framework of Quantum Mechanics, the physical properties of a quantum system are described in terms of a Hilbert space. In particular, the one dimensional projectors are the pure states of the system and the physical content of the theory is given by the transition probabilities between such states. ∗ Dedicated
to Francesca. 893
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In this context, a symmetry is a bijective map from the set of pure states onto itself preserving the transition probabilities; the set S of all symmetries is a group under the composition of maps. A group G is a symmetry group for a quantum system if there exists a group homomorphism from G to S. We call such homomorphisms symmetry actions of G. Given a symmetry group G (for example, the covariance group of space-time is a symmetry group for the free particles), one can pose the mathematical problem of describing all possible symmetry actions of G. For the problem to be well-posed, one has to specify a suitable notion of equivalence between symmetry actions. Taking into account the physical meaning of the theory, a symmetry action α1 , acting on a Hilbert space H1 , is equivalent to another symmetry action α2 , acting on a Hilbert space H2 , if there exists a bijective map β from the set of pure states of H1 onto the set of pure states of H2 such that β preserves the transition probabilities and intertwines α1 and α2 . Using the Wigner theorem on the characterisation of symmetries in terms of unitary operators [1], one can analyse the symmetry actions of G in the framework of projective representations, i.e., maps g 7→ Ug from G to the group of unitary operators satisfying Ug Uh = µ(g, h)Ugh g, h ∈ G , where µ(g, h) belongs to the circle group T and the map (g, h) 7→ µ(g, h) is called a multiplier of G. However, the natural notion of unitary equivalence among projective representations does not correspond to the one among symmetry actions. The classification of projective representations for finite groups, up to unitary equivalence, was given by Schur [2, 3] who first introduced the concept of projective representation. In particular, he showed the existence of a finite group with the property that there is a one to one correspondence between its irreducible unitary representations and the irreducible projective representations of G and this correspondence preserves the unitary equivalence, (see, for example, [4, Sec. 14.2] for a modern exposition of these results). The first to solve this problem for an infinite group was Wigner in his celebrated paper [5]. He classified the projective representations of the Poincar´e group considering the unitary representations of its universal covering group. Bargmann, in his seminal paper [6], considered the case of a connected Lie group G, reducing the problem of classifying its projective representations to the one of classifying the projective representations of the universal covering G∗ . He proved that the multipliers of G∗ can always be chosen smooth and that its projective representations are in one to one correspondence with a family of unitary representations of a set of Lie groups associated with G∗ . The fundamental contribution of Mackey to this problem was a complete analysis in the case of G being a locally compact topological group [7]. In particular, he showed that there is a one to one correspondence between the projective representations of G and the unitary representations of a family of locally compact topological groups, namely the central extensions of T by G, parametrised by the group H 2 (G, T) of multipliers of G.
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The structure of H 2 (G, T) was exploited completely by Moore in the context of the cohomology of locally compact topological groups [8–11]. In particular, he proved that there exists a central extension H of an abelian group by G, called splitting group, such that there is a surjective map from the set of irreducible representations of H and the set of irreducible projective representations of G, preserving again the notion of unitary equivalence [9]. The construction of the splitting group given by Moore requires a careful analysis of the cohomology groups H 2 (G, Z) and H 2 (G, R) (see Sec. 2 of [9]). Since G, in general, is not simply connected, one cannot use linear methods to study these cohomology groups. A clear and complete exposition of the theory of projective representations and multipliers can be found in Varadarajan’s book [12], which we use in the following as a standard reference. In this paper we consider the case of a connected Lie group G and we prove that there exist a Lie group G, namely the universal extension of G, and a notion of equivalence among unitary representations of G, called physical equivalence, such that there is a one to one correspondence between the equivalence classes of irreducible unitary representations of G (with respect to physical equivalence) and the equivalence classes of irreducible symmetry actions of G. The explicit construction of G, which is a splitting group in the sense of Moore, requires the knowledge of H 2 (G∗ , R), where G∗ is the universal covering group of G. Moreover, G is a central extension of an abelian Lie group K by G and H 2 (G, T) b , where V is a is isomorphic, as a topological group, to the quotient group K/V b of K and V is completely subgroup (not necessarily closed) of the dual group K defined in terms of H 1 (G∗ , T), i.e. the group of characters of G∗ . Since G∗ is simply connected, the study of H 2 (G∗ , R) and H 1 (G∗ , T) can be done by the use of linear methods. Finally, we show that every symmetry action of G is induced by a representation of G satisfying an admissibility condition and vice versa. This admissibility condition is at the root of the existence of superselection rules in the case of reducible representations (this topic has been recently considered also by Divakaran [13]). In the paper we assume that G is connected, otherwise one can consider only the connected component of the identity of G. We do not consider the problem of discrete symmetries. 2. Preliminary Results In this section we introduce the notations and we review some results on the theory of multipliers for simply connected Lie groups. 2.1. Notations 1. By Hilbert space we mean a complex separable Hilbert space with scalar product h·, ·i linear in the second argument. 2. If H is a Hilbert space, we denote by PH (or P) the set of one dimensional projectors on H.
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3. We use the word representation to mean a unitary representation of a topological group, acting on a Hilbert space and continuous with respect to the strong operator topology. 4. If α is a Lie group homomorphism we denote by α˙ the differential of α at the identity. Let H be the Hilbert space associated with a quantum system. Then P is the set of pure states of the system and the transition probability between two pure states P1 , P2 ∈ P is given by tr P1 P2 , where tr · denotes the trace. Definition 1. A map α : P → P is a symmetry of P if α is bijective and tr P1 P2 = tr α(P1 )α(P2 ) , P1 , P2 ∈ P . We denote by S the set of symmetries of P. The set S is a group with respect to the usual composition of maps and it is a topological space with respect to the initial topology given by the family of functions S 3 α 7→ tr P1 α(P2 ) ∈ R , labelled by P1 , P2 ∈ P. Using the Wigner theorem [1], we can obtain a useful characterisation of S. In fact, let U denote the set of unitary operators on H, U the set of antiunitary operators on H, and let T = {z ∈ C : |z| = 1} denote the circle group. We identify T with {zI : z ∈ T} ⊂ U. The set U ∪ U is a group under the usual composition of operators and it is a topological space with respect to the restriction of the strong operator topology. Then we have the following results, whose proofs can be found, for example, in [12] (for a review see also [14]). Proposition 1. With the above notations: 1. the group U ∪ U is a second countable metrisable topological group, U is the connected component of the identity and T is its centre; 2. the group S is a second countable metrisable topological group; 3. the map π : U ∪ U → S defined as π(U )(P ) = U P U −1 ,
P ∈P,
is a continuous surjective open group homomorphism and its kernel is the group T; 4. there exists a measurable map s from S0 , the connected component of the identity of S, to U such that s(I) = I, π(s(α)) = α , We call such a map a section for π.
α ∈ S0 .
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We are now in a position to define a symmetry action. Let G be a connected Lie group. Definition 2. A symmetry action of G on H is a continuous group homomorphism α : G → S. Two symmetry actions α1 and α2 of G acting, respectively, on the Hilbert spaces H1 and H2 are equivalent if there is a bijective map β from PH1 onto PH2 such that tr P Q H = tr β(P )β(Q) H , 1
βα1g
2
=
α2g β
,
for all P, Q ∈ PH1 and for all g ∈ G. A symmetry action α is irreducible if for all P1 , P2 ∈ P there is g ∈ G satisfying tr P2 αg (P1 ) 6= 0 . Since G is connected all symmetry actions take values in S0 . Moreover, if α is an irreducible symmetry action, then every symmetry action equivalent to α is irreducible too. Finally, the condition on the continuity of the symmetry actions can be weakened, using the standard result that a group homomorphism α : G → S is continuous if and only if it is measurable (see, for example, Lemma 5.28 of [12]). 2.2. Multipliers for Lie groups: A review In this section we give a brief review of the theory of multipliers for a connected, simply connected Lie group. For this class of groups the problem of the classification (up to equivalence) of multipliers can be reduced to a finite-dimensional linear problem on the Lie algebra of the group. We stress that the theory works only in the case of simply connected Lie groups; nevertheless, as we are going to show in the next section, this is enough to study the symmetry actions also for Lie groups which are not simply connected. All the proofs of this section can be found, for example, in Chap. 7 of [12] where a systematic study of multipliers is presented. Let H be a connected Lie group and A an abelian Lie group (for the moment we do not assume that H is simply connected). We denote by e and 1 the unit elements of H and A, respectively. Definition 3. An A-multiplier of H is a measurable map τ from H × H to A such that τ (e, g) = τ (g, e) = 1 , τ (g1 , g2 g3 )τ (g2 , g3 ) = τ (g1 , g2 )τ (g1 g2 , g3 ) ,
g∈H, g1 , g2 , g3 ∈ H .
Two A-multipliers τ1 and τ2 of H are equivalent if there is a measurable map b from H to A such that τ2 (g1 , g2 ) =
b(g1 g2 ) τ1 (g1 , g2 ) , b(g1 )b(g2 )
g1 , g2 ∈ H .
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An A-multiplier τ is exact if it is equivalent to the multiplier 1, that is, τ (g1 , g2 ) =
b(g1 g2 ) , b(g1 )b(g2 )
g1 , g2 ∈ H
for some measurable map b : H → A. The set of A-multipliers is an abelian group under the pointwise multiplication and the set of exact A-multipliers is a subgroup. We denote by H 2 (H, A) the corresponding quotient group. Remark 1. There is a natural topology on H 2 (H, A) converting it into a locally compact group (in general not Hausdorff), see [11]. If A = T it coincides with the (quotient) topology of uniform convergence on compact sets defined on the set of T-multipliers (see Theorem 6 of [11]). We will always consider H 2 (H, T) endowed with this topology. From now on, assume that H is simply connected. If A is the group T, we have the following result. Lemma 1. Each T-multiplier of H is similar to one of the form eiτ , where τ is a smooth R-multiplier of H. Moreover, τ is exact if and only if eiτ is exact. If A is the vector group Rn , we denote it additively. The set of Rn -multipliers is a real vector space under the pointwise operations and the set of exact Rn -multipliers is a subspace of it, so that H 2 (H, Rn ) is a vector space. Moreover, we have: Lemma 2. Any Rn -multiplier of H is equivalent to a smooth one. Let Lie (H) be the Lie algebra of H. Definition 4. A bilinear skew symmetric map F from Lie (H) × Lie (H) to Rn such that F (X, [Y, Z]) + F (Z, [X, Y ]) + F (Y, [Z, X]) = 0 ,
X, Y, Z ∈ Lie (H) ,
is called a closed Rn -form. A closed Rn -form F is exact if there is a linear map q from Lie (H) to Rn such that F (X, Y ) = q([X, Y ]) ,
X, Y ∈ Lie (H) .
The set of closed Rn -forms is a finite dimensional real vector space and the set of exact Rn -forms is a subspace. We denote by H 2 (Lie (H), Rn ) the corresponding quotient space.
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Theorem 1. The vector spaces H 2 (H, Rn ) and H 2 (Lie (H), Rn ) are canonically isomorphic. We exhibit explicitly the above isomorphism. Let F be a closed Rn -form, and denote by Rn ⊕F Lie (H) the Lie algebra defined by the following Lie bracket: (v1 , X1 ), (v2 , X2 ) := F (X1 , X2 ), X1 , X2 , for all X1 , X2 ∈ Lie (H) and v1 , v2 ∈ Rn . Let α : Rn → Rn ⊕F Lie (H) be the natural injection and β : Rn ⊕F Lie (H) → Lie (H) be the natural projection. Both these maps are Lie algebra homomorphisms and Ker β = Im α. By the general theory of Lie groups, there exist a unique (up to an isomorphism) connected, simply connected Lie group HF , such that Lie (HF ) = Rn ⊕F Lie (H), and two Lie group homomorphisms a : Rn → HF , b : HF → H such that a˙ = α and b˙ = β. Moreover, one can prove that a is a homeomorphism from Rn onto a(Rn ) and HF /a(Rn ) is isomorphic to H. By a lemma of Malˇcev (see, for example, Lemma 7.26 of [12]) there exists a smooth map c from H to HF such that c(e) = e and b(c(h)) = h for all h ∈ H. If we define τF (h1 , h2 ) = c(h1 )c(h2 )c(h1 h2 )−1 ,
h1 , h2 ∈ H ,
then τF is a smooth Rn -multiplier. The equivalence class of τF is the image of the equivalence class of F under the isomorphism of the above theorem. Since τF is smooth, one can easily check that HF is isomorphic, as a Lie group, to Rn ×τF H, which is a Lie group with respect to the product (v1 , g1 )(v2 , g2 ) = (v1 + v2 + τF (g1 , g2 ), g1 g2 ) ,
v1 , v2 ∈ Rn , g1 , g2 ∈ H .
3. Main Results In this section we introduce the notion of universal central extension G for a connected Lie group G and the notions of physical equivalence and admissibility for representations of G. By the use of these concepts, we then state the main results of the paper. Moreover, we discuss the physical equivalence for induced representations in the case that G is a semidirect product with abelian normal factor. 3.1. Universal central extension Let G be a connected Lie group. We denote by G∗ its universal covering group and by δ the corresponding covering homomorphism. Let H 2 (G∗ , R)δ be the set of equivalence classes [τ ] ∈ H 2 (G∗ , R) such that τ (k, g ∗ ) = τ (g ∗ , k) ,
k ∈ Ker δ, g ∗ ∈ G∗ .
(1)
Since Ker δ is central in G∗ , Eq. (1) holds for all R-multipliers equivalent to τ and, hence, the definition of H 2 (G∗ , R)δ is well-posed. Moreover, H 2 (G∗ , R)δ is a subspace of H 2 (G∗ , R), so that, due to Theorem 1, it has finite dimension N .
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By Lemma 2, we can fix N smooth R-multipliers of G∗ , τ1 , . . . , τN , such that the equivalence classes [τ1 ], . . . , [τN ] form a basis of H 2 (G∗ , R)δ . Let τ : G∗ × G∗ → RN be defined as τ (g1∗ , g2∗ )i = τi (g1∗ , g2∗ ) ,
g1∗ , g2∗ ∈ G∗ , i = 1, . . . , N ,
then τ is a smooth RN -multiplier of G∗ . The restriction of τ to Ker δ×Ker δ is an RN -multiplier of the discrete group Ker δ, hence it is exact (see Proposition 2, Sec. 4, Chap. 1 of [15]). Then, without loss of generality, we can always assume that τ is smooth and that τ (k1 , k2 ) = 0 ,
k1 , k2 ∈ Ker δ .
(2)
Definition 5. Let G = RN ×τ G∗ be the product manifold. Since τ is smooth, G is a Lie group with the product (v1 , g1∗ )(v2 , g2∗ ) = (v1 + v2 + τ (g1∗ , g2∗ ), g1∗ g2∗ ) ,
g1∗ , g2∗ ∈ G∗ , v1 , v2 ∈ Rn .
We call it the universal central extension of G and we denote by σ the smooth map from G to G given by σ(v, g ∗ ) = δ(g ∗ ) ,
v ∈ RN , g ∗ ∈ G∗ .
We denote by K the closed subgroup of G defined as K = {(v, k) ∈ G : v ∈ RN ,
k ∈ Ker δ} .
The main properties of G are stated in the following lemma. Lemma 3. Let G be the universal central extension of G. 1. The restriction of a character of G to the subgroup RN is the identity character. Any character of G∗ extends naturally to a character of G. 2. The map σ is a surjective group homomorphism whose kernel is K, which is central in G. Moreover, the group K is the direct product of RN and Ker δ. 3. There is a measurable map c : G → G such that c(e) = e and σ(c(g)) = g for all g ∈ G (we call such a map a section for σ). 4. Given a section c for σ, let Γc from G × G to K be the map Γc (g1 , g2 ) = c(g1 )c(g2 )c(g1 g2 )−1 ,
g1 , g2 ∈ G,
then Γc is a K-multiplier of G and its equivalence class does not depend on the choice of the section c. 5. If we consider RN as a subgroup of K in a natural way, then the K-multiplier Γc ◦ (δ × δ) of G∗ is equivalent to τ .
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Proof. 1. Let χ be a character of G. The restriction of χ to RN is of the form χ(v, e∗ ) = eiw·v
v ∈ RN ,
for some w ∈ RN . Then, if g1∗ , g2∗ ∈ G∗ , χ((0, g1∗ ))χ((0, g2∗ )) = χ((τ (g1∗ , g2∗ ), e∗ ))χ((0, g1∗ g2∗ )) ∗
∗
= eiw·τ (g1 ,g2 ) χ((0, g1∗ g2∗ )) . Hence, by Lemma 1, the R-multiplier w · τ is exact, so that w = 0. The other statement is evident. 2. The fact that K is central in G follows taking into account that, by definition of H 2 (G∗ , R)δ , τ (k, g ∗ ) = τ (g ∗ , k) for all k ∈ Ker δ and g ∗ ∈ G∗ . Using Eq. (2), one has K = RN × Ker δ. The other facts are evident. 3. By the previous item, G is isomorphic, as a Lie group, to the quotient G/K. The existence of a section is thus a standard result (see, for example, Theorem 5.11 of [12]). 4. If g1 , g2 ∈ G, then σ(Γc (g1 , g2 )) = e, so that Γc (g1 , g2 ) ∈ K. By direct computation one checks that Γc is a K-multiplier. Let c0 be another section, then, for all g ∈ G, c(g) = b(g)c0 (g) for some measurable map b from G to K. Hence, for all g1 , g2 ∈ G b(g1 g2 ) Γc (g1 , g2 ) . Γc0 (g1 , g2 ) = b(g1 )b(g2 ) 5. Let i : G∗ → G be the natural immersion and a be the measurable map from G to G a(g ∗ ) = c(δ(g ∗ ))i(g ∗ )−1 , g ∗ ∈ G∗ . ∗
Since σ(a(g ∗ )) = e, then a takes values in K. Then, if g1∗ , g2∗ ∈ G∗ , Γc (δ(g1∗ ), δ(g2∗ )) = c(δ(g1∗ ))c(δ(g2∗ ))c(δ(g1∗ )δ(g2∗ ))−1 = a(g1∗ )i(g1∗ )a(g2∗ )i(g2∗ )i(g1∗ g2∗ )−1 a(g1∗ g2∗ )−1 = a(g1∗ )a(g2∗ )a(g1∗ g2∗ )−1 i(g1∗ )i(g2∗ )i(g1∗ g2∗ )−1 = a(g1∗ )a(g2∗ )a(g1∗ g2∗ )−1 (τ (g1∗ , g2∗ ), e∗ ), i.e., Γc ◦ (δ × δ) is equivalent to τ .
The following theorem describes the group H 2 (G, T) in terms of the characters of K and the characters of G∗ . We first observe that: Lemma 4. Let c : G → G be a section for σ and Γc the corresponding Kmultiplier of G defined in statement 4 of Lemma 3. Let χ be a character of K, then the map µχ from G × G to T defined as µχ (g1 , g2 ) = χ (Γc (g1 , g2 )) ,
g1 , g2 ∈ G ,
is a T-multiplier of G and its equivalence class [µχ ] does not depend on the choice of the section c.
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Proof. It is a simple consequence of the properties of Γc given in statement 4 of Lemma 3. b be the dual group of K (with the topology of uniform convergence on compact Let K sets) and V the subgroup of characters that extend to characters of G, which, due to the statement 1 of Lemma 3, can be identified with characters of G∗ . Theorem 2. The mapping b 3 χ 7→ [µχ ] ∈ H 2 (G, T) K is a surjective homomorphism whose kernel is V. Moreover, H 2 (G, T) is isomorphic, b as a topological group, to the quotient group K/V. Proof. By direct computation, one can check that χ 7→ [µχ ] is a group homomorphism. To show its surjectivity, we notice that, since the equivalence class [µχ ] does not depend on the specific form of the section c, we can choose for c the particularly simple form c(g) = (0, c˜(g)) g ∈ G , c(g)) = g for all where c˜ : G → G∗ is measurable and satisfies c˜(e) = e∗ and δ(˜ g ∈ G. With this choice, a straightforward calculation shows that c(g1 ), c˜(g2 )) − τ (γ(g1 , g2 ), c˜(g1 g2 )) , γ(g1 , g2 )) , Γc (g1 , g2 ) = (τ (˜
(3)
where g1 , g2 ∈ G and c(g2 )˜ c(g1 g2 )−1 ∈ Ker δ . γ(g1 , g2 ) = c˜(g1 )˜ Let now µ be a T-multiplier of G and µ∗ the T-multiplier of G∗ µ∗ (g1∗ , g2∗ ) = µ(δ(g1∗ ), δ(g2∗ )) ,
g1∗ , g2∗ ∈ G∗ .
Applying Lemma 1 to µ∗ , we have that µ∗ (g1∗ , g2∗ ) =
a(g1∗ g2∗ ) iτ (g1∗ ,g2∗ ) e , a(g1∗ )a(g2∗ )
g1∗ , g2∗ ∈ G∗ ,
(4)
for some smooth R-multiplier τ of G∗ and some measurable function a from G∗ to T. We claim that τ (k, g ∗ ) = τ (g ∗ , k) k ∈ Ker δ , g ∗ ∈ G∗ . In fact, let k ∈ Ker δ and g ∗ ∈ G∗ , since µ∗ (k, g ∗ ) = µ∗ (g ∗ , k) = 1, then eiτ (k,g
∗
)
=
∗ a(k)a(g ∗ ) a(k)a(g ∗ ) = = eiτ (g ,k) . ∗ ∗ a(kg ) a(g k)
(5)
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Hence τ (k, g ∗ ) = τ (g ∗ , k) + 2πn(k, g ∗ ) where n(k, g ∗ ) is an integer. By continuity of τ (k, ·) and since G∗ is connected, the map n(·, ·) depends only on k, and, choosing g ∗ = k, we conclude that n(k, g ∗ ) = 0 for all k ∈ Ker δ, g ∗ ∈ G∗ . Due to (5), the equivalence class of τ belongs to H 2 (G∗ , R)δ and, by definition of τ , there is w ∈ RN such that, up to equivalence, τ = w · τ . Hence (4) becomes µ∗ (g1∗ , g2∗ ) =
a(g1∗ g2∗ ) iw·τ (g1∗ ,g2∗ ) e , a(g1∗ )a(g2∗ )
g1∗ , g2∗ ∈ G∗ .
(6)
The previous equality implies that the map χ from K to T χ(v, k) := eiw·v a(k) ,
v ∈ RN , k ∈ Ker δ ,
is, in fact, a character of K. Hence, by Lemma 4, χ defines a T-multiplier µχ of G. We will show that µχ is equivalent to µ. In fact, using Eq. (3), one has µχ (g1 , g2 ) = χ(Γc (g1 , g2 )) = eiw·(τ (˜c(g1 ),˜c(g2 ))−τ (γ(g1 ,g2 ),˜c(g1 g2 ))) a(γ(g1 , g2 )) Using twice Eq. (6) we obtain eiw·τ (˜c(g1 ),˜c(g2 )) =
a(˜ c(g1 ))a(˜ c(g2 )) µ(g1 , g2 ) a(˜ c(g1 )˜ c(g2 ))
e−iw·τ (γ(g1 ,g2 ),˜c(g1 g2 )) =
a(˜ c(g1 )˜ c(g2 )) a(γ(g1 , g2 ))a(˜ c(g1 g2 ))
so that µχ (g1 , g2 ) =
a(˜ c(g1 ))a(˜ c(g2 )) µ(g1 , g2 ) , a(˜ c(g1 g2 ))
which shows the equivalence of µ and µχ . Suppose now that χ is a character of K that extends to a character of G (still denoted by χ). Then µχ (g1 , g2 ) = χ(c(g1 )c(g2 )c(g1 g2 )−1 ) = χ(c(g1 ))χ(c(g2 ))χ(c(g1 g2 )−1 ), showing that µχ is exact. Conversely, assume that µχ (g1 , g2 ) =
a(g1 g2 ) a(g1 )a(g2 )
for some measurable function a : G → T. Observe that, for all h ∈ G, hc(σ(h))−1 ∈ K and define χ0 : G → T as χ0 (h) = χ(hc(σ(h))−1 )a(σ(h))−1
h ∈ G.
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Then χ0 is a character of G. Indeed, χ0 is measurable, and if h1 , h2 ∈ G, χ0 (h1 )χ0 (h2 ) =
χ(h1 c(σ(h1 ))−1 h2 c(σ(h2 ))−1 ) a(σ(h1 ))a(σ(h2 ))
=
χ(h1 h2 c(σ(h2 ))−1 c(σ(h1 ))−1 )µχ (g1 , g2 ) a(σ(h1 h2 ))
=
χ(h1 h2 c(σ(h2 ))−1 c(σ(h1 ))−1 c(σ(h1 ))c(σ(h2 ))c(σ(h1 h2 ))−1 ) a(σ(h1 h2 ))
= χ(h1 h2 c(σ(h1 h2 ))−1 )a(σ(h1 h2 ))−1 = χ0 (h1 h2 ) . Moreover, since a(e) = 1, χ0 (k) = χ(k) for all k ∈ K. Hence, H 2 (G, T) is isomorphic, as an abstract group, to the quotient group K/V . This completes the proof but to observe that G is a splitting group for T in the sense of Moore (see Definition 3 of [11]) so that, using Theorem 6 of [11], H 2 (G, T) is isomorphic, as a topological group, to the quotient group K/V . As a consequence of the previous result, one has that H 2 (G, T) is Hausdorff if and b The following example, inspired by Moore [9], shows that only if V is closed in K. this is not always the case. Example 1. Let G = T2 × R2 be the Lie group with product 0 0 0 0 2π (z, ζ, x, y)(z 0 , ζ 0 , x0 , y 0 ) = zz 0 ei α (xy −yx ) , ζζ 0 ei2π(xy −yx ) , x + x0 , y + y 0 , where α ∈ R, α 6= 0. The universal covering group G∗ of G is G∗ = R2 × R2 with product (v, w, x, y)(v 0 , w0 , x0 , y 0 ) = (v + v 0 + (xy 0 − yx0 ), w + w0 + (xy 0 − yx0 ), x + x0 , y + y 0 ) , and with covering homomorphism δ from G∗ to G 2π δ(v, w, x, y) = ei α v , ei2πw , x, y . We have Ker δ = {(αn, m, 0, 0) : n, m ∈ Z} . A simple algebraic calculation shows that any R-multiplier of G∗ is equivalent to one of the form τ ((v, w, x, y), (v 0 , w0 , x0 , y 0 )) = Q((v, w), (x0 , y 0 )) , where Q(·, ·) is a bilinear form on R2 × R2 . It follows that H 2 (G∗ , R)δ is {0} so that b is isomorphic the universal central extension of G is G∗ and K = Ker δ. Then K to T × T.
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Moreover, one checks that the characters of G∗ are of the form (v, w, x, y) 7→ eia(v−w)+bx+cy , where a, b, c ∈ R. Hence we have V = (eiαa , e−ia ) ∈ T × T : a ∈ R , so that V is closed in T × T if and only if α is rational. If V is closed, we can give a better description of H 2 (G, T). Define K0 = {(v, k) ∈ K : b(k) = 1 for any character b of G∗ } , then K0 is an abelian closed subgroup of K. Since V is closed, a standard result on abelian locally compact groups (see, for example, Theorem 4.39 of [16]) shows that c0 b c0 of K0 . In particular, any element χ ∈ K K/V is isomorphic to the dual group K b extends to an element χ b ∈ K and χ b is uniquely defined by χ, up to an element of V . Let µχ be the T-multiplier of G defined by b(Γc (g1 , g2 )) , µχ (g1 , g2 ) = χ
g1 , g2 ∈ G ,
where Γc is defined in statement 4 of Lemma 3. As a consequence of Theorem 2, the equivalence class [µχ ] depends only on χ and not on the particular extension chosen. Corollary 1. If V is closed , the mapping c0 3 χ 7→ [µχ ] ∈ H 2 (G, T) K is an isomorphism of topological groups. We now turn to the representations of G. Definition 6. A representation U of G satisfying the condition Uh ∈ T
for all h ∈ K ,
(7)
is called admissible. Definition 7. Let U be an admissible representation, then its restriction to K is a character of K and, due to statement 2 of Lemma 3, is of the form U(v,k) = eiw·v (k) ,
v ∈ RN , k ∈ Ker δ ,
for some w ∈ RN and some character of Ker δ. We call w the algebraic charge of U and the topological charge.
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In the above definition we have followed the terminology of Divakaran [13]. Definition 8. Let U and U 0 be two representations of G acting respectively in H and H0 . We say that U and U 0 are physically equivalent if there exist a unitary or antiunitary operator B : H → H0 and a measurable map b : G → T such that BUh = b(h)Uh0 B ,
h ∈ G.
(8)
We notice the following facts concerning these definitions. 1. The notion of physical equivalence preserves condition (7) and the usual notion of irreducibility of representations. The case of unitary equivalence is a particular instance of the physical equivalence. 2. Since K is central in G, every irreducible representation of G is admissible. 3. Since in (8) U and U 0 are representations, the map b is, in fact, a character of G and, by statement 1 of Lemma 3, a character of G∗ . We are now in a position to state the main property of G. Given an admissible representation U of G, define, for all g ∈ G, αU g = π(Uh ) ,
(9)
where π is defined in Proposition 1 and h ∈ G is such that σ(h) = g. The following theorem is then obtained. Theorem 3. With the above notations, αU is a symmetry action of G and the correspondence [U ] 7→ [αU ] between the physical equivalence classes of admissible representations of G and the equivalence classes of symmetry actions of G is a bijection. The representation U of G is irreducible if and only if αU is an irreducible symmetry action of G. Proof. In the following we fix a section c : G → G for σ (cf. item 3 of Lemma 3) and a section s : S0 → U for π : U → S0 (cf. item 4 of Proposition 1). Due to condition (7), if h1 , h2 ∈ G are such that σ(h1 ) = σ(h2 ) = g, then π(Uh1 ) = π(Uh2 ), showing that αU g is well-defined. In particular, we have αU g = π(Uc(g) ) g ∈ G . First we show that g 7→ αU g is a symmetry action of G. Indeed, if g1 , g2 ∈ G then U αU g1 αg2 = π Uc(g1 ) π Uc(g2 ) = π Uc(g1 ) Uc(g2 ) = π Uc(g1 )c(g2 )c(g1 g2 )−1 π Uc(g1 g2 ) = π Uc(g1 g2 ) = αU g1 g2 ,
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where we used the fact that c(g1 )c(g2 )c(g1 g2 )−1 ∈ K and condition (7). Since c is U measurable, αU is measurable too, and αU e = I. Hence α is a symmetry action of G. Now let U and U 0 be two physically equivalent admissible representations of G acting on H and H0 , respectively, then the corresponding symmetry actions αU and 0 αU are equivalent too. Indeed, in this case BUh = b(h)Uh0 B,
h ∈ G,
for some unitary or antiunitary B : H → H0 and a character b : G → T. Define β from PH to PH0 as β(P ) = BP B −1 . Then β is bijective, preserves the transition U0 U probabilities and satisfies βαU g = αg β for all g ∈ G, which is just to say that α 0 U U and α are equivalent. This shows that the map [U ] 7→ [α ] is well-defined. We now show its surjectivity. Let α be a symmetry action of G, define µ : G × G → U as µ(g1 , g2 ) := s(αg1 )s(αg2 )s(αg1 g2 )−1 ,
g1 , g2 ∈ G .
Since π(µ(g1 , g2 )) = I then µ(g1 , g2 ) ∈ T. Moreover, µ is measurable and by a direct computation one confirms that µ is, in fact, a T-multiplier of G. By Theorem 2, there are a character χ of K and a measurable function a : G → T such that µ(g1 , g2 ) =
a(g1 g2 ) µχ (g1 , g2 ) a(g1 )a(g2 )
g1 , g2 ∈ G .
Define a map U α : G → U as Uhα := χ(hc(σ(h))−1 )a(σ(h))s(ασ(h) ) ,
h ∈ G.
Then U α is a representation of G. Indeed, 1. U α is measurable as a composition of measurable maps; α 2. U(0,e ∗ ) = I, since a(e) = 1 and s(I) = I; 3. for any h1 , h2 ∈ G, Uh1 Uh2 = χ(h1 c(σ(h1 ))−1 h2 c(σ(h2 ))−1 ) ×a(σ(h1 ))a(σ(h2 ))s(ασ(h1 ) )s(ασ(h2 ) ) = χ h1 h2 c(σ(h2 ))−1 c(σ(h1 ))−1 ×a(σ(h1 ))a(σ(h2 ))µ(g1 , g2 )s(ασ(h1 h2 ) ) = χ h1 h2 c(σ(h2 ))−1 c(σ(h1 ))−1 ×χ c(σ(h1 ))c(σ(h2 ))c(σ(h1 h2 ))−1 ×a(σ(h1 h2 ))s(ασ(h1 h2 ) ) = χ h1 h2 c(σ(h1 h2 ))−1 a(σ(h1 h2 ))s(ασ(h1 h2 ) ) = U h1 h2 .
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Since π ◦ s = id S0 and σ ◦ c = id G , one readily verifies that αU = α, proving the surjectivity of the map [U ] 7→ [αU ]. 0 Assume next that αU and αU are equivalent symmetry actions. By the definition of equivalence of symmetry actions, there is a bijective map β, preserving the transition probabilities, such that, for all g ∈ G, 0 β = βπ Uc(g) . π Uc(g) Applying the Wigner theorem [1], we deduce that for some unitary or antiunitary operator B and for some measurable map b : G → T, 0 = b(g)BUc(g) B −1 . Uc(g)
Let h ∈ G, g = σ(h), and k = hc(g)−1 , then k ∈ K and 0 Uh0 = Uk0 Uc(g)
= Uk0 b(c(g))BUc(g) B −1 = Uk0 b(c(g))BUk−1 Uh B −1 = ˆb(h)BUh B −1 , taking into account that, due to (7), Uk0 and Uk−1 are phase factors that we have collected in ˆb. This shows that U and U 0 are physically equivalent representations of G, proving the injectivity of the map [U ] 7→ [αU ]. Finally, due to (7), an admissible representation U is irreducible if and only if
φ1 , Uc(g) φ2 = 0 ∀ g ∈ G =⇒ (φ1 = 0 or φ2 = 0) . This last condition is equivalent to the fact that αU is irreducible.
This theorem shows that the equivalence classes of admissible representations of G classify the different (with respect to the given symmetry group G) quantum systems. In particular, the irreducible representations of G are always admissible and describe the elementary systems. Let us now consider the case of reducible representations of G. For the sake of simplicity, let U = U1 ⊕ U2 where U1 and U2 are irreducible representations. Since the representations Ui are irreducible, they are admissible and we denote by wi and i the corresponding algebraic and topological charges. However, in general, U is not admissible. A simple calculation shows that U is admissible if and only if w1 = w2 and 1 = 2 . This fact is at the root of the existence of superselection rules for non-elementary systems, as it will be discussed in more detail in the examples. Furthermore, the relation between the decomposition into irreducible representations and the notion of physical equivalence requires some special care. One can easily show that, if b is a nontrivial character of G that is 1 on K, then U1 ⊕ U2 and U1 ⊕ b U2 are physically inequivalent admissible representations, even though U2 and bU2 are physically equivalent. In the same way, if the algebraic charge w of
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U1 and U2 is zero and their topological charge is such that 2 ∈ V , then U1 ⊕ U2 and U1 ⊕ bBU2 B −1 (where b is any character of G that extends 2 and B is any antiunitary operator) are physically inequivalent admissible representations, even though U2 and bBU2 B −1 are physically equivalent. This kind of phenomenon does not occur if one considers the unitary equivalence instead of the physical one. We add some comments about the relation between the admissible representations of G and the projective representations of G and G∗ . Let U be an admissible representation of G, w and its algebraic and topological charges. 1. The map G∗ 3 g ∗ 7→ U(0,g∗ ) ∈ U is a projective representation of G∗ with ∗ ∗ T-multiplier µ∗ (g1∗ , g2∗ ) = eiw·¯τ (g1 ,g2 ) . 2. If c : G → G is a section for σ then the map G 3 g 7→ Uc(g) ∈ U is a projective representation of G and its T-multiplier is µχ where χ(v, k) = eiw·v (k) and µχ is defined in Lemma 4. As a consequence of statement 5 of Lemma 3, µ∗ and µχ ◦ (δ × δ) are equivalent, nevertheless, even if µ∗ is exact, µχ could be nonexact (see remark 1 in Sec. 4.1 and 1 in Sec. 4.2). 3.2. The physical equivalence for semidirect products According to Theorem 3, the irreducible inequivalent symmetry actions of a group G are completely described by the irreducible physically inequivalent representations of its universal central extension G. In the examples we consider in the next section, the universal central extension is a regular semidirect product with abelian normal subgroup, so that any irreducible representation is unitarily equivalent to some induced one [17]. In this way, the problem of characterising physically inequivalent irreducible representations is reduced to the analogous problem for the induced ones. In the present section we describe the solution in terms of properties of the orbits in the dual space and of the inducing representations. Let G = A ×0 H be a Lie group with A an abelian normal closed subgroup and H a closed subgroup. We denote by Aˆ the dual group of A and by (g, ·) 7→ g[·] ˆ If x ∈ A, ˆ let Gx both the inner action of G on A and the dual action of G on A. be the stability subgroup of G at x and G [x] the corresponding orbit. We assume that each orbit in Aˆ is locally closed (i.e. the semidirect product is regular) and, to simplify the exposition, that it has a G-invariant σ-finite measure. Moreover, given x ∈ Aˆ and a representation D of Gx ∩ H acting in a Hilbert G (xD) the representation of G unitarily induced space K, we denote by U = IndG x
by the representation xD of Gx , (xD)ah = xa Dh ,
a ∈ A, h ∈ Gx ∩ H .
Explicitly, let ν be a G-invariant σ-finite measure on G [x] and c a measurable map from G [x] to G such that c(x) = e and c(y)[x] = y for all y ∈ G [x] (we call such a map a section for G [x]), then U acts on the Hilbert space L2 ( G [x], ν, K) as (Ug f )(y) = (xD)(c(y)−1 gc(g−1 [y])) f (g −1 [y]) , where y ∈ G [x], f ∈ L2 ( G [x], ν, K), and g ∈ G.
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We are now in a position to classify all the equivalence classes (with respect to the notion of physical equivalence) of irreducible representations of G in the case of regular semidirect products. ˆ i.e., Let Aˆs be the set of singleton G-orbits in A, Aˆs = y ∈ Aˆ : g[y] = y ,
g∈G .
Define for all x ∈ Aˆ the orbit class ex := yg[x ] : y ∈ Aˆs , O
g ∈ G, = ±1 .
ex , G [x0 ] ⊂ O ex and O ex = O ex0 , so that we can choose a Obviously, for all x0 ∈ O ˆ ˆ exi . family {xi }i∈I of elements in A such that A is the disjoint union of the sets O Theorem 4. Let G = A ×0 H be a regular semidirect product. 1. Every irreducible representation of G is physically equivalent to one of the (xi D) for some index i and some irreducible representation D of form Ind G G xi
Gxi ∩ H. 2. If i 6= j and D, D0 are two representations of Gxi ∩ H and Gxj ∩ H, (xi D) and Ind G (xj D0 ) are physically inequivalent. respectively, then Ind G G G xi
xj
(xD) 3. Let x ∈ Aˆ and D, D0 be two representations of Gx ∩ H. Then Ind G G x
(xD0 ) are physically equivalent if and only if one of the following and Ind G Gx two conditions is satisfied: (a) there exist y ∈ Aˆs , a character χ of H, and a unitary operator M such that G [x] = yG [x] , 0 −1 , Dhsh −1 = χs M Ds M
s ∈ Gx ∩ H ,
where h ∈ H is such that x = yh[x]; (b) there exist y ∈ Aˆs , a character χ of H, and an antiunitary operator M such that G [x] = yG [x]−1 , 0 −1 , Dhsh −1 = χs M Ds M
s ∈ Gx ∩ H ,
where h ∈ H is such that x = yh[x−1 ]. Motivated by the above theorem, if U is a representation of G physically equiv(xD) we say, with slight abuse of teralent to some induced representation Ind G Gx e minology, that U lives on the orbit class Ox . The proof of the theorem is based on the following lemma.
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ˆ Let D be a representation of Gx ∩ H acting in K and Lemma 5. Let x, x0 ∈ A. D a representation of Gx0 ∩ H acting in K0 . The induced representations Ind G G 0
x
(xD) and Ind G (x0 D0 ) are physically equivalent if and only if there exist h ∈ G, G x0
a character χ e of G and a unitary or antiunitary operator M from K onto K0 such that 1. Gx0 = hGx h−1 ; eg M (xD)g M −1 for all g ∈ Gx . 2. (x0 D0 )hgh−1 = χ Moreover, every character of G is of the form (a, h) 7→ χ ˆ a χh
a ∈ A, h ∈ H
where χ ˆ ∈ Aˆs and χ is a character of H. e is a character Proof. First we prove the statement on the characters of G. If χ ˆ and χ be its restrictions to A and H, respectively. Then χ is a character of G, let χ of H and, by definition of dual action, χˆ ∈ Aˆs . The proof of the converse implication is similar. We now turn to the first statement. To simplify the notations, denote U = (xD) and U 0 = Ind G (x0 D0 ). The representations U and U 0 are physically Ind G G G x
x0
equivalent if and only if there exist a character χ e of G and a unitary or antiunitary operator B such that eB −1 U B . U0 = χ As a first step we define in terms of U and χ e two induced representations U + and − U of G such that eW±−1 U W± , U± = χ where W+ [resp. W− ] is unitary [resp. antiunitary]. In particular, U + and U − are physically equivalent to U . By the previous result χ e = χχ, ˆ where χ ˆ ∈ Aˆs and χ is a character of H. ˆ ˆ ˆ ±1 . The maps ψ± are Define the maps ψ+ and ψ− from A onto A as ψ± (x) := χx measurable isomorphisms that commute with the action of G, so that ψ± maps the orbit G [x] onto the orbit G [ψ± (x)] and one has Gx = Gψ± (x) . If ν is an invariant measure on G [x], the image measure ν ± with respect to ψ± is an invariant measure −1 is a section on G [ψ± (x)] and if c is a section for the orbit G [x], then c± = c ◦ ψ± for the action of G on the orbit G [ψ± (x)]. Fix a unitary operator L+ and an antiunitary operator L− on K. Consider the representations of Gx , g 7→ χ eg L± (xD)g L−1 ± , and observe that their restriction to A are exactly the elements x± := ψ± (x). Since Gx± = Gx we can define the induced representations of G, U ± := Ind G G acting in L2 ( G [x± ], ν ± , K).
x±
(e χL± xDL−1 ± ),
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Moreover, define the operators W± from L2 ( G [x± ], ν ± , K) onto L2 ( G [x], ν, K) −1 e±1 (W± f )(y) = χ c(y) L± f (ψ± (y)) ,
y ∈ G [x] .
It is easy to show that W+ [resp. W− ] is unitary [resp. antiunitary]. We have eW±−1 U W± . U± = χ In fact, let g ∈ G, f ∈ L2 ( G [x± ], ν ± , K), and y ∈ G [x± ] −1 eg χ e−1 χ eg W±−1 Ug W± f (y) = χ c± (y) L± (Ug W± f ) (ψ± (y)) −1 −1 ± e−1 [ψ± (y)]) =χ eg χ c± (y) L± (xD)γ (g,y) (W± f )(g −1 −1 ± e−1 e±1 [y]) =χ eg χ c± (y) L± (xD)γ (g,y) χ c± (g−1 [y]) L± f (g −1 [y]) = (e χL± xDL−1 ± )γ ± (g,y) f (g
= (Ug± f )(y) , −1 −1 (y))−1 gc(g −1 [ψ± (y)]). where γ ± (g, y) = c± (y)−1 gc± (g −1 [y]) = c(ψ± To conclude the proof of the lemma, observe first that there always exist a unitary operator V such that either B = W+ V or B = W− V , according to the fact that B is unitary or antiunitary. Hence U and U 0 are physically equivalent if and only if U 0 is unitarily equivalent either to U + or to U − . Due to a theorem of Mackey (see, for example, Theorem 6.42 of [16]), this is possible if and only if there exist h ∈ G such that Gx0 = hGx h−1 and a unitary or antiunitary operator M (depending on the fact that B is unitary or antiunitary) such that (x0 D0 )hgh−1 = χ eg M (xD)g M −1 for all g ∈ Gx .
We turn to the proof of Theorem 4. Proof of Theorem 4. 1. Since the semidirect product is regular, a theorem of Mackey (see, for example, Theorem 6.42 of [16]) implies that each irreducible unitary representation of G is unitarily (hence physically) equivalent to one of the (xD0 ) for some x ∈ Aˆ and some irreducible representation D0 of Gx ∩ H. form Ind G Gx exi and, by definition of orbit class, there exist There is an index i such that x ∈ O ˆ y ∈ As and h ∈ G such that x = yh[x ] where = ±1. Hence Gx = hGxi h−1 and i
we can define a representation D of Gxi ∩ H either as Dg = Dh0 −1 gh ,
g ∈ Gxi ∩ H ,
if = 1, or as Dg = M Dh0 −1 gh M −1 ,
g ∈ Gxi ∩ H ,
(xD0 ) if = −1, where M is a fixed antiunitary operator. Then, by Lemma 5, Ind G G x
(xi D). is physically equivalent to Ind G G xi
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2. If Ind G (xi D) and Ind G (xj D0 ) are physically equivalent, the condition (2) G G xi
xj
of Lemma 5 with the choice g = a ∈ A implies that xj = yh[xi ] for some y ∈ Aˆs and = ±1, so that, by definition of xi , i = j. 3. Apply Lemma 5 with x = x0 , taking into account the form of the characters of G. We observe that if D0 is unitarily equivalent to D, the conditions (a) of item 3 of Theorem 4 are satisfied with y = 1, χ e = 1, and h = e and this is exactly the case of unitary equivalence of the induced representations. However, in general, there are other possibilities apart from the unitary equivalence. There are even situations in which both conditions (a) and (b) hold. 4. Examples In this section we give a brief review of the classification of the free quantum particles for the Poincar´e group and for the Galilei group using the framework introduced in the previous section. We consider the case of the Galilei group in 2 + 1 dimensions and we confront our results with the ones obtained by Bose [19]. 4.1. The Poincar´ e group Let G be the connected component of the Poincar´e group, which is the semidirect product of A = R4 and the connected component H of the Lorentz group. The covering group G∗ of G is the semidirect product of A = R4 and SL(2, C). It is a standard result (see, for example, [12]) that each multiplier of G∗ is exact. Hence, the universal central extension of G is its universal covering group and our Theorem 3 reduces, in this case, to Theorem 7.40 of [12]. The classification of relativistic free quantum particles is thus traced back to the problem of classifying the irreducible representations of G∗ . This problem was first solved by Wigner [5], in terms of two parameters: the mass, labelling the orbits in the dual group of A, and the spin, labelling the irreducible representations of the stability group at the origin of each orbit. We add some comments. 1. Since every multiplier of G∗ is exact, we have that K = Ker δ = Z2 . Moreover, G∗ has only the trivial character, since the only singleton orbit in Aˆ is the origin and the semisimple group SL(2, C) has no nontrivial characters. b ' Z2 . Explicitly, any Hence, by Theorem 2, H 2 (G, T) is isomorphic to K T-multiplier of G is either exact or equivalent to (g, g 0 ) 7→ (c(g)c(g 0 )c(gg 0 )−1 ) , where c is a section for the covering homomorphism δ and is the nontrivial character of K.
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2. Since G∗ has only the trivial character, two representations of G∗ are physically equivalent if and only if they are either unitarily or antiunitarily equivalent. 3. For reducible representations the admissibility condition (7) gives rise to the superselection rule that does not allow the superposition among fermions and bosons. 4.2. The Galilei group We discuss this case in more details since it presents a nontrivial application of the notion of universal central extension. Let V := (R3 , +) be the group of velocity transformations and let SO(3) be the rotation group in R3 . The group SO(3) acts on V in a natural way and we can consider the corresponding semidirect product, which is the homogeneous Galilei group, G0 := V ×0 SO(3) . The elements of G0 are denoted by (v, R). Let Ts := (R3 , +) be the group of space translations and Tt := (R, +) the group of time translations; we denote the group of space-time translations by T := Ts × Tt and its elements by (a, b). The action of G0 on T is defined by (v, R)[(a, b)] := (Ra + bv, b) ,
(v, R) ∈ G0 , (a, b) ∈ T ,
and the corresponding semidirect product G := T ×0 G0 is the Galilei group. For any g ∈ G we write g = (a, b, v, R). The covering group of G is G∗ = T ×0 (V ×0 SU (2)) , where SU (2) acts on V and V ×0 SU (2) acts on T in a natural way using the covering homomorphism δ from SU (2) onto SO(3). We denote again by δ the covering homomorphism G∗ → G (this is a small abuse of notation that does not cause any confusion) and we notice that Ker δ = {(0, 0, 0, ±I)}. The corresponding Lie algebra is, as a vector space, Lie (G∗ ) = Lie (T ) ⊕ Lie (V) ⊕ Lie (SU (2)) = R4 ⊕ R3 ⊕ su (2) , and we denote its elements by (a, b, v, A), with b ∈ R, a, v ∈ R3 and A ∈ su(2). We apply the results of Sec. 2.2 to compute the multipliers of G∗ . A classical result of Bargmann [6], shows that H 2 (Lie (G∗ ), R) is one dimensional and a nonexact closed R-form is given by F ((a1 , b1 , v1 , A1 ), (a2 , b2 , v2 , A2 )) = v1 · a2 − v2 · a1 ,
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where · denotes the scalar product on R3 . To compute the R-multiplier τF corresponding to F , we have to exhibit the simply connected Lie group G∗F such that Lie (G∗F ) = R ⊕F Lie (G∗ ) . Denote the elements of Lie (G∗F ) by (c, a, b, v, A). By a direct computation one can confirm that {(v, A) ≡ (0, 0, 0, v, A) : (v, A) ∈ Lie (V) ⊕ su (2)} is a subalgebra of Lie (G∗F ) isomorphic to Lie (V ×0 SU (2)), and that {(c, a, b) ≡ (c, a, b, 0, 0) : (c, a, b) ∈ R ⊕ Lie (T )} is an abelian ideal of Lie (G∗F ) isomorphic to Lie (R × T ). Hence, Lie (G∗F ) is isomorphic to the semidirect sum of Lie (R × T ) and Lie (V ×0 SU (2)). Explicitly, if (v, A) ∈ Lie (V ×0 SU (2)) and (c, a, b) ∈ Lie (R × T ), one has ˙ [(v, A), (c, a, b)] = (v · a, δ(A)a + bv, 0, 0, 0) =: ρ(v, ˙ A)(c, a, b) , with ρ(v, ˙ A) denoting the 5 × 5 real matrix 0 v 0 ˙ ρ(v, ˙ A) = 0 δ(A) v, 0 0 0 which acts on the (column) vector (c, a, b) ∈ Lie (R × T ) ' R × T . Let ρ be the representation of V ×0 SU (2) on R × T such that its differential at the identity is ρ. ˙ Then G∗F is the semidirect product of R × T and V ×0 SU (2) with respect to the action defined by ρ. We denote the elements of G∗F by (c, a, b, v, h) where b, c ∈ R, a, v ∈ R3 and h ∈ SU (2). To compute explicitly ρ, if A ∈ su (2), then ˙ ρ(0, eA ) = eρ(0,A)
=
∞ X 1 ρ(0, ˙ A)n n ! n=0
1
0
0
˙ = 0 eδ(A) 0 . 0 0 1 Thus, for all h ∈ SU (2)
1 0 0 ρ(0, h) = 0 δ(h) 0 . 0 0 1
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In a similar way, if v ∈ V, one gets 1 2 v 2 ρ(v, I) = 0 I v . 0 0 1
1 v
Hence, the action of V ×0 SU (2) on R × T is explicitly given by 1 ρ(v, h)[(c, a, b)] = (c + v · δ(h)a + bv2 , δ(h)a + bv, b) , 2 with (v, h) ∈ V ×0 SU (2) and (c, a, b) ∈ R × T , and the multiplication law in G∗F is 1 g1 g2 = (c1 +c2 +v1 ·δ(h1 )a2 + b2 v2 , a1 +δ(h1 )a2 +b2 v1 , b1 +b2 , v1 +δ(h1 )v2 , h1 h2 ) , 2 for all gi = (ci , ai , bi , vi , hi ) ∈ G∗F , i = 1, 2. The corresponding R-multiplier τF for G∗ is 1 τF (g1∗ , g2∗ ) = v1 · δ(h1 )a2 + b2 v12 g1∗ , g2∗ ∈ G∗ . 2 We notice that the usual way to deduce τF from F is not so direct and requires more computations. It is evident that, if k ∈ Ker δ and g ∗ ∈ G∗ , τF (k, g ∗ ) = τF (g ∗ , k) , so that dim H 2 (G∗ , R)δ = 1. Since τF (k1 , k2 ) = 0 k1 , k2 ∈ Ker δ , we can choose τ = τF , and we have G = G∗F and K = R × Z2 . Since G is the semidirect product of A := R × T and H = V ×0 SU (2), we can use the results of Sec. 3.2 to classify the physically inequivalent irreducible representations of G. We identify the dual group Aˆ of A with R × R3 × R by the pairing h(m, p, E), (c, a, b)i = mc − p · a + Eb . With this identification, the dual action of G on Aˆ is 1 2 g[(m, p, E)] = m, δ(h)p + mv, E + mv + v · δ(h)p , 2 with g = (c, a, b, v, h) ∈ G. With respect to this action Aˆ splits into three kinds of orbits: 1. for each E0 ∈ R, m ∈ R, m 6= 0, p2 3 G [(m, 0, E0 )] = (m, p, E) : p ∈ R , E = E0 + ; 2m
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2. for each r ∈ R, r > 0, G [(0, pr , 0)] = {(0, p, E) : p ∈ R3 , p2 = r2 , E ∈ R} , where pr = (0, 0, r); 3. for each E0 ∈ R, G [(0, 0, E0 )] = {(0, 0, E0 )} . ˆ the semidirect product is regular and we can Since these orbits are closed in A, apply Theorem 4. To do this, we observe that the set of singleton orbits is Aˆs = {(0, 0, E0 ) : E0 ∈ R} , and the orbit classes are the following: 1. for any m > 0, e(m,0,0) = O
[
G [(m, 0, E)] ∪ G [(−m, 0, E)] ;
E∈R
2. for any r > 0, e(0,0,0) = 3. O
S E∈R
e(0,(0,0,r),0) = G [(0, (0, 0, r), 0)] ; O G [(0, 0, E)] .
e(m,0,0) , that have a direct physical interWe consider only the set of orbit classes O pretation. Define, for each m > 0, e(m,0,0) , pm := (m, 0, 0) ∈ O then the stability subgroup at pm is A ×0 SU (2) and the irreducible unitary representations of SU (2) are unitarily equivalent to those of the form Dj acting on the Hilbert space C2j+1 , with 2j ∈ N. Moreover, 1. if y ∈ Aˆs , y 6= 0, then yG [pm ] 6= G [pm ]; 2. G [pm ] 6= G [pm ]−1 3. H has only the character 1, 0 4. the representations Dj and Dj , with j 6= j 0 , act on Hilbert spaces with different dimension, so that they are unitarily inequivalent.
Applying Theorem 4, we conclude that every irreducible representation of G living epm is physically equivalent to an induced representation of the on an orbit class O form j U m,j := Ind G A×0 SU(2) (pm D ) , where the inducing representation pm Dj of A ×0 SU (2) is (a, h) 7→ eihpm , ai Dj (h) .
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Moreover, the set {U m,j : m ∈ R, m > 0, 2j ∈ N} is a family of physically inequivalent irreducible representations of G. Hence, by Theorem 3, the inequivalent irreducible symmetry actions of G are classified by two parameters m > 0 and 2j ∈ N. This result was obtained by Bargmann [6]. We end with some comments. 1. The characters of G∗ are of the form G∗ 3 (a, b, v, h) 7→ eiEb ∈ T , where E ∈ R. When restricted to K, any character is trivial. Hence, by b ' R × Z2 . The elements Theorem 2, the group H 2 (G, T) is isomorphic to K b of K are the maps R × Z2 3 (c, ξ) 7→ eimc (ξ) ∈ T where m ∈ R and is a character of Z2 . If s is a section for δ : SU (2) → SO(3), we have that any T-multiplier of G is equivalent to one of the form 0
1 0
((a, b, v, R), (a0 , b0 , v0 , R0 )) 7→ eim(v·Ra + 2 b v ) (s(R)s(R0 )s(RR0 )−1 ) . 2
2. Let U be an admissible representation and m ∈ R, ∈ Z2 be the corresponding algebraic and topological charges. Then the algebraic charge m parametrises the orbits in the dual group (as in the Poincar´e case) and it has the physical meaning of a mass. On the other hand, the topological charge is connected with the spin of the particles: the case = 1 characterises the bosonic representations, while = −1 corresponds to the fermionic ones. 3. If we consider a direct sum of irreducible representations, the admissibility condition (7) gives rise to two superselection rules. Namely, it does not allow superposition among particles with different masses (Bargmann superselection rule) and superposition among bosons and fermions. 4.3. The Galilei group in 2 + 1 dimensions From the physical point of view, the interest in the Galilei group in 2 + 1 dimensions arises in solid state physics where some genuine examples of two dimensional systems can be found. The analysis of the multipliers of this group has been done by Bose [18]. The classification of the representations of the corresponding central extensions has been done in [19]. In the latter paper no discussion of the physical equivalence is given and this leads to misleading conclusions regarding the spin of elementary particles. For these reasons we consider anew this case here as a nontrivial application of our theory. The Galilei group in 2 + 1 dimensions is G = T ×0 (V ×0 SO(2)) ,
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where T = Ts × Tt , Ts = R2 , Tt = R, and V = R2 . The semidirect product structure is the analogous of the 3 + 1 dimensional case. The covering group is G∗ = T ×0 (V ×0 R) and we denote its elements as (a, b, v, r), where a, v ∈ R2 , b ∈ R and r ∈ R. The kernel of the covering homomorphism δ is {(0, 0, 0, 2πk) : k ∈ Z} . The Lie algebra of G∗ is, as a vector space, Lie (G∗ ) = Lie (T ) ⊕ Lie (V) ⊕ Lie (R) = R3 ⊕ R2 ⊕ R , and we denote its elements by (a, b, v, r), with b, r ∈ R, a, v ∈ R2 . A result of Bose [18], shows that H 2 (Lie (G∗ ), R) is a three dimensional vector space and a basis is given by the equivalence classes of the following closed R-forms: F1 ((a1 , b1 , v1 , r1 ), (a2 , b2 , v2 , r2 )) = r1 b2 − r2 b1 , F2 ((a1 , b1 , v1 , r1 ), (a2 , b2 , v2 , r2 )) = v1 · a2 − v2 · a1 , F3 ((a1 , b1 , v1 , r1 ), (a2 , b2 , v2 , r2 )) = v1 ∧ v2 , where v1 ∧ v2 is a shorthand notation for v1x v2y − v2x v1y . Define F as the closed R3 -form F = (F1 , F2 , F3 ). To compute the corresponding R3 -multiplier τF of G∗ , we have to determine the simply connected Lie group G∗F with Lie algebra Lie (G∗F ) = R3 ⊕F Lie (G∗ ) . The algebra Lie (G∗F ) is, in fact, a semidirect sum. This can be shown as follows. Write Lie (G∗F ) = R2 ⊕Lie (G∗ )⊕R and its elements as (c1 , c2 , X, x) with c1 , c2 , x ∈ R and X ∈ Lie (G∗ ) in such a way that [(c1 , c2 , X, x), (c01 , c02 , X 0 , x0 )] = (F1 (X, X 0 ), F2 (X, X 0 ), [X, X 0 ], F3 (X, X 0 )) . By direct computation, the set {(v, r, x) ≡ (0, 0, 0, 0, v, r, x) : (v, r, x) ∈ Lie (V) ⊕ Lie (R) ⊕ R} is a subalgebra of Lie (G∗F ) with Lie brackets 0 ˙ ˙ 0 )v, 0, v ∧ v0 ) − δ(r [(v, r, x), (v0 , r0 , x0 )] = (δ(r)v
where (v, r, x), (v0 , r0 , x0 ) ∈ Lie (V)⊕Lie (R)⊕R. By this equation, Lie (V)⊕Lie (R)⊕ R is isomorphic to the Lie algebra of the covering group H of the diamond group, i.e. the Lie group H = V × R × R with product (v, r, x)(v0 , r0 , x0 ) = (v + δ(r)v0 , r + r0 , x + x0 + v ∧ δ(r)v0 ) with (v, r, x), (v0 , r0 , x0 ) ∈ H.
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Moreover, the set {(c1 , c2 , a, b) ≡ (c1 , c2 , a, b, 0, 0, 0) : (c1 , c2 , a, b) ∈ R2 ⊕ Lie (T )} is an abelian ideal of Lie (G∗F ) isomorphic to Lie (R2 × T ). Taking into account the previous results and the fact that, as a vector space, Lie (G∗F ) = R2 ⊕ Lie (T ) ⊕ (Lie (V) ⊕ Lie (R) ⊕ R) , then Lie (G∗F ) is isomorphic to the semidirect sum of Lie (R2 × T ) and Lie (H). Explicitly, if (v, r, x) ∈ Lie (H) and (c1 , c2 , a, b) ∈ Lie (R2 × T ) one has ˙ + bv, 0) [(v, r, x), (c1 , c2 , a, b)] = (rb, v · a, δ(r)a =: ρ(v, ˙ r, x)(c1 , c2 , a, b) , where ρ(v, ˙ r, x) is the 5 × 5 matrix
0 0
0 0 ρ(v, ˙ r, x) = 0 0 0 0 0 0
0
r
0 , v
v 0 −r r
0 0
0
which acts on the column vector (c1 , c2 , a, b) ∈ Lie (R2 × T ) ' R2 × T . If ρ is the representation of H such that its differential at the identity is ρ, ˙ ∗ 2 then GF is the semidirect product of R × T and H with respect to ρ. A simple calculation shows that 1 0 0 0 1 2 0 1 v v 2 ρ(v, 0, 0) = 0 0 1 0 v 0 0 0 1 0 0
1 0 ρ(0, r, 0) = 0 0
0
0
0
1
0
0 0
δ(r)
r
1
0 0
0 0
0
1
1 0
0
0
0 1 ρ(0, 0, a) = 0 0 0 0 0 0
0 1 0 0 1 0
0 . 0 1
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Hence the action of H on R2 × T is given by bv2 , δ(r)a + bv, b . (v, r, x)[(c1 , c2 , a, b)] = c1 + br, c2 + v · δ(r)a + 2 If g = (c1 , c2 , a, b, v, r, x) and g 0 = (c01 , c02 , a0 , b0 , v0 , r0 , x0 ) are in G∗F , then b0 v 2 , a + δ(r)a0 + b0 v, b + b0 , gg 0 = c1 + c01 + b0 r, c2 + c02 + v · δ(r)a0 + 2 v + δ(r)v0 , r + r0 , x + x0 + v ∧ δ(r)v0 , so that the explicit form of τF = (τ1 , τ2 , τ3 ) is τ1 (g, g 0 ) = b0 r τ2 (g, g 0 ) = v · δ(r)a0 + b0 v2 /2 τ3 (g, g 0 ) = v ∧ δ(r)v0 . By Theorem 1, the equivalence classes [τ1 ], [τ2 ], [τ3 ] form a basis of H 2 (G∗ , R). Moreover τ2 and τ3 satisfy the condition τi (k, g ∗ ) = τi (g ∗ , k) ,
k ∈ Ker δ, g ∗ ∈ G∗ ,
while τ1 does not. It follows that dim H 2 (G∗ , R)δ = 2 and we can choose τ = (τ2 , τ3 ) (notice that τ (k1 , k2 ) = 0 if k1 , k2 ∈ Ker δ) and the universal central extension G of G can be identified with the semidirect product of the vector group A = R × T and the Lie group H with respect to the action of H on A given by bv2 , δ(r)a + bv, b , (v, r, x)[(c, a, b)] = c + v · δ(r)a + 2 where the elements of A are denoted by (c, a, b), with c ∈ R, a ∈ Ts and b ∈ Tt , and the ones of H by (v, r, x), with x, r ∈ R and v ∈ V. As usual, we denote the elements of G as (c, a, b, v, r, x). Finally, one has that K = {(c, 0, 0, 0, 2πn, x) : c, x ∈ R, n ∈ Z} ' R2 × Z . Since G is a semidirect product we apply the results of Sec. 3.2 to classify the irreducible physically inequivalent representations of G. Let Aˆ be the dual group of A. We identify Aˆ with R4 using the pairing h(m, p, p0 ), (c, a, b)i = mc − p · a + p0 b . The dual action of G on Aˆ is 1 2 g[(m, p, p0 )] = m, δ(r)p + mv, p0 + δ(r)p · v + mv , 2
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where g = (c, a, b, v, r, x) ∈ G. We have the following orbits for the dual action. 1. For each l ∈ R, l > 0, G [(0, pl , 0)] = {(0, p, p0 ) : p2 = l2 } , where pl = (0, l). 2. For each E ∈ R, G [(0, 0, E)] = {(0, 0, E)} . 3. For each m ∈ R, E ∈ R, m 6= 0, p2 =E . G [(m, 0, E)] = (m, p, p0 ) : p0 − 2m ˆ hence the semidirect product is regular and Theorem 4 All the orbits are closed in A, holds. The set of singleton orbits is Aˆs = {(0, 0, E) : E ∈ R} , and the orbit classes of G are the following: 1. for each l ∈ R, l > 0,
e(0,p ,0) = G [(0, pl , 0)] ; O l e(0,0,0) = O
2.
[
G [(0, 0, E)] ;
E∈R
3. for any m > 0, e(m,0,0) = O
[
G [(m, 0, E)] ∪ G[(−m, 0, E)] .
E∈R
In the sequel we will exploit in detail only the third case, which presents some interesting physical features. e(m,0,0) . We have that Let m > 0 and pm = (m, 0, 0) ∈ O Gpm ∩ H = {(v, r, x) ∈ H : v = 0} is isomorphic to R2 and its irreducible representations are its characters. Explicitly, λ, µ ∈ R define the character of Gpm ∩ H (0, r, x) 7→ eiλx eiµr . Now we observe that 1. if y ∈ Aˆs , y 6= 0, then yG [pm ] 6= G [pm ]; 2. G [pm ] 6= G [pm ]−1 ; 3. the characters of H are of the form (v, r, x) 7→ eiµr .
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According to Theorem 4, every irreducible representation of G living on an orbit epm is equivalent to one of the form U m,λ = IndG (Dm,λ ) where Dm,λ is class O G pm
the representation of Gpm (c, a, b, 0, r, x) 7→ ei(mc+λx) . Moreover, the set {U m,λ : m, λ ∈ R, m > 0} is a family of physically inequivalent representations of G. To compute explicitly U m,λ , we observe that the orbit p2 =0 G[pm ] = (m, p, p0 ) : p0 − 2m can be identified with R2 using the map p2 2 ∈ G[pm ] . R 3 p ←→ m, p, 2m With respect to this identification the action of G on the orbit becomes (c, a, b, v, r, x)[p] = δ(r)p + mv so that the Lebesgue measure dp on R2 is G-invariant. We consider the section β : R2 → G p p 7→ 0, 0, 0, , 0, 0 m for the action of G on R2 . The representation U m,λ of G acts in L2 (R2 , dp) as 2 b 1 m,λ f (p) = ei( 2m p −p·a+mc) eiλ(x+ m v∧p) f (δ(−r)(p − mv)) . U(c,a,b,v,r,x) From the explicit form of U m,λ one readily gets that the angular momentum, i.e. the selfadjoint operator that generates the 1-parameter subgroup of rotations, has only the orbital part, so that the elementary particles in 2 + 1 dimensions have no spin. However, they acquire a new charge λ which is not of a space-time origin, but arises from the structure of the multipliers. If λ 6= 0, the two generators of velocity transformations do not commute. We add some final comments. 1. The characters of G∗ are G∗ 3 (a, b, v, r) 7→ eiEb eiµr ∈ T , where E, µ ∈ R. The set V of characters of K that extend to G∗ is V = {(c, 0, 0, 0, 2πn, x) 7→ z n : z ∈ T} ' T .
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b = R2 × T and K0 = R2 . Applying The group V is a closed subgroup of K 2 Corollary 1, H (G, T) is isomorphic to R2 . In particular, any T-multiplier of G is equivalent to one of the form 0
1 0
0
((a, b, v, R), (a0 , b0 , v0 , R0 )) 7→ eim(v·Ra + 2 b v ) eiλ(v∧Rv ) 2
where (m, λ) ∈ R2 . 2. From the explicit form of the characters of G∗ one has that, for all E, µ ∈ R, the representation m,λ (c, a, b, v, r, x) 7→ ei(Eb+µr) U(c,a,b,v,r,x)
is physically equivalent to U m,λ . Hence the angular momentum and the energy are both defined up to an additive constant. For the energy this phenomenon is well known in 3 + 1 dimensions, while it does not occur for the angular momentum. 3. The admissibility condition (7) gives rise to two superselection rules that do not allow superposition among states with different mass m and among states with different charge λ. However, there is no superselection rule connected with the spin. References [1] E. P. Wigner, Group Theory and Its Application to the Quantum Theory of Atomic Spectra, Academic Press Inc., New York, 1959, pp. 233-236. [2] I. Schur, J. Reine Angew. Math. 127 (1904) 20–50. [3] I. Schur, J. Reine Angew. Math. 132 (1906) 85–137. ´ ements de la th´eorie des repr´esentations, Editions ´ [4] A. Kirillov, El´ MIR, Moscou, 1974. [5] E. P. Wigner, Ann. Math. 40 (1939) 149–204. [6] V. Bargmann, Ann. Math. 59 (1954) 1–46. [7] G. W. Mackey, Acta Math. 99 (1958) 265–311. [8] C. C. Moore, Trans. Amer. Math. Soc. 113 (1964) 40–63. [9] C. C. Moore, Trans. Amer. Math. Soc. 113 (1964) 64–86. [10] C. C. Moore, Trans. Amer. Math. Soc. 221 (1976) 1–33. [11] C. C. Moore, Trans. Amer. Math. Soc. 221 (1976) 35–58. [12] V. S. Varadarajan, Geometry of Quantum Theory, Second ed., Springer-Verlag, Berlin, 1985. [13] P. T. Divakaran, Rev. Math. Phys. 6 (1994) 167–205. [14] G. Cassinelli, E. De Vito, P. Lahti and A. Levrero, Rev. Math. Phys. 9 (1997) 921. [15] J. Braconnier, J. Math. Pures Appl. 27 (1948) 1–85. [16] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, 1995. [17] G. W. Mackey, Ann. Math. 55 (1952) 101–139. [18] S. K. Bose, Commun. Math. Phys. 169 (1995) 385–395. [19] S. K. Bose, J. Math. Phys. 36 (1995) 875–890.
STABILITY AND INSTABILITY OF THE WAVE EQUATION SOLUTIONS IN A PULSATING DOMAIN J. DITTRICH∗ Nuclear Physics Institute, Academy of Sciences of the Czech Republic, CZ-250 68 Rez, Czech Republic E-mail: [email protected]
P. DUCLOS and N. GONZALEZ† Centre de Physique Th´ eorique‡ CNRS Luminy, Case 907, F-13288 Marseille - Cedex 9, France and PhyMat, Universit´ e de Toulon et du Var, La Garde, France E-mail: [email protected] Received 17 June 1997 Revised 8 December 1997 The behavior of energy is studied for the real scalar field satisfying d’Alembert equation in a finite space interval 0 < x < a(t); the endpoint a(t) is assumed to move slower than the light and periodically in most parts of the paper. The boundary conditions are of Dirichlet and Neumann type. We give sufficient conditions for the unlimited growth, the boundedness and the periodicity of the energy E. The case of unbounded energy without infinite limit (0 < lim inf t→+∞ E(t) < lim supt→+∞ E(t) = +∞) is also possible. For the Neumann boundary condition, E may decay to zero as the time tends to infinity. If a is periodic, the solution is determined by a homeomorphism F¯ of the circle related to a. The behavior of E depends essentially on the number theoretical characteristics of the rotation number of F¯ .
Contents 1. Introduction 2. Preliminaries 2.1. Notations, definitions and preliminary results 2.2. Existence and unicity of the solution 2.3. Some useful lemmas 3. Dirichlet Problem 3.1. Stability 3.1.1. A universal lower bound 3.1.2. Periodicity 3.1.3. Absence of strong instability 3.1.4. A sufficient condition of stability ∗ Also
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member of the Doppler Institute of Mathematical Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Prague, Czech Republic. † Also Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Prague, Czech Republic. ‡ Unite Propre de Recherche 7061.
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3.2. Instability 3.2.1. A universal upper bound 3.2.2. A sufficient condition of instability 3.2.3. A sufficient condition of strong instability 3.2.4. Asymptotics 3.2.5. Instability: strong instability is not the rule 3.2.6. Perturbation of the boundary 4. Neumann Problem 4.1. Stability 4.1.1. An upper bound 4.1.2. Periodicity 4.1.3. Asymptotics 4.1.4. Absence of strong instability 4.1.5. A sufficient condition of stability 4.2. Instability 4.2.1. Universal lower and upper bounds 4.2.2. Singular initial data 4.2.3. Sufficient conditions for the decay of the energy A. Appendix B. Glossary
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1. Introduction The study of the so-called Fermi accelerators becomes more and more extensive. The name comes from Fermi’s considerations on the possible mechanism of cosmic rays acceleration [1]. In the later studies up to contemporary ones, they serve as simple prototypes of the externally driven dynamical systems, mainly in the connection with the deterministic and chaotic behavior of the classical and quantum systems. The first mechanical models were proposed by Ulam [2], the rigorous results in Newtonian mechanics (Pustyl’nikov [3, 4]) and in special-relativistic classical mechanics (Pustyl’nikov [5, 6, 4]) were obtained much later. Only as a sample ˇ of papers in nonrelativistic quantum mechanics let us mention Karner [7], Seba [8], Dembi´ nski, Makowski and Peplowski [9], Dodonov, Klimov and Nikonov [10], ˇˇtov´ıˇcek [11]. Similar problems for classical wave equation (Balazs [12], Duclos and S ˇ Cooper [13], Dittrich, Duclos and Seba [14], Cooper and Koch [15], M´eplan and Gignoux [16]) and Maxwell equations (Cooper [17]) were also considered. An analogous model in quantum field theory was treated, for example, by Moore [18], Calucci [19], Dodonov, Klimov and Nikonov [20], Johnston and Sarkar [21]. In the present paper, we continue and extend the study for the classical d’Alembert equation. Let us consider the one-dimensional wave equation in a domain with one spatial boundary fixed and the second one moving slower than the wave velocity. Let us assume that the boundary motion is described by a Lipschitz continuous function a and assume that the field satisfies either Dirichlet or Neumann boundary conditions. We describe the behavior of the energy E of the field in more details and for a wider class of functions a than the papers [13, 14, 15, 16] which treat only the case a ∈ C k (R) (k ≥ 2) and a periodic. The key to the results is that the orbits of the characteristics of the wave equation are given by a Lipschitz homeomorphism F of R which depends only on a and becomes the lift of a homeomorphism of the circle when a is periodic. According to the arithmetic properties of the rotation number of F and to the regularity of a,
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E will behave differently. It is worth remarking that our results are nonperturbative (in other words there are no assumptions on the smallness of time variations of a). Section 2 is devoted to the formulation of the problems (Dirichlet and Neumann), the link between a and F , the existence and unicity of a weak solution in the space of finite energy fields. Section 3 is divided into two subsections: Stability (E is bounded from below and above) and Instability (E is unbounded), both for the Dirichlet problem. In Subsec. 3.1, we give a universal lower bound (a needs just be bounded), a necessary and sufficient condition of periodicity and sufficient conditions for the existence of an upper bound. In Subsec. 3.2, we distinguish two types of instability: a weaker one (the limit superior is infinite) and a stronger one (the limit inferior is infinite). We first give a sufficient condition of instability and we state a sufficient condition of strong instability. The proof introduces the condition for occurence of “wandering characteristics” and therefore generalizes the condition of “periodic characteristics” given by Cooper [13]. We compute an asymptotics which shows the exponential increase of E due to the presence of periodic characteristics; in this case, under any small periodic perturbation of the boundary motion a, E keeps increasing exponentially. We also show on examples that E may have an infinite limit superior but a finite limit inferior in the case where there are no periodic characteristics and the boundary a is smooth; this answers negatively to a conjecture of Cooper [17]. Cooper and Koch [15] already used diffeomorphisms of the circle in the Dirichlet problem and they show that the spectrum of the evolution operator on one period depends on the rotation number. Section 4 is devoted to the Neumann problem. To our knowledge, this is the first treatment in the literature. As we shall see, E behaves in a completely different manner. In Subsec. 4.1, we prove that E is universally bounded from above except if the initial conditions are singular (in which case we show that E may diverge exponentially). The periodicity and lower bounds are proven like for the Dirichlet problem. We also propose conditions for an asymptotically periodic energy and give explicit asymptotics for this case. In Subsec. 4.2, we show that for appropriate initial conditions E decays (and even exponentially fast) to 0. Let a be a strictly positive real function to be precised later. The problems we consider are the Dirichlet problem: ϕtt − ϕxx = 0 ,
t ∈ R,
0 < x < a(t) ,
(1)
ϕ(x, 0) = ϕ0 (x) ,
0 < x < a(0) ,
(2)
ϕt (x, 0) = ϕ1 (x) ,
0 < x < a(0) ,
(3)
ϕ(0, t) = 0 ,
t ∈ R,
(4)
ϕ(a(t), t) = 0 ,
t ∈ R,
(5)
and the Neumann problem for which ϕx (0, t) = 0 , ϕx (a(t), t) = 0 ,
t ∈ R,
(6)
t∈R
(7)
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are required instead of (4)–(5). Since it will not play a role in the mathematical analysis, the wave velocity of the field ϕ is normalized to 1. In addition, if a is periodic, then by a rescaling in the parameters, one can also take the period equal to 1; this will simplify our notations. The energy of the field ϕ is given by the standard expression E(t) :=
1 2
Z
a(t)
(|ϕt (x, t)|2 + |ϕx (x, t)|2 ) dx .
(8)
0
2. Preliminaries 2.1. Notations, definitions and preliminary results We start with some notations and known results. Not to be very formal in trivialities, some equivalent spaces of functions (like functions on a circle and their liftings on a line) are indentified in our notation explained below. Let X be either the set Z (the integers) or N (the nonnegative integers) or Q (the rational numbers) or R (the real numbers). Then X ∗ := X \ {0}, X+ := {x ∈ ∗ := X ∗ ∩ X+ . For any set X, we denote by X˙ or Int X its X; x ≥ 0} and X+ interior. Denote by T the 1-dimensional torus (the circle of unit length) and by X either T or R. Let C 0 (X) be the space of the continuous functions and let C k (X), k ∈ N∗ , be the space of the k-times continuously differentiable functions; the kth derivative of a function F is denoted by F (k) or Dk F . Denote by C k (T) the set of 1-periodic and k-times continuously differentiable functions on R. One defines the norms kF k0 := supx∈T |F (x)| on C 0 (T) and kF kk := max0≤i≤k kDi F k0 on C k (T) for finite k ∈ N. By C ∞ (X) (resp. C ω (X)) one denotes the space of infinitely differentiable (resp. R-analytic) functions on X. For a measurable function F : X → R, we shall denote by Fmin and Fmax its essential infimum and its essential older continuous functions supremum respectively. Let Lipβ (X) be the space of H¨ with exponent β ∈ (0, 1]. By definition, if β ∈ (0, 1), C β (X) := Lipβ (X). If β = 1, Lip(X) := Lip1 (X) is the set of Lipschitz functions. We shall denote (y) the Lipschitz constant of a function F by L(F ) := supx,y∈X,x6=y F (x)−F . Let x−y π : R → T, x 7→ x + Z be the canonical projection. For any continuous map F¯ : T → T, the function F satisfying F¯ ◦ π = π ◦ F is called a lift of F¯ to R. Denote by Diffk (R) the C k -diffeomorphisms on R. Let [x] be the integer part of a real number x. One calls Dk (T), k ∈ R+ ∪ {+∞, ω}, the set of lifts of the orientationpreserving C k -diffeomorphisms of T, i.e. Dk (T) := {F ∈ Diff[k] (R); F − Id ∈ C ◦ (T), D[k] (F −Id) ∈ C k−[k] (T), D[k] (F −1 −Id) ∈ C k−[k] (T)}. One calls F ∈ D0 (T) a Lipschitz homeomorphism if F and F −1 are Lipschitz continuous. Recall that a 1periodic function F : R → R is of bounded variation on T if F is of bounded variation on [0, 1] and one denotes by BV(T) the set of functions of bounded variation on T. For any F ∈ D0 (T), the rotation number ρ(F ) is defined by F n (x) − x , n→+∞ n
ρ(F ) := lim
x ∈ R,
(9)
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where F n := F ◦ F ◦ . . . ◦ F is the nth iterate of F . In Herman [22, Prop. II.2.3, p. 20], the limit (9) is proven to exist (it is a real number independent of x) and to be uniform w.r.t. x. If in the sequel ρ(F ) = pq for p ∈ Z, q ∈ N∗ , it is always assumed that p and q are relatively primes. Definition 2.1. A point x0 ∈ R is said to be a periodic point of period q ∈ N∗ of F ∈ D0 (T) if there exists p ∈ N such that F q (x0 ) = x0 + p. The point x0 is said to be attracting if there exists a neighborhood U of x0 such that for all x ∈ U , F nq (x) − np tends to x0 as n tends to +∞. If x0 is an attracting periodic point of F −1 , then x0 is called a repelling periodic point of F . F ∈ D1 (T) is said to be a Morse–Smale diffeomorphism if F has a finite nonzero number of periodic points ak (with period q ∈ N∗ ), all of them hyperbolic (i.e. either DF q (ak ) < 1 and in this case the point is attracting or DF q (ak ) > 1 and in this case the point is repelling). One can show (e.g. Herman [22, Prop. II.5.3, p. 24]) that the existence of a periodic point x0 for F ∈ D0 (T), F q (x0 ) = x0 + p, is equivalent to ρ(F ) = pq ∈ Q. It means that if the rotation number is irrational then there are no periodic points. The minimal assumptions that we shall make on the moving boundary described by a are the following: Assumption 2.2. The function a is strictly positive, Lipschitz continuous on R with L(a) ∈ [0, 1). Other assumptions will be always given explicitly. In particular if a is 1-periodic, then a ∈ Lip(T). Let Id be the identity on R. Define h := Id − a, k := Id + a on R. Under Assumption 2.2, it is easy to see that h, k, h−1 , k −1 are Lipschitz homeomorphisms 1 . Hence k ◦ h−1 , h ◦ k −1 on R and L(k), L(h) ≤ 1 + L(a), L(h−1 ), L(k −1 ) ≤ 1−L(a) are also Lipschitz homeomorphisms on R and L(k ◦ h−1 ), L(h ◦ k −1 ) ≤ 1+L(a) 1−L(a) . If moreover a is 1-periodic, then h, k, k ◦ h−1 , h ◦ k −1 ∈ D0 (T). The proof of the following lemma is left to the reader. L(a) ∈ [0, 1)} Lemma 2.3. The sets An := {a ∈ C n (T); a > 0, a ∈ Lip(T), and Fn := F ∈ Dn (T); F > Id, F ∈ Lip(R), F −1 ∈ Lip(R) equipped with C n topologies are homeomorphic for n ∈ N ∪ {+∞, ω}. If n ≥ 1, An and Fn are open subsets of C n (T) and Dn (T) respectively. More generally there exists a bijection between the set of functions a satisfying Assumption 2.2 and the set of Lipschitz homeomorphisms F on R with F > Id. We will need the relations: F := (Id + a) ◦ (Id − a)−1 = Id + 2a ◦ (Id − a)−1 , −1 F + Id F − Id . ◦ a = 2 2
(10) (11)
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In the sequel there is a deep interplay between the invariant measures of diffeomorphisms of the circle and the physics of the problems. Let us remind some definitions. Let X be a compact metric space (for instance, X = T) and F : X → X be a continuous map. The measure µ is said to be an invariantmeasure of F iff µ 0 belongs to the set of probability measures on X (i.e. µ ∈ C 0 (X) the dual space of C 0 (X), µ ≥ 0 and µ(X) = 1) and for every µ-measurable set A, µ(F −1 (A)) = µ(A). According to Krylov and Bogolyubov’s theorem (e.g. Katok and Hasselblatt [23, Theorem. 4.1.1, p. 135]), for any continuous map of X, there exists at least one invariant measure. For the particular case of F ∈ D0 (T): if ρ(F ) ∈ R \ Q, the invariant measure µ is unique (F is said uniquely ergodic, see Herman [22, Prop. II.8.5, p. 28]). If ρ(F ) ∈ Q, the invariant measure is in general not unique and it may be atomic. As an illustration we prove the following lemma: Lemma 2.4. Let a ∈ A0 and F defined by (10). Let p, q ∈ N∗ , x0 ∈ R. Then F q (x0 ) = x0 + p ⇔
q−1 X
a ◦ h−1 ◦ F k (x0 ) =
k=0
p . 2
(12)
Proof. Assume that F q (x0 ) = x0 + p. Thus ρ(F ) = pq . Clearly 1X δF¯ k (x0 ) q q−1
µ :=
k=0
is an invariant probability measure for F¯ (δF¯ k (x0 ) is the Dirac measure at F¯ k (x0 )). Let ψ := F − Id ∈ C 0 (T). By formula (10), ψ = 2a ◦ h−1 . According to Herman [22, Prop. II.2.3, p. 20], Z ψ dµ ,
ρ(F ) = T
Z p = 2 a ◦ h−1 dµ . (13) q T Pq−1 Inserting µ into (13), the equality k=0 a ◦ h−1 ◦ F k (x0 ) = p2 is proven. Conversely, assume the formula on the right-hand side of the equivalence holds Pn−1 and let ψ := F − Id ∈ C 0 (T). Then F n = Id + k=0 ψ ◦ F k , so so that
F q (x0 ) = x0 + 2
q−1 X
a ◦ h−1 ◦ F k (x0 ) = x0 + p .
k=0
Remark 2.5. Let x0 ∈ I0 := [−a(0), a(0)). Then {(a ◦ h−1 ◦ F n (x0 ), h−1 ◦ F n (x0 )); n ∈ N} is the set of intersections between the characteristic t + x = x0 , reflecting against the boundaries, and the moving boundary in the (x, t) plane. The physical explanation of Lemma 2.4 is the following: the relation (12) expresses the existence of a periodic characteristic of period p after q bounces.
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In order to apply Denjoy’s theorem we will use the functions of class P (Herman [22, Def. VI.4.1, p. 74]). Definition 2.6. Let F ∈ D0 (T). Then F is of class P if F is differentiable up to a countable set (u.c.s.) and if its derivative is equal u.c.s. to a (1-periodic) function which is bounded from below by a positive number and is of bounded variation on T. Let us give a slight generalisation of the variation of a function and define a suitable class of functions a. Definition 2.7. Let f be a function defined u.c.s. in T. One denotes d ) := inf {Var(b); b : T → R, b(x) = f (x) u.c.s. in T} . Var(f A function a is said to be of class Q if a ∈ A0 , a is differentiable u.c.s. in T and d 0 ) < +∞. Var(a A link between the classes P and Q is established in the following lemma. Lemma 2.8. If a is of class Q, then F := (Id + a) ◦ (Id − a)−1 is of class P and 2 d 0) . d 0) ≤ Var(a Var(F (1 − L(a))2 Proof. For any > 0 there exists b1 : T → R such that b1 = a0 u.c.s. in T and d 0 ) + . Defining Var(b1 ) < Var(a L(a) if b1 (x) > L(a) ∀x ∈ T, b(x) := b1 (x) if −L(a) ≤ b1 (x) ≤ L(a) −L(a) if b1 (x) < −L(a) , d 0 ) + . We have F 0 = 1+a00 ◦h−1 then b = a0 u.c.s. in T and Var(b) ≤ Var(b1 ) < Var(a 1−a ◦h−1 exists u.c.s. and is bounded in the L∞ -norm. Since h−1 ∈ D0 (T), b ◦ h−1 ∈ BV(T) 1+b◦h−1 0 u.c.s. and and Var(b ◦ h−1 ) = Var(b). Let G := 1−b◦h −1 . Then G = F n X 1 + b ◦ h−1 (tk ) 1 + b ◦ h−1 (tk−1 ) Var(G) = sup 1 − b ◦ h−1 (tk ) − 1 − b ◦ h−1 (tk−1 ) k=1
= sup
n X k=1
2|b ◦ h−1 (tk ) − b ◦ h−1 (tk−1 )| |1 − b ◦ h−1 (tk )| · |1 − b ◦ h−1 (tk−1 )|
≤
2 Var(b ◦ h−1 ) (1 − L(a))2
=
2 2 d 0) + Var(a Var(b) < (1 − L(a))2 (1 − L(a))2
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where the supremum is taken over all 0 ≤ t0 ≤ t1 ≤ . . . ≤ tn ≤ 1. Since this inequality holds for any > 0, d 0 ) ≤ Var(G) ≤ Var(F
2 d 0) . Var(a (1 − L(a))2
We recall that F := k ◦ h−1 and we define ∀n ∈ Z,
In := [F n (−a(0)), F n (a(0))) .
(14)
For convenience, denote xn := F n (a(0)), for all n ∈ Z. Since F (−a(0)) = a(0), In = [xn−1 , xn ). If a is bounded from below and from above, then ∀n ∈ Z,
2amin ≤ xn+1 − xn ≤ 2amax .
It can be easily seen that [ In = R
and ∀n 6= m, Im ∩ In = ∅ .
(15)
(16)
n∈Z
Clearly {In }n∈Z is a partition of R. The aim of the present paper is to study the energy E of the field: 1 2 2 kϕx (·, t)k(0,a(t)) + kϕt (·, t)k(0,a(t)) , E(t) = 2 where we denote by k · kX the L2 -norm on the measurable set X with respect to the Lebesgue measure m. It will appear useful to give the following definition. Definition 2.9. We shall say that the model is stable if E is bounded from below and from above by strictly positive constants. Equivalently the model is unstable if the limit superior (resp. limit inferior) of E is infinite (resp. zero). Moreover if the limit of E exists and is infinite or zero, then the model is said strongly unstable. Let Σ ⊂ Rn , n ≥ 1, be an open set. Let D(Σ) := C0∞ (Σ) and D0 (Σ) be its dual. 1 1 (Σ) and H0,loc (Σ) We will use also the standard Sobolev spaces H 1 (Σ), H01 (Σ), Hloc n 1 defined as the space of distributions φ such that for all ψ ∈ D(R ), ψφ ∈ H0 (Σ). 1 (Ω) is Definition 2.10. Let Ω := {(x, t) ∈ R2 ; 0 < x < a(t)}. Then ϕ ∈ Hloc called a weak solution of (1), (4)–(5) or of (1), (6)–(7) if
∀ψ ∈ D(Ω),
(ϕt , ψt )Ω − (ϕx , ψx )Ω = 0
(17)
(or simply ϕtt − ϕxx = 0 in D0 (Ω)) and the boundary conditions (4)–(5) or (6)–(7) are satisfied. Remark 2.11. The traces needed for the initial and boundary conditions do 1 (Ω). However they exist if ϕ satisfies (17); see not exist for general ϕ in Hloc Theorems 2.12 and 2.13 below.
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2.2. Existence and unicity of the solution We were not able to find in the literature a proof of the existence and unicity of a solution with finite energy to our problems when the boundary a is not smooth. Indeed the standard technique consists in transforming the problems into new ones with fixed boundaries and then to apply the general theory of hyperbolic partial differential equations with variable coefficients (cf. Ladyzhenskaya [25], Lions and Magenes [26], etc.). But this requires roughly that a is C 2 . That is why we give in details the proof for more general a. Theorem 2.12 (Dirichlet problem). If a ∈ Lip(R), L(a) ∈ [0, 1), a > 0 and (ϕ0 , ϕ1 ) ∈ H01 ((0, a(0))) × L2 ((0, a(0))) , then there exists a unique weak solution ϕ of the Dirichlet problem satisfying the 1 (R) ∩ L∞ (R) such that initial conditions (2)–(3). Moreover there exists f ∈ Hloc ϕ(x, t) = f (t + x) − f (t − x)
a.e. in Ω
(18)
1 (Ω). and ϕ ∈ L∞ (Ω) ∩ H0,loc
Proof. First step: the form of general solution and the regularity of traces. By 1 (R) the assumptions on Ω, if a weak solution ϕ exists, then there exist f , g ∈ Hloc such that ϕ(x, t) = f (t + x) + g(t − x) a.e. in Ω. Moreover ϕ can be extended ¯ (so that the last relation holds in Ω). ¯ continuously to Ω The proof is given in appendix (Lemma B). Functions of such a form have traces on the boundaries of Ω and on the intersections of Ω with lines t = constant which are given by continuous 1 (∂Ω) and in H 1 ((0, a(t))) respectively. extension; moreover these traces are in Hloc Second step: the fixed boundary. Since ϕ has a trace ϕ(0, t) = f (t) + g(−t) on x = 0, ϕ(0, t) = 0 implies that g = −f . Then ϕ(x, t) = f (t + x) − f (t − x) a.e. in Ω. Third step: the initial conditions. Since ϕ has a trace on t = 0 given by ϕ(x, 0) = f (x) − f (−x) and ϕt (x, 0) = f 0 (x) − f 0 (−x) a.e. in (0, a(0)), the initial conditions (2)–(3) give Z x 1 ϕ1 (y) dy , (19) ϕ0 (x) + ∀x ∈ [0, a(0)], f (x) := f (0) + 2 0 Z x 1 ϕ1 (y) dy . (20) −ϕ0 (x) + f (−x) := f (0) + 2 0 The constant f (0) is arbitrary. Clearly f ∈ H 1 (I0 ), since ϕ0 (0) = 0 by assumption. Fourth step: the moving boundary. The function ϕ has a trace on x = a(t) and ϕ(a(t), t) = 0; this implies that f ◦F = f
on R
with F := (Id + a) ◦ (Id − a)−1 .
(21)
Fifth step: construction of a solution. By iteration of the formula (21): ∀n ∈ Z, f ◦ F n = f . Thus the function f is known on In for all n ∈ Z, and f ∈ C 0 (I˙n ) for
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all n ∈ Z. Since ϕ0 (a(0)) = 0 and F (−a(0)) = a(0), f is continuous at ±a(0), hence f ∈ C 0 (R) by iterations. Since F is Lipschitz continuous, f ∈ H 1 (In ) for all n ∈ Z, 1 1 (R) and ϕ ∈ H0,loc (Ω). Note that kf kL∞(R) = kf kL∞ (I0 ) < +∞. thus f ∈ Hloc Obviously ϕ ∈ L∞ (Ω), since kϕkL∞ (Ω) ≤ 2kf kL∞(I0 ) < +∞. Sixth step: unicity. Up to a constant, f is uniquely determined on I0 and by (21), f is then unique on R. Thus ϕ is unique, since the difference f (t+ x)− f (t− x) is independent of the constant f (0). Theorem 2.13 (Neumann problem). If a ∈ Lip(R), L(a) ∈ [0, 1), a > 0 and (ϕ0 , ϕ1 ) ∈ H 1 ((0, a(0))) × L2 ((0, a(0))) , then there exists a unique weak solution ϕ of the Neumann problem satisfying the 1 (R) such that initial conditions (2)–(3). Moreover there exists f ∈ Hloc ϕ(x, t) = f (t + x) + f (t − x) a.e. in Ω .
(22)
The proof, very similar to that for the Dirichlet problem, is given in appendix. One just needs to notice that the moving boundary condition yields the essential functional equation (23) f 0 ◦ F = f 0 a.e. in R . In the sequel the initial conditions are assumed to satisfy the requirements for the existence and unicity of the weak solution if other assumptions are not mentioned. Let us give also conditions for the existence of classical C 2 -solutions. The proof is left to the reader. Theorem 2.14. Let a ∈ C 2 (R), |a0 | < 1, a > 0 and (ϕ0 , ϕ1 ) ∈ C 2 ([0, a(0)]) × C ([0, a(0)]). If 1
ϕ0 (0) = 0, ϕ1 (0) = 0,
ϕ0 (a(0)) = 0,
ϕ000 (0) = 0 ,
ϕ1 (a(0)) + a0 (0)ϕ00 (a(0)) = 0 ,
(1 + a02 (0))ϕ000 (a(0)) + 2a0 (0)ϕ01 (a(0)) + a00 (0)ϕ00 (a(0)) = 0 , then there exists a unique classical C 2 -solution of the Eqs. (1)–(5). If ϕ00 (0) = 0 ,
ϕ00 (a(0)) = 0 ,
ϕ01 (0) = 0 ,
ϕ01 (a(0)) + a0 (0)ϕ000 (a(0)) = 0 ,
then there exists a unique classical C 2 -solution of the Eqs. (1)–(3), (6)–(7). 2.3. Some useful lemmas We start by some considerations on the energy E. Lemma 2.15. Under the assumptions of Theorem 2.12 (Dirichlet problem) or Theorem 2.13 (Neumann problem) the following assertions hold:
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∀t ∈ R, E(t) = kf 0 k2(h(t),k(t)) < +∞. If E(0) = 0, then for all t ∈ R, E(t) = 0. E is absolutely continuous on R. Finally, E 0 = −2a0
k0 0 |f ◦ k|2 h0
E 0 = 2a0 |f 0 ◦ k|2
a.e. in R
a.e. in R
(Dirichlet problem),
(Neumann problem).
(24) (25)
Proof. Let us show 1) for the Dirichlet problem (the proof is similar for the Neumann problem). Since ϕ(x, t) = f (t + x) − f (t − x) a.e. in Ω, 1 kϕx (·, t)k2(0,a(t)) + kϕt (·, t)k2(0,a(t)) 2 1 0 2 2 kf (t + ·) + f 0 (t − ·)k(0,a(t)) + kf 0 (t + ·) − f 0 (t − ·)k(0,a(t)) = 2
E(t) =
= kf 0 k(h(t),k(t)) . 2
For all t ∈ R, k(t) − h(t) = 2a(t) is finite, since a is continuous. The function f 0 is in L2loc(R), therefore E(t) < +∞ for all t ∈ R. This proves 1). If E(0) = 0, then f 0 (x) = 0 a.e. on I0 . By the relations (21) or (23), f 0 (x) = 0 a.e. in R and 2) is proved. Part 3) follows from 1), the Lipschitz property of h and k and the fact that 0 f ∈ L2loc (R). Then the differentiation of E (the derivatives exist a.e. by 3)) and the use of the relations (21) (Dirichlet problem) or (23) (Neumann problem) give 4). Remark 2.16. Statement 4) of Lemma 2.15 says that E and a vary in opposite sense in the Dirichlet problem and in the same sense in the Neumann problem. This is a manifestation of the Doppler effect. In Lemma 2.17 we give a useful formula for E. We recall that the intervals {In }n∈Z are defined by (14). Lemma 2.17. Let a ∈ Lip(R), L(a) ∈ [0, 1), a > 0. Then for all t in R, there exists a unique n(t) in Z such that h(t) ∈ In(t) . This defines a function n : R → Z with n(0) = 0. For the Dirichlet problem there exists a function αD such that ∀t ∈ R,
2
f0
E(t) = αD (t) √
n(t) DF I0
(26)
and for the Neumann problem there exists a function αN such that ∀t ∈ R,
p
2
E(t) = αN (t) f 0 DF n(t) . I0
(27)
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Moreover αD (0) = αN (0) = 1 and 1 1 ≤ αD (t) ≤ 0 , 0 Fmax Fmin
∀t ∈ R,
0 0 Fmin ≤ αN (t) ≤ Fmax .
If in addition a is 1-periodic, then ∀t ∈ R∗ ,
a(t) 1 + a(0) n(t) 1 a(t) 1 + a(0) 1 − − < < − + . ρ(F ) ρ(F )t ρ(F )|t| t ρ(F ) ρ(F )t ρ(F )|t|
(28)
Proof. The first statement about the existence of the function n is true since {In }n∈Z is a partition of R. Assume E(0) > 0 (the opposite case is trivial), then f 0 6= 0 (as a function in
0 2
f 2 , since In is mapped by F −n onto L2 (I0 )) and for all n ∈ Z, kf 0 kIn = √DF n I0
I0 and DF n · f 0 ◦ F n = f 0 a.e. on R. Therefore by Lemma 2.15, kf 0 kIn > 0. It is well defined. To find the bounds on αD we use follows that αD (t) := kf E(t) 0 k2 In(t)
again Lemma 2.15 and the relation: ∀t ∈ R, kf 0 k(h(t),k(t)) = kf 0 k(h(t),xn(t) ) + kf 0 k(xn(t) ,k(t)) 2
2
=
2
2 kf 0 k(h(t),xn(t) )
0 2
f
+
√F 0
.
(xn(t)−1 ,h(t))
Similar considerations hold for the Neumann case. If a is 1-periodic, according to Herman [22, Prop. II.2.3, p. 20], ∀n ∈ Z∗ , ∀x ∈ R,
−
F n (x) − x 1 1 < − ρ(F ) < . |n| n |n|
Then, choosing x = ±a(0) and taking into account that F n (−a(0)) ≤ h(t) < F n (a(0)), we have ∀n ∈ Z,
∀t ∈ h−1 (In ),
nρ(F ) − 1 − a(0) < h(t) < nρ(F ) + 1 + a(0) .
Since ρ(F ) > 0 under our assumptions, (28) is proven.
3. Dirichlet Problem 3.1. Stability 3.1.1. A universal lower bound We prove that the energy cannot become arbitrarily small. Note that a is not necessarily periodic. We recall that f 0 (x) = 12 (ϕ00 (x) + ϕ1 (x)) a.e. in (0, a(0)) and f 0 (x) = 12 (ϕ00 (−x) − ϕ1 (−x)) a.e. in (−a(0), 0). Let us denote Mc := {x ∈ 4 f0 ˆ0 := supc∈R∗ k kM2c . It is easy to see that I0 ; 0 < |f 0 (x)| ≤ c} for c ∈ R∗+ and E 2a(0)c + ˆ0 ≤ E(0) if E(0) > 0. 0<E
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Theorem 3.1. Assume that E(0) > 0, a ∈ Lip(R), L(a) ∈ [0, 1), a > 0 and amax < +∞. Then a(0) ˆ0 . E (29) ∀t ∈ R, E(t) ≥ 0 amax Fmax If in addition ϕ00 , ϕ1 ∈ L∞ ((0, a(0))), then ∀t ∈ R,
E(t) ≥
E(0)2 . 0 Fmax 2amax kf 0 k2L∞ (I0 ) 1
(30)
Proof. By Cauchy–Schwarz inequality, for any c ∈ R∗+ :
√
f0
f0 p
2 0 n
√ √ DF ≤ 2amax c2 , ∀n ∈ Z, kf 0 kMc ≤
DF n f
DF n Mc Mc Mc
√
2
2
√
since DF n ≤ DF n = F n (a(0)) − F n (−a(0)) ≤ 2amax (see the formula Mc
I0
(15)). By Lemma 2.17, ∀t ∈ R,
E(t) ≥
1 0 Fmax
2 0
kf 0 k4Mc
√ f
≥ 1 .
0 Fmax 2amax c2 DF n(t) Mc
This proves (29). If moreover ϕ00 , ϕ1 ∈ L∞ ((0, a(0))), then f 0 ∈ L∞ (I0 ) and choosing c := kf 0 kL∞ (I0 ) , we have kf 0 k2Mc = E(0) and (30) follows. Remark 3.2. The condition amax < +∞ is necessary for Theorem 3.1, since the energy may vanish asymptotically otherwise. To see that, assume that a ∈ C 1 (R) is stricly increasing and a0 ≥ γ > 0 on R+ . Then amax = +∞ and limt→+∞ E(t) = 0. In fact, F 0 ≥ 1+γ 1−γ on [−a(0), +∞). Then, by Lemma 2.17, n(t) R n(t) 1−γ 0 2 |f (x)| dx = E(0) for t ≥ 0. Since h(t) tends to E(t) ≤ 1−γ 1+γ I0 1+γ infinity as t tends to infinity and h(t) ≤ F n(t) (a(0)), n(t) tends to infinity as t tends to infinity. Thus limt→+∞ E(t) = 0. 3.1.2. Periodicity Special case of bounded energy is the energy periodic in time. We find the necessary and sufficient condition for the energy to be periodic for all initial data. Theorem 3.3. 1) Let a ∈ A0 . If F q = Id + p for some q ∈ N∗ and p ∈ N∗ , then ∀(x, t) ∈ Ω,
ϕ(x, t + p) = ϕ(x, t)
and
∀t ∈ R,
E(t + p) = E(t) .
2) Let a ∈ Lip(R) such that L(a) ∈ [0, 1), a > 0. If there exists p ∈ R∗+ such that for all (ϕ0 , ϕ1 ) ∈ H01 ((0, a(0))) × L2 ((0, a(0))), the relation E(t + p) = E(t)
(31)
holds for all t ∈ R, then a is periodic of period p and if a is not constant in R, there exists q ∈ N, q ≥ 2, such that F q = Id + p.
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Proof. 1) Let a ∈ A0 , F q = Id + p with q ∈ N∗ , p ∈ N∗ . Then the relation f ◦ F = f implies f ◦ (Id + p) = f and the first statement follows from Lemma 2.15 1). 2) Let the assumptions of the second part hold now. First step. Using the relations (19), (20) and (21), it is not difficult R to see that for any t ∈ R and any ψ ∈ L2 (I(t)), I(t) := (h(t), k(t)), such that I(t) ψ dm = 0, there exist initial conditions (ϕ0 , ϕ1 ) ∈ H01 ((0, a(0))) × L2 ((0,R a(0))) such that the restriction of f 0 to I(t) is ψ. It is also easy to prove that if I(t) |ψ|2 g dm = 0 for some g ∈ L∞ (I(t)) and all ψ of the above properties, then g = 0 a.e. in I(t). To see that, choose e.g. ψ(x) := ±1 at the points where g(x) ≥ 0 and 0 elsewhere; the R signs can be chosen in such a way that I(t) ψ dm = 0. Then we see that g(x) > 0 only on a set of zero measure. The same holds for g(x) < 0. Second step. We recall the relation equivalent to (24): E 0 (t) = −2
h0 (t) 0 |f (h(t))|2 a0 (t) k 0 (t)
a.e. on R .
Substituting this expression into the derivative of Eq. (31) we obtain h0 (t) 0 h0 (t + p) 0 2 0 |f |f (h(t))|2 a0 (t) (h(t + p))| a (t + p) = k 0 (t + p) k 0 (t)
a.e. on R .
(32)
For any t ∈ R there exist t0 ∈ R and q ∈ N such that h(t) ∈ I(t0 ) and h(t + p) ∈ F q (I(t0 )). Here the integer q depends on t and t0 but can be chosen as constant in a neighborhood of a given t. Using the relation f 0 ◦ F −q · DF −q = f 0 a.e. on R, Eq. (32) reads h0 (t + p) DF −q (h(t + p))2 |f 0 (F −q (h(t + p)))|2 a0 (t + p) k 0 (t + p) =
h0 (t) 0 |f (h(t))|2 a0 (t) k 0 (t)
a.e. on R.
(33)
Here all the factors are nonzero except of a0 values and f 0 values corresponding to the arguments in the same interval I(t0 ). Let M0 := {x ∈ R; a0 (x) = 0}, M1 := {x ∈ R; a0 (x) exists, a0 (x) 6= 0 and (33) holds}. The relation (33) is valid by assumption for all initial conditions (ϕ0 , ϕ1 ) from the required set and consequently a0 (t) = a0 (t + p) = 0 for almost every t ∈ M0 and F −q (h(t + p)) = h(t) for t ∈ M1 ,
(34)
since otherwise we could simply construct (ϕ0 , ϕ1 ) such that one side of the equation (33) is zero while the other is nonzero. At almost every points of M1 which are accumulation points of M1 we can differentiate Eq. (34) and substitute the result into Eq. (33). Choosing the initial conditions such that f 0 (h(t)) 6= 0 and realizing that |a0 | < 1, a simple calculation gives: a0 (t + p) = a0 (t) at almost every considered accumulation points. Since the isolated points of M1 form a countable and therefore
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zero-measure set, the last relation holds a.e. in M1 and M0 , i.e. a.e. in R. So a(t + p) = a(t) + c with a constant c ∈ R. However, a is positive by assumption. Therefore c = 0 and p is a period of a, and also the relations h(t + p) = h(t) + p, k(t + p) = k(t) + p and F (t + p) = F (t) + p hold. If a is not constant in R, then m(M1 ) > 0 and there exists a point t such that (34) holds. The relation E(t + p) = E(t) expressed with the help of Lemma 2.15 1) leads to Z 1 2 1− |f 0 | dm = 0 . DF q I(t) According to the first step, the relation DF q (x) = 1 follows for almost every x ∈ I(t) and then F q (x) = x + p , taking into account (34). By application of F n we now see that the last relation holds in every interval F n (I(t)) with n ∈ Z and therefore in R. Example 3.4. When q = 1 the condition of the part 1) of the above theorem is satisfied iff a is constant and equal to a = p2 . For q = 2 we are able to prove that p p = . F 2 = Id + p ⇔ ∀t ∈ R, a(t) + a t + 2 2 p An example of such function is a(t) := 4 + α sin(2πt) for some p ∈ N∗ , p odd, and 1 . It would be interesting to find such an explicit characterization for any q. |α| < 2π 3.1.3. Absence of strong instability Proposition 3.5. Assume that a is of class Q and ρ(F ) ∈ R \ Q. Let V := d F 0 ). Then there exists a subsequence {I±qn }n∈N of the sequence {In }n∈Z deVar(ln fined by (14) such that qn tends to infinity as n tends to infinity and ∀n ∈ N,
∀t ∈ h−1 (I±qn ),
e−V E(0) eV E(0) ≤ E(t) ≤ . 0 0 Fmax Fmin
(35)
Proof. By the conditions assumed on a, F is of class P (see Definition 2.6 and Lemma 2.8). Then the Denjoy’s inequality holds (Herman [22, Prop. VI.4.4, p. 75]): ∀n ∈ N, e−V ≤ DF ±qn ≤ eV u.c.s. The rationals pqnn ∈ Q are the convergents of ρ(F ). The integers qn ∈ N satisfy the following properties: qn+1 > qn (n ≥ 1), qn ≥ n 2 2 (n ≥ 2). For further details see Herman [22, Sec. V, p. 57] or Lang [27]. It follows from (26) that ∀n ∈ N, ∀t ∈ h−1 (I±qn ), e−V E(0)αD (t) ≤ E(t) ≤ eV E(0)αD (t). Remark 3.6. The behavior described by Proposition 3.5 is in fact more general. Let m, n ∈ N and define the sequence of intervals Jn,m := ∪m k=−m Iqn +k . Then −1 we have for all t ∈ h (Jn,m ), m+1 1 0 , 0 e−V E(0) ≤ E(t) min Fmin Fmax m+1 1 0 eV E(0) . ≤ max Fmax , 0 Fmin
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Notice that under the condition of Proposition 3.5, the case of strong instability is excluded. The sequence {qn }n∈N used in the proof diverges at least exponentially. 3.1.4. A sufficient condition of stability Let us first give some definitions. If F ∈ Dl (T), 0 ≤ l ≤ ω, then F is said to be C -conjugate to Rρ(F ) := Id + ρ(F ), 0 ≤ k ≤ l, if there exists g ∈ Dk (T) such that g ◦ F = Rρ(F ) ◦ g. The function g is called the conjugacy of F . We start with a rather abstract but general theorem. k
Theorem 3.7. Assume a ∈ A0 , ρ(F ) ∈ R \ Q and denote by µ the unique invariant measure of F¯ . Assume that there exist λ1 , λ2 ∈ R such that the Radon– dµ ≤ λ2 < +∞. Then Nikodym derivative of µ w.r.t. m satisfies: 0 < λ1 ≤ dm ∀t ∈ R,
λ2 E(0) λ1 E(0) ≤ E(t) ≤ . 0 0 λ2 Fmax λ1 Fmin
(36)
Conversely, if a ∈ A0 , E(0) > 0 and E is bounded from above by a strictly positive constant, then there exists an invariant measure of F¯ which is not singular w.r.t. m. Rx dµ be the Radon–Nikodym derivative and set g(x) := 0 dµ. Proof. Let φ := dm By Herman [22, Sec. II, p. 19], g◦F = g+ρ(F ). Obviously g is absolutely continuous ≤ Dg −1 ≤ λ−1 and λ1 ≤ g 0 = φ ≤ λ2 a.e. It follows that g ∈ D0 (T) and λ−1 2 1 −1 n −1 a.e. Now F = g ◦ Rρ(F ) ◦ g, hence DF = Dg ◦ Rnρ(F ) ◦ g · g 0 a.e. Using Lemma 2.17, we have Z Z 2 2 1 |f 0 (x)| |f 0 (x)| 1 dx ≤ E(t) ≤ dx ∀t ∈ R, 0 0 n(t) (x) n(t) (x) Fmax Fmin I0 DF I0 DF and (36) follows immediately. The statement of the second part of the theorem is equivalent to: if all the invariant measures of F¯ are singular w.r.t. m, then supt∈R E(t) = +∞. Since by Corollary 3.15 (which is proven independently), the supremum of E is infinite under this condition, the proof is complete. Remark 3.8. From the proof of Theorem 3.7, it can be seen that if F ∈ F0 is Lipschitz conjugate to the translation Rρ(F ) (i.e. the conjugacy and its inverse are Lipschitz continuous), then the model is stable. Let us give an explicit example of stability. Example 3.9. Assume that a ∈ A0 is defined on one period by b b (λ + 1)t0 + b λ−1 t+ if ≤ t ≤ λ+1 λ+1 2 2 a(t) := −β −β b (λ + 1)t0 + b λ − 1t + 1 + b − λ ≤t≤ +1 if λ−β + 1 1 + λ−β 2 2
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−β
1−λ −1 for b, β ∈ R∗+ , λ > 1 and t0 := λ−λ , −β ∈ (0, 1). Computing F := (Id+a)◦(Id−a) it can be shown that F satisfies assumptions of Herman [22, Prop. VI.7.7.1, p. 82] β ∈ Zρ(F ) ( mod 1), and we have: if ρ(F ) ∈ R \ Q (by a suitable choice of b) and 1+β then this model is stable.
By adding some conditions, we can prove a general result of stability. First we recall the definition: Definition 3.10. Let β ∈ R+ . Then α ∈ R is a β-Diophantine number if there exists C > 0 such that p C p ∀ ∈ Q , α − ≥ 2+β . q q q The set of Diophantine numbers is of full measure and meager in the real numbers. For definitions and details see Herman [22, Sec. I, p. 11 and Sec. V, p. 57] or Lang [27]. Theorem 3.11. Assume (i) a ∈ C k (T), k ∈ R, k > 2, |a0 | < 1 and a > 0 and (ii) ρ(F ) is β-Diophantine with β < k − 2. Then (36) holds. Proof. The assumption (i) implies that F ∈ Dk (T). A result of Katznelson and Ornstein [28] gives: let F ∈ Dk (T) with ρ(F ) a β-Diophantine number and k > 2 + β. Then F is C 1 -conjugate to Id + ρ(F ). By Remark 3.8, the proof is complete. 3.2. Instability 3.2.1. A universal upper bound It is not assumed here that a is periodic. We recall that n(t) is defined in Lemma 2.17. Proposition 3.12. If a ∈ Lip(R), L(a) ∈ [0, 1), a > 0, then ∀t ∈ R+ ,
E(t) ≤ E(0)
1 0 Fmin
n(t)+1 .
(37)
If moreover amin > 0 and E(0) > 0, then the maximal rate of exponential increase satisfies of E defined by r := lim supt→+∞ ln E(t) t r≤
1 1 ln 0 . 2amin Fmin
Proof. The first statement follows from Lemma 2.17,
2 n(t)+1
1 f0 2
≤ √ kf 0 kI0 . ∀t ∈ R+ , E(t) = αD (t) 0
n(t) F DF min I0
(38)
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Since amin ≤ a ≤ amax , then Id + 2amin ≤ F ≤ Id + 2amax (since F = Id + 2a ◦ h−1 ), hence Id + 2namin ≤ F n ≤ Id + 2namax. Combining these inequalities and the fact that by definition F n(t) (−a(0)) ≤ t − a(t) < F n(t) (a(0)), it is easy to prove that for all t ∈ R+ , h(t) + a(0) h(t) − a(0) ≤ n(t) ≤ 2amax 2amin (with the convention that if amax = +∞, then a−1 max = 0). Using (37), (38) follows. Remark 3.13. We will see in Remark 3.26 f) that this bound is almost optimal. Notice that in the case where a is periodic, one can deduce from the spectral analysis in Cooper and Koch [15] an other estimate, namely r ≤ 2 ln rU with rU the spectral radius of the evolution operator on one period. This estimate is more accurate if there exists no attracting hyperbolic periodic point, since in this case rU = 1 and therefore r = 0. It then follows that the rate of increase of E is subexponential. 3.2.2. A sufficient condition of instability We recall that if F ∈ D0 (T), then F¯ denotes its projection on the circle. Let ˜ := us identify [−a(0), −a(0) + 1] =: J ⊂ R with T and denote for any X ⊂ J, X S n ( m,n∈Z (F (X) + m)) ∩ I0 . Theorem 3.14. Let a ∈ A0 . Assume that for every invariant measure µ of F¯ , there exists a Borel set B in T such that µ(B) = 0 and >0 kϕ00 + ϕ1 k(B∩(0,a(0))) ˜
or
kϕ00 − ϕ1 k((−B)∩(0,a(0))) > 0, ˜
(39)
then m(B) > 0 (i.e. m is not absolutely continuous w.r.t. µ) and lim sup E(t) = +∞ . t→+∞
Proof. The relation m(B) > 0 follows from (39) and from the fact that F is a Lipschitz homeomorphism. It is also clear that there exist two integers m0 , n0 ∈ Z such that kf 0 kL1 (M) > 0 for the set M := (F n0 (B) + m0 ) ∩ I0 . Let us introduce a sequence of probability measures on T defined for all m-measurable set X ⊂ T by Z mn (X) := X
n−1 1 X ¯k DF dm . n
(40)
k=0
Since the probability measures form a weak* compact subset of C 0 (T)0 , there exists a subsequence {mni }i∈N which converges in the weak* topology to a Borel measure µ. It is easily seen that µ is an invariant measure of F¯ . By assumption, there exists a Borel set B in T such that µ(B) = 0 and m(B) > 0. Due to the regularity of µ and m, for every > 0, there exists an open set V in J such that V contains B and µ(V ) < ; there also exists a compact set K in J such that B contains K and
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m(B \ K ) < . According to Urysohn’s lemma, there exists a function g ∈ C 0 (T) such that the support of g is contained in V , the range of g is included in [0, 1] and g is equal to 1 on K . With the help of the function g it is easily seen that there exists i ∈ N such that for all i ≥ i , Z mni (K ) = K
ni −1 1 X DF¯ k dm < . ni k=0
Let us denote N := F n0 (K ) + m0 and M := N ∩ I0 . We will extend the measure mn to the Borel sets in R by formula (40) with F¯ replaced by F . A straightforward calculation shows that ! Z nX nX n0 +n i −1 0 −1 Xi −1 1 k k k DF¯ − DF¯ + DF¯ dm . mni (N ) = ni K k=0
k=0
k=ni
The integral of the last two sums is bounded by n0 . So there exists an integer i0 n0 such that for all i ≥ i0 , mni (N ) < 2. Since m(M \ M ) ≤ DFmax , we have: 1 0 0 kf kL1 (M ) > 2 kf kL1 (M) > 0 for sufficiently small . Now we will establish a useful inequality using Cauchy–Schwarz inequality. Let A ⊂ I0 be an m-measurable arbitrary set. Then kf 0 kL1 (A) =
n−1 1X 0 kf kL1 (A) n k=0
≤
n−1
0
√ 1 X
√ f
DF k
n A DF k A k=0
≤
2 ! 12 n−1 0
f 1 X
√
n DF k A k=0
n−1 √
2 1 X
DF k n A
! 12 .
k=0
Let {tk }k∈N be a sequence of real numbers such that tk ∈ h−1 (Ik ) for all k ∈ N and choose A := M ,
n−1 n−1
f 0 2 1X 1X
E(tk ) = αD (tk ) √
n n DF k I0 k=0
k=0
2 n−1 f0 1 X
≥ 0
√ k Fmax n DF M k=0 1
≥
1 0 Fmax
kf 0 k2L1 (M )
√
2 1 Pn−1 k k=0 DF n
M
≥
1
kf 0 k2L1 (M)
0 Fmax 4mn (N )
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and then for all i ≥ i0 ,
ni −1 kf 0 k2L1 (M) 1 1 X E(tk ) > 0 ni 8Fmax k=0
and since can take arbitrary small values, ni −1 1 X E(tk ) = +∞ . i→+∞ ni
lim
k=0
It follows that the limit superior of E is infinite as t tends to infinity.
Corollary 3.15. Let a ∈ A0 and assume that E(0) > 0. 1) If every invariant measure of F¯ is singular w.r.t. m, then lim sup E(t) = +∞ . t→+∞
2) If ρ(F ) ∈ R \ Q and the (unique) invariant measure of F¯ is not absolutely continuous w.r.t. m, then lim sup E(t) = +∞ . t→+∞
Proof. Assume that the assumptions of 1) hold: then the Borel set B of ˜ = 0. Theorem 3.14 can be chosen such that m(B) = 1 and µ(B) = 0, so m(I0 \ B) 0 Since E(0) > 0, kf kB˜ > 0 and the conclusion of Theorem 3.14 holds. Assume that the assumptions of 2) hold: then there exists a Borel set B in T such that m(B) = 0 and µ(B) > 0. Denote C := ∪n∈Z F¯ n (B). Since F ∈ F0 , F¯ (C) = C (i.e. C is F¯ -invariant) and m(C) = 0. Now we will prove that µ(C) = 1 which will show that µ and m are unusually singular. Then the first part of this corollary ends the proof. Let g be the conjugacy of F ; g is continuous and nondecreasing (see Herman [22, Prop. II.7.1, p. 25]) and g¯ ◦ F¯ (C) = g¯(C) + ρ(F ), hence g¯(C) is invariant by the irrational rotation Id + ρ(F ) and since m is its unique invariant measure, it follows that m(¯ g(C)) = 0 or m(¯ g(C)) = 1. But m(¯ g (B)) = µ(B) > 0, thus µ(C) = m(¯ g (C)) = 1. Example 3.16. If in Example 3.9, β = 1 and ρ(F ) ∈ R \ Q, then the limit superior of the energy as the time tends to infinity is infinite, whenever E(0) > 0 (see Herman [22, Theorem VI.7.4, p. 79] and Corollary 3.15 2)). Remark 3.17. In view of Theorems 3.7, 3.14 and Corollary 3.15, we put forward the question: whether there exists an example of F ∈ F0 with an invariant measure equivalent to m and leading to an infinite limit superior of the energy. Presently we do not know the answer.
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3.2.3. A sufficient condition of strong instability We give sufficient condition for the unlimited growth of energy in this section. The main results are contained in Theorem 3.20 and its corollaries. Definition 3.18. A point x ∈ R is called a wandering point for F ∈ F0 if there exists an open neighborhood U of x in R such that ∀n ∈ N∗ ,
∀m ∈ Z,
(F n (U ) + m) ∩ U = ∅ .
(41)
The set of wandering points in I˙0 will be denoted by W (F ) := {x ∈ I˙0 ; x is a wandering point for F } . Clearly, W (F ) is an open subset of I˙0 and W (F ) = W (F −1 ). Lemma 3.19. Let a ∈ A0 . Then for all x ∈ W (F ) there exists an open set U such that x ∈ U ⊂ W (F ) and lim kDF n kL1 (U) = 0,
n→±∞
+∞ X
kDF n kL1 (U) ≤ 1 .
(42)
n=−∞
Proof. Since x ∈ W (F ) is a wandering point, there exists an open set U 3 x in I˙0 such that U ⊂ W (F ) and (F n (U ) + m) ∩ U = ∅ for all n ∈ N∗ and m ∈ Z. It P+∞ P n n follows easily that +∞ n=−∞ kDF kL1 (U) = n=−∞ m(F (U )) ≤ 1. Theorem 3.20. If a ∈ A0 and kϕ00 + ϕ1 k(W (F )∩(0,a(0))) > 0
or
kϕ00 − ϕ1 k(−W (F )∩(0,a(0))) > 0 ,
then lim E(t) = +∞ .
t→±∞
(43)
Moreover, for all 0 < c < +∞ there exists an increasing sequence of integer numbers {nk }+∞ k=−∞ such that ∀t ∈ h−1 (Ink ),
E(t) ≥ c
|t| |t| ln . ρ(F ) ρ(F )
(44)
Proof. Since kf 0 kW (F ) > 0 by assumption, W (F ) is nonempty, so by Lemma 3.19, there exists a subset U ⊂ W (F ) such that lim kDF n kL1 (U) = 0
n→+∞
(45)
and kf 0 kU > 0. Since f 0 ∈ L2loc (R) and U is a bounded set, 0 < kf 0 kL1 (U) < +∞ .
(46)
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By Cauchy–Schwarz inequality, 0
kf kL1 (U)
√
f0 n
≤ √ DF
. U DF n
(47)
U
Then by Lemma 2.17, ∀t ∈ R,
E(t) ≥
1 0 Fmax
2 0 kf 0 k2L1 (U)
√ f
≥ 1
0 Fmax
DF n(t) 2 1 DF n(t) U L (U)
according to (47). Now, (45) and (46) imply (43). Due to Lemma 3.19, for any d > 0 there exists an increasing sequence of integer d nk (otherwise the series kL1 (U) ≤ numbers {nk }+∞ k=−∞ such that kDF |nk | ln |nk | −1 nk of nintegrals in (42) would diverge). If t ∈ h (Ink ), then h(t) ≤ F (a(0)) and F k (a(0)) − a(0) 1 as in proof of (28). Assume that t ≥ h−1 (a(0)), − ρ(F ) < nk |nk | < nk . Now for all t ∈ h−1 (Ink ), t > h−1 (a(0)) + 1, then nk > 0 and h(t)−a(0)−1 ρ(F ) E(t) ≥
kf 0 kL1 (U) |t − a(t) − a(0) − 1| |t − a(t) − a(0) − 1| ln . 0 Fmax d ρ(F ) ρ(F ) 1
2
The same arguments can be repeated for t < 0. The required relation (44) follows. Corollary 3.21. Assume a ∈ A0 , ρ(F ) = pq ∈ Q∗ and F q 6= Id + p. Denote by A := {x ∈ I0 ; F q (x) 6= x + p} the set of nonperiodic points. If kϕ00 + ϕ1 k(A∩(0,a(0))) 6= 0 or kϕ00 − ϕ1 k(−A∩(0,a(0))) 6= 0, then limt→±∞ E(t) = +∞. Proof. We shall verify that W (F ) consists of the complement of the set of periodic points (thus W (F ) = A) and m(W (F )) > 0. Since the set of wandering points is open it is sufficient to prove that at least one wandering point exists. By assumption, there exists a point x1 ∈ I0 such that F q (x1 ) 6= x1 + p. Evidently, F q (x1 ) 6= x1 + n for any n ∈ Z (otherwise ρ(F ) would have a value different from p p q q ). There also exists a point x0 ∈ I0 such that F (x0 ) = x0 + p as ρ(F ) = q . Let us pass from I0 and F to T and the projected map F¯ : T → T. Let us call ¯0 := π(x0 ). There exists a maximal (connected) open interval x ¯1 := π(x1 ) and x U in T containing x ¯1 such that F¯ q (x) 6= x for all x ∈ U . The interval U does not / U ). The endpoints of U are fixed points of F¯ q , one cover the whole T (e.g. x ¯0 ∈ is attracting and the second is repelling. Since F¯ q is orientation-preserving and ¯1 , F¯ (¯ x1 ), . . . , F¯ q (¯ x1 ) continuous, F¯ q (U ) = U . It is easily seen that all the points x are different. So there exists an open interval V such that x ¯1 ∈ V ⊂ U and the sets V, F¯ (V ), . . . , F¯ q (V ) are pairwise disjoint. Looking for the ordering of endpoints of the intervals F¯ nq (V ) in U it is seen that V ∩ F¯ nq (V ) = ∅ for n ∈ Z∗ (since F¯ q is monotone in U ). Now we prove that V ∩ F¯ nq+l (V ) = ∅ for n ∈ Z∗ , l ∈ {0, 1, . . . , q − 1}. Let us assume the opposite: there exist y ∈ V and z ∈ V such that y = F¯ nq+l (z) for some
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n ∈ Z∗ and l ∈ {0, 1, . . . , q − 1}. Then F¯ mq (y) = F¯ l (F¯ (n+m)q (z)) for any m ∈ Z. In the limit m → +∞, we have F¯ mq (y) → b, F¯ (n+m)q (z) → b where b is the attracting endpoint of U . So F¯ l (b) = b which contradicts the value of ρ(F ) = pq . As a result, we have proved that V ∩ F¯ m (V ) = ∅ for all m ∈ Z∗ and x1 is a wandering point. Since x1 is an arbitrary nonperiodic point, then A ⊂ W (F ) (and even A = W (F )). Thus assumptions of Theorem 3.20 are valid. Corollary 3.22. If a ∈ A0 , F has a finite nonzero number of periodic points and E(0) is strictly positive, then limt→±∞ E(t) = +∞. Proof: By assumption, there exist p, q ∈ N∗ such that ρ(F ) = pq . Moreover [−a(0), a(0)] \ W (F ) consists of a finite number of points; this can be seen in the proof of Corollary 3.21. Thus m(W (F )) = m(I0 ) = 2a(0) > 0, which implies that kf 0 kW (F ) > 0. Since the assumptions of Theorem 3.20 are satisfied, E tends to infinity as t tends to infinity. Corollary 3.23. Assume a ∈ A0 , ρ(F ) ∈ R \ Q, F is not C 0 -conjugate to Id+ρ(F ) and ϕ00 +ϕ1 6= 0 a.e. in W (F )∩(0, a(0)) or ϕ00 −ϕ1 6= 0 a.e. in (−W (F ))∩ (0, a(0)). Then limt→±∞ E(t) = +∞. Proof. To apply Theorem 3.20, let us prove that m(W (F )) > 0. As ρ(F ) ∈ R \ Q, there exists a unique invariant measure µ of F¯ in T and the support of µ satisfies supp µ = Ω(F¯ ) (Ω(F¯ ) is the set of nonwandering points on T). Since F is not C 0 -conjugate to Id + ρ(F ), supp µ 6= T. For these two facts see Herman [22, Sec. II.7.2, p. 25]. Since Ω(F¯ ) is closed, its nonempty complement in T is open and therefore of positive Lebesgue measure. The lift to R of this complement can be mapped by F into I0 (see Lemma 2.17). Thus m(W (F )) > 0. Remark 3.24. Corollary 3.23 is nonempty as show for instance Katok and Hasselblatt [23, Prop. 12.2.2., p. 405]: for all l ∈ [0, 1) and all β ∈ (0, 1), there exists F ∈ D1+β (T) such that ρ(F ) ∈ R \ Q and m(Ω(F¯ )) = l. Theorem 3.20 gives a new proof of divergence of the energy (for the previous ˇ results of that type see Cooper [13], Dittrich, Duclos and Seba [14], Cooper and Koch [15], M´eplan and Gignoux [16]). Assumptions based on the wandering points (instead of the periodic points like in [13, 14, 16]) give simplifications in the proof and more general results. Corollary 3.22 was already obtained by Cooper [13]. Corollary 3.21 is a slight generalization of Corollary 3.22: it allows to treat an infinite number of periodic points.a Corollary 3.23 was unexpected since ρ(F ) ∈ R \ Q: although there are no periodic points, E is increasing without bound. But classical solutions are excluded: if a ∈ A2 , then F is C 0 -conjugate to Id + ρ(F ) by Denjoy’s theorem (see Herman [22, Theorem VI.5.5, p. 76]). Note that Theorem 3.20 works with a boundary a differentiable a.e. In particular this includes the piecewise linear a Consider for instance the function a ∈ A equal to p (p ∈ N∗ ) on a half period and nonconstant 0 2 on the other half period. Then F has full intervals of periodic points.
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model. In Corollaries 3.21 and 3.22, there are no assumptions made on the nature of the periodic points (hyperbolic or not): this is the reason why we cannot have precise estimates on the rate of increase of E (see Subsec. 3.2.4 for a particular case). Compared to Theorem 3.14 the improvement is the following: if there exists a wandering set of strictly positive Lebesgue measure, then the model is strongly unstable. Finally we can extend the condition of Cooper for the growth of E without bound: “there exists a finite number of periodic characteristics” by “there exists a set of wandering characteristics which has an intersection of positive Lebesgue measure with {x ∈ (0, a(0)); ϕ00 (x) + ϕ1 (x) 6= 0} ∪ {x ∈ (−a(0), 0); ϕ00 (−x) − ϕ1 (−x) 6= 0}”. 3.2.4. Asymptotics In this section we will improve results given by Cooper [13], Dittrich, Duclos ˇ and Seba [14], Cooper and Koch [15]; see also Remark 3.26. For the asymptotics of the solution, see Cooper [13, Theorem 2, p. 79]. We recall that f is related to the initial conditions on I0 by the formulas (19)– (20). Although we have also worked out the case ρ(F ) := pq ∈ Q∗ , we present here only the case q = 1. Assume that there are only two periodic points of F in I0 : a1 := −a(0) which is attracting and hyperbolic and a2 which is repelling. Then a(0) is also an attracting hyperbolic periodic point with F 0 (a(0)) = F 0 (a1 ). For 2 simplicity take a(0) := p2 . Then In = I0 + np and E(np) = kf 0 kIn . We define for every x ∈ I0 , +∞ Y F 0 (a1 ) . l(x) := F 0 ◦ F k (x) k=0
Theorem 3.25. Assume a ∈ A1 , a00 ∈ L∞ (R), ρ(F ) := p ∈ N∗ and E(0) > 0. Assume also that F has only two periodic points in I0 : a1 := −a(0) and a2 ∈ I˙0 such that F 0 (a1 ) < 1 and F 0 (a2 ) ≥ 1. Take a(0) := p2 . Then l ∈ C 0 (I0 \ {a2 }), l > 0 on I0 \ {a2 } (l(a2 ) = 0) and
√ 2
lf 0 E(np) =
I0
F 0 (a1 )n
(1 + o(1))
as n → +∞ .
(48)
Proof. We use the following notations: J := [a1 , a2 ], for all n ∈ N, Kn :=
0 2 0
√ f ˆ n := 0 1 n , rn := Kn , ln (x) := Qn−1 F0 (ak1 ) > 0. Thus rn =
DF n , K ˆn k=0 F ◦F (x) F (a1 ) K J
√ 0 2
ln f . We first prove that ln (x) converges pointwise to l(x) (first step) and J that it is uniformly bounded in order to apply the Dominated Convergence Theorem and conclude that limn→+∞ rn =: L1 ∈ R+ (second step). It follows that Kn = ˆ n (L1 + o(1)) as n → +∞. Notice that by symmetry the proof needs just to be K done on one side, for instance [a1 , a2 ], and similar results hold for [a2 , a(0)].
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◦F (x) First step: pointwise convergence. Let vk (x) := F (aF1 )−F > −1. Then for 0 ◦F k (x) Qn−1 0 all x ∈ [a1 , a2 ], ln (x) = k=0 (1 + vk (x)). By continuity of F , there exists a set 0 J− := [a1 , a1 + δ− ], δ− > 0, such that for all x ∈ J− , F 0 (x) ≤ F (a21 )+1 < 1. By assumption, for all x ∈ [a1 , a2 ), there exists n0 (x) ∈ N such that for all n ≥ n0 (x), F n (x) ∈ J− . By the Mean Value Theorem, for any x in [a1 , a2 ),
1 k 00
and |vk (x)| ≤ |a1 − F (x)| · kF kL∞ (J) ·
F0 ∞ L (J) k
∀k ≥ n0 (x), |a1 − F k (x)| = F k−n0 (x) ◦ F n0 (x) (a1 ) − F k−n0 (x) ◦ F n0 (x) (x) ≤ |a1 − x| · kF 0 kL0∞ (J) · kF 0 kL∞ (J0 − ) . n (x)
k−n (x)
Q+∞ P It follows that +∞ k=0 |vk (x)| is convergent, therefore k=0 (1 + vk (x)) is absolutely convergent. The pointwise limit of ln (x) is l(x) (l > 0 on [a1 , a2 )). Since ln is clearly uniformly convergent on any compact included in [a1 , a2 ), l ∈ C 0 ([a1 , a2 )). Second step: uniform bound. Let J+ := [a2 − δ+ , a2 ] with δ+ small enough such that minJ+ F 0 ≥ F 0 (a1 ). For x ∈ J+ \ {a2 }, denote by nx ∈ N the greatest integer such that n < nx implies that F n (x) ∈ J+ . We know that for all k < nx , F 0 (a1 ) ≤ 1: it follows that for all x ∈ [a2 − δ+ , a2 ) and for all n ≤ nx , ln (x) ≤ 1. F 0 ◦F k (x) If J+ = J (this is the case if F 0 (a1 ) = minJ F 0 ) then nx = +∞ and immediately for all x ∈ J and for all n ∈ N, 0 ≤ ln (x) ≤ 1. If J+ 6= J, then nx < +∞ and for all x ∈ J+ \ {a2 }, for all n > nx , ln (x) = ln−nx (F nx (x))lnx (x) ≤ ln−nx (F nx (x)) ≤ sup sup ln (y) =: M < +∞ , n∈N y6∈J+
since the convergence of ln is uniform on [a1 , a2 − δ+ ]. We have proved that ln (x) ≤ max{1, M } on J and for all x ∈ J \ {a2 }, limn→+∞ ln (x) = l(x) > 0. By the Dominated Convergence Theorem we have that
√ 2
lim rn = lf 0 =: L1 ≥ 0 . n→+∞
J
Summing the √ results from both intervals [−a(0), a2 ] and [a2 , a(0)], the relation (48) follows with k lf 0 kI0 > 0. Remark 3.26. a) This proof can be generalized to an arbitrary finite number −1 of periodic points {ai }1≤i≤N by making a partition ∪N i=1 [ai , ai+1 ) of I0 by the periodic points, then considering E as a sum of terms over every interval [ai , ai+1 ) and finally applying the results proved in Theorem 3.25. b) Let us compare our result with previous ones (for p = 1): in Cooper [13], ˇ ≥ A + nc for Dittrich, Duclos and Seba [14], it is proven that for all n ∈ N∗ , ln E(n) n 0 ∗ = some constants 0 < A < − ln F (a1 ) and c ∈ R ; in Cooper and Koch [15], ln E(n) n
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− ln F 0 (a1 ) + o(1) as n → +∞ and now we have
√ 2
as n → +∞, with c := ln lf 0 .
ln E(n) n
= − ln F 0 (a1 ) +
c n
+o
1 n
I0
c) If F 0 (a2 ) > 1 and f 0 is L∞ in aneighborhood of a2 , we are even able to prove 1 1 that the remainder is not only o n but O n2 . We do not give the proof here [29]. d) The reader can deduce the behavior of E(t) from E(np) by Lemma 2.17. e) Even for nonsmooth boundary a, there might be an exponential increase of E. The following result can be proven: assume that a ∈ A0 , ρ(F ) = pq ∈ Q∗ and there exists a periodic point of F in I0 called a1 and satisfying lim sup x→a+ 1
F q (x) − F q (a1 ) =: γ < 1 . x − a1
Let a2 > a1 be the nearest periodic point of F and assume that kf 0 k(a1 ,a2 ) > 0. Then for all > 0, there exists a constant c > 0 such that ∀t ∈ R+ ,
E(t) ≥ c
1 γ+
pt .
f) By Proposition 3.12, we know that the maximal rate of exponential increase of 0 =: r0 . Assume p = 1, then in Theorem 3.25 we obtain E satisfies r ≤ − 2a1min ln Fmin 0 r = − ln F (a1 ). Since p = q = 1, then 12 ∈ Ran a by Lemma 2.4. Now assume |r 0 −r| r between the estimate of is e = 2a1min − 1 , which can be
0 , then the relative error e := that F 0 (a1 ) := Fmin
Proposition 3.12 and the result of Theorem 3.25 made arbitrarily small if amin is close to 12 . 3.2.5. Instability: strong instability is not the rule
Proposition 3.27. 1) Let G ∞ := D∞ (T) \ Int ρ−1 (Q). Then G ∞ is a closed set with empty interior and there exists A ⊂ G ∞ ∩ ρ−1 (R \ Q) such that A is a dense Gδ set in G ∞ . Moreover, for all F ∈ A ∩ F∞ , lim sup E(t) = +∞ , t→+∞
whenever E(0) > 0. 1 , β > |α| and α is fixed. Let 2) Let Fβ (x) := x + α sin(2πx) + β, 0 < |α| < 2π Kα := [|α|, |α| + 1] \ Int {β; ρ(Fβ ) ∈ Q}. Then Kα is a Cantor set and there exists B ⊂ Kα ∩ ρ−1 (R \ Q) such that B contains a dense Gδ set in Kα . Moreover, for all F ∈ B ∩ F∞ , lim supt→+∞ E(t) = +∞, whenever E(0) > 0. Proof. A direct consequence of Herman [22, Theorem XII.1.11, p. 168 and Prop. XII.1.13, p. 169] and Corollary 3.15. Remark 3.28. This is an application of Corollary 3.15 which shows that instability can be even generic. The example given in part 2) is the well known map
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of Arnol’d [30]. We see that the instability is possible even if there are no periodic points for F and if a is smooth. This answers negatively to a conjecture of Cooper [17]. By Proposition 3.5, one knows that the limit inferior of energy E is finite under the assumptions of Proposition 3.27. 3.2.6. Perturbation of the boundary In the following proposition we will prove that the instability of our model is in general the rule after perturbation. Let us denote Ea := E the energy of the model with a moving boundary a. Proposition 3.29. Let a ∈ Ak , k ≥ 1, and Ea (0) > 0. The following two assertions are true: 1) There exists a ˜ ∈ Ak arbitrarily near a in the C k -topology such that Ea˜ is increasing exponentially fast as t tends to infinity, whenever Ea˜ (0) > 0. 2) Assume that F := (Id + a) ◦ (Id − a)−1 is a Morse–Smale diffeomorphism. ˜kk < , Ea˜ Then there exists > 0 such that for all a ˜ ∈ Ak satisfying ka − a is still increasing exponentially fast, whenever Ea˜ (0) > 0. Proof. Let a ∈ Ak , k ≥ 1; then F := (Id +a)◦ (Id− a)−1 ∈ Fk . By a theorem of Peixoto (e.g. Nitecki [31, Sec. 1.2, p. 50]), one knows that there exists a dense, open subset of Dk (T) which consists of diffeomorphisms G with the following properties: the nonwandering set Ω(G) is finiteb and all periodic points are hyperbolic.c Consequently, there exists F˜ ∈ Dk (T) arbitrarily near F in the C k -topology. Since Fk is open for k ≥ 1, we can take F˜ ∈ Fk . By Lemma 2.3, Ak and Fk are homeomorphic and ! !−1 F˜ + Id F˜ − Id ◦ ∈ Ak (49) a ˜ := 2 2 (see formula (11)). Then Ea˜ is increasing exponentially fast by Theorem 3.25 or Remark 3.26 e). This proves 1). If we assume that F ∈ Fk , k ≥ 1, is a Morse–Smale diffeomorphism, then if F˜ is a sufficiently small perturbation of F in the C k -topology, F˜ is still a Morse–Smale ˜ by the formula (49), we can apply diffeomorphism and F˜ ∈ Fk . Thus, defining a Theorem 3.25 or Remark 3.26 e), which say that Ea˜ is increasing exponentially if Ea˜ (0) > 0. 4. Neumann Problem What will be remarkable is the opposite behavior of the Neumann problem w.r.t. the Dirichlet problem. Technicalities in the proofs are often the same, so we omit them. b Hence Ω(G) consists only of periodic points. c G is a Morse–Smale diffeomorphism.
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4.1. Stability 4.1.1. An upper bound Note that here a is not necessarily periodic. We recall that f 0 (x) =
1 0 (ϕ (x) + ϕ1 (x)) 2 0
f 0 (x) =
1 (−ϕ00 (−x) + ϕ1 (−x)) 2
a.e. in (0, a(0)) , a.e. in (−a(0), 0) .
(50) (51)
Proposition 4.1. If a ∈ Lip(R), L(a) ∈ [0, 1), a > 0, amax < +∞ and ϕ1 ∈ L∞ ((0, a(0))) ,
ϕ00 ∈ L∞ ((0, a(0))) ,
then ∀t ∈ R,
0 kf 0 kL∞ (I0 ) . E(t) ≤ 2amax Fmax 2
Moreover E ∈ Lip(R) and L(E) ≤ 2L(a) kf 0 kL∞ (I0 ) . 2
Proof. By the conditions on ϕ0 and ϕ1 , f 0 ∈ L∞ (I0 ). Then by Lemma 2.17, ∀t ∈ R,
p
2
0
0 E(t) ≤ Fmax
f DF n(t)
I0
2 0 ≤ Fmax kf 0 kL∞ (I0 ) F n(t) (a(0)) − F n(t) (−a(0)) 0 kf 0 kL∞ (I0 ) 2amax . ≤ Fmax 2
By the relation (23), we have f 0 ◦F = f 0 a.e. So by successive iterations we can show that f 0 ∈ L∞ (R) and kf 0 kL∞ (R) = kf 0 kL∞ (I0 ) . Using the formula E 0 = 2a0 |f 0 ◦ k|2 a.e. (see Lemma 2.15 4)), we get that E is Lipschitz continuous on R and L(E) ≤ 2L(a)kf 0 k2L∞ (I0 ) . Remark 4.2. The assumption that ϕ00 , ϕ1 ∈ L∞ ((0, a(0))) is crucial in the Neumann problem. There are examples for which the energy diverges if it is not the case as is shown in Sec. 4.2.2. Note also that if a is strictly increasing on R+ with a0 ≥ γ > 0, then amax = +∞ and limt→+∞ E(t) = +∞. The proof is similar as the one in Remark 3.2. 4.1.2. Periodicity With respect to the Dirichlet case (cf. Theorem 3.3) only minor changes are necessary to establish the following theorem: Theorem 4.3. 1) Let a ∈ A0 . If F q = Id + p for some q ∈ N∗ and p ∈ N∗ , then ∀t ∈ R, E(t + p) = E(t) .
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2) Let a ∈ Lip(R) such that L(a) ∈ [0, 1), a > 0. If there exists p ∈ R∗+ such that for all (ϕ0 , ϕ1 ) ∈ H 1 ((0, a(0))) × L2 ((0, a(0))), the relation E(t + p) = E(t) holds for all t ∈ R, then a is periodic of period p and if a is not constant in R, there exists q ∈ N, q ≥ 2, such that F q = Id + p. 4.1.3. Asymptotics In the next theorem we give an asymptotics in a simplified case: instead of a general rational rotation number ρ(F ) = pq , we consider ρ(F ) = p ∈ N∗ ; moreover we restrict the number of periodic points to two. This simplification has the advantage to propose a readable formula for the asymptotics. The general formula can be also worked out. Remember that f 0 is given on I0 by the formulas (50)–(51). Theorem 4.4. Let p ∈ N∗ . Assume that (i) a ∈ A1 , (ii) p2 ∈ Ran a, (iii) F has just two fixed points a1 , a2 in I0 such that a1 is attracting and a2 is repelling; take a1 := −a(0), (iv) (ϕ0 , ϕ1 ) ∈ C 1 ([0, a(0)]) × C 0 ([0, a(0)]) and ϕ00 (0) = ϕ00 (a(0)) = 0. Then E is asymptotically 1-periodic and more precisely lim E(t) − 2|f 0 (a2 )|2 a(t) = 0 . t→+∞
p Proof. p pBy assumption0 (ii), we can choose a(0)0 = 2 and then for all n ∈ Z, In = − 2 , 2 + np. Let fn be the restriction of f to In . Then Eq. (23) implies that fn0 = f00 ◦ F −n . By (iv), f 0 ∈ C 0 (I0 ), thus f 0 ∈ C 0 (R). Obviously, kf 0 kL∞ (R) = kf 0 kL∞ (I0 ) < +∞. Let φ : I0 → R be the function equal to f 0 (a1 ) if x = a1 and equal to f 0 (a2 ) otherwise. Assumption (iii) implies that for all x ∈ I0 , f 0 (x + np) converges pointwise to φ(x) as n → +∞. Let τ ∈ I0 and define the sequence tn := tn (τ ) := h−1 (τ ) + np, n ∈ N. Then h(tn ) = τ + np ∈ In and by obvious changes of variables:
√ 2 2 2
+ kf 0 k(τ +np,xn ) E(tn (τ )) = kf 0 k(h(tn (τ )),k(tn (τ ))) = f 0 F 0 (xn−1 ,τ +np)
√
2 = f 0 (· + np) F 0
(−a(0),τ )
+ kf 0 (· + np)k(τ,a(0)) . 2
Since f 0 ∈ L∞ (R) and limn→+∞ f 0 (x + np) = φ(x) for all x ∈ I0 , we can apply the Dominated Convergence Theorem, lim E(tn (τ )) = |f 0 (a2 )| (F (τ ) − F (−a(0))) + |f 0 (a2 )|2 (a(0) − τ ) 2
n→+∞
= |f 0 (a2 )| (F (τ ) − τ ) =: E∞ (τ ) . 2
It is easy to see that
|E(tn (τ )) − E∞ (τ )| ≤ kF 0 kL∞ (I0 ) |f 0 (· + np)|2 − |f 0 (a2 )|2 L1 (I
0)
.
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The right-hand side tends to zero when n tends to infinity and is independent of τ , thus E(tn ) converges uniformly w.r.t. τ to E∞ (τ ). Since F (τ ) − τ = 2a(h−1 (τ )) = 2a(tn (τ )), the proof is complete. 4.1.4. Absence of strong instability Like for the Dirichlet case (cf. Proposition 3.5) we have, by similar arguments: Proposition 4.5. Assume that a is of class Q and ρ(F ) ∈ R \ Q. Let V := d F 0 ). Then there exists a subsequence {I±qn }n∈N of the sequence {In }n∈Z deVar(ln fined by (14) such that qn tends to infinity as n tends to infinity and ∀n ∈ N,
∀t ∈ h−1 (I±qn ),
0 0 e−V E(0)Fmin ≤ E(t) ≤ eV E(0)Fmax .
4.1.5. A sufficient condition of stability The proofs of the following theorems are similar to Theorems 3.7 and 3.11. Theorem 4.6. Assume a ∈ A0 , ρ(F ) ∈ R \ Q and denote by µ the unique invariant measure of F¯ . Assume that there exist λ1 , λ2 ∈ R such that the Radon– dµ ≤ λ2 < +∞. Then Nikodym derivative of µ w.r.t. m satisfies: 0 < λ1 ≤ dm ∀t ∈ R,
λ2 λ1 0 0 E(0)Fmin ≤ E(t) ≤ E(0)Fmax . λ2 λ1
(52)
Theorem 4.7. If a ∈ C k (T), k ∈ R, k > 2, |a0 | < 1, a > 0 and ρ(F ) is β-Diophantine with β < k − 2, then (52) holds. Remark 4.8. Like for the Dirichlet problem, if F ∈ F0 is Lipschitz conjugate to Id + ρ(F ), then the Neumann model is stable. 4.2. Instability 4.2.1. Universal lower and upper bounds Note that here a is not necessarily periodic and n(t) is defined in Lemma 2.17. The more general bounds of the Neumann problem are given by the following proposition. The proof is similar to the one of Theorem 3.12. Proposition 4.9. If a ∈ Lip(R), L(a) ∈ [0, 1) and a > 0, then ∀t ∈ R+ ,
0 E(0) (Fmin )
n(t)+1
0 ≤ E(t) ≤ E(0) (Fmax )
n(t)+1
.
If moreover E(0) > 0, then the following two assertions are true: 1) If amin > 0, then the maximal rate of exponential increase of E defined by satisfies r+ := lim supt→+∞ ln E(t) t r+ ≤
1 0 ln Fmax . 2amin
(53)
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2) If limt→+∞ a(t) = 0, then the minimal rate of exponential increase of E t satisfies defined by r− := lim inf t→+∞ ln E(t) t r− ≥
1 0 ln Fmin , 2amax
(54)
with a−1 max = 0 if amax = +∞.
One could think that these estimates are unrealistic by the already existing bound given in Proposition 4.1. The results given in Proposition 4.10 and Theorem 4.13 show that this is not the case. 4.2.2. Singular initial data Proposition 4.10. There exist a ∈ Ak (1 ≤ k ≤ +∞ or k = ω) and initial conditions (ϕ0 , ϕ1 ) ∈ H 1 ((0, a(0))) × L2 ((0, a(0))) with ϕ00 6∈ L∞ ((0, a(0))) (or ϕ1 6∈ L∞ ((0, a(0)))) such that lim E(t) = +∞ t→+∞
and the rate of increase is exponential. Proof. Let F ∈ Fk , k ≥ 1, such that F (0) = p for some p ∈ N∗ , F 0 (0) > 1 and there exists x0 ∈ (0, a(0)] such that for all x ∈ [0, x0 ], F 0 (x) = F 0 (0). It is obvious that ρ(F ) = p and that we can build such F . To this F there corresponds a unique a ∈ Ak as stated in Lemma 2.3. For simplification we introduce F˜ := F − p. Then F˜ (0) = 0 and DF n = DF˜ n . For any n ∈ N∗ , let us look for x ∈ (0, x0 ] such that F˜ n (x) ≤ x0 . We define now the sequence y0 := x, yn := F˜ (yn−1 ), n ≥ 1. Immediately yn = F 0 (0)n x. Then for all n ∈ N, n ≥ 1, Fn (x) ≤ x0 ⇔ x ≤
x0 0 F (0)n
=: n → 0 as n → +∞ .
Assume for instance that ϕ0 ∈ H 1 ((0, a(0))) and ϕ1 ∈ L2 ((0, a(0))) satisfy ϕ0 (x) := 8 34 0 − 14 3 x and ϕ1 (x) := 0 for x ∈ (0, x0 ] and are arbitrary otherwise. Then f (x) = x on (0, x0 ]. Thus f 0 ∈ L2 ((0, x0 )) \ L∞ ((0, x0 )). Hence, by Lemma 2.17, ∀t ∈ R∗+ ,
p
2
0
0 E(t) ≥ Fmin
f DF˜ n(t)
(0,n(t) )
0 = Fmin F 0 (0)n(t) kf 0 k(0,n(t) ) 2
√ n(t) x0 0 0 √ = 2Fmin F 0 (0)n(t) p = 2Fmin x0 F 0 (0) 2 . F 0 (0)n(t) This construction is valid for k ≤ +∞. We can also easily build examples of F ∈ Dω (T). This lower bound and Proposition 4.9 show that E grows without bound exponentially fast.
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Remark 4.11. If in the above proof we assume more generally that ϕ0 (x) behaves on (0, x0 ] like xα , 12 < α < 1, E(t) grows faster than cF 0 (0)2(1−α)n(t) . The more singular is ϕ0 (i.e. α smaller and smaller) the faster the energy increases. In particular, the bound (53) on the maximal rate of exponential increase of E is almost reached for α close to 12 . The physical explanation of this proposition is simple: singularities of the initial data are trapped by an attractor and amplified by the Doppler effect. 4.2.3. Sufficient conditions for the decay of the energy As an immediate consequence of Theorem 4.4, we have: Corollary 4.12. If all the conditions of Theorem 4.4 hold and f 0 (a2 ) = 0, then limt→+∞ E(t) = 0. In the next theorem we give conditions to have an exponential decay of the energy. We recall that for every x ∈ I0 , we defined in Theorem 3.25, l(x) :=
+∞ Y k=0
F 0 (a1 ) . F 0 ◦ F k (x)
Theorem 4.13. Assume that (i) a ∈ A1 , a00 ∈ L∞ (R). (ii) ρ(F ) = pq ∈ Q∗ and F has just two periodic points a1 , a2 on I0 such that DF q (a1 ) < 1 and DF q (a2 ) ≥ 1. Take a1 := −a(0). (iii) (ϕ0 , ϕ1 ) ∈ C 1 ([0, a(0)]) × C 0 ([0, a(0)]), ϕ00 (0) = ϕ00 (a(0)) = 0 and E(0) > 0. (iv) There exists an open neighborhood of a2 denoted by V := (a2 − , a2 + ) for some > 0, such that for all x ∈ V , f 0 (x) = 0. Then E decreases exponentially to 0 and more precisely:
0 2
f q m
E(mp) =
√l DF (a1 ) (1 + o(1)) as m → +∞ , I0 where l is continuous and strictly positive on [−a(0), a(0)] \ V . Proof. Since the proof is very similar to the one of Theorem 3.25, we only sketch it. By (ii), a(0) is an attracting periodic point, DF q (−a(0)) = DF q (a(0)). Since for all t ∈ R+ , there exist a unique m(t) ∈ N and a unique r(t) ∈ {0, 1, . . . , q − 1} such that n(t) = m(t)q + r(t), it is possible to rewrite the energy as follows: p
p
2
2
0
0
n(t) n(t) + f DF E(t) = αN (t) f DF
(a1 ,a2 −)
p
2
= β(t) f 0 DF qm(t)
(a1 ,a2 −)
= β(t)E(m(t)p)
(a2 +,a(0))
p
2
+ f 0 DF qm(t)
(a2 +,a(0))
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for some strictly positive and bounded function β (which is obtained similarly as = p1 . Now similar arguments αN in Lemma 2.17). By Lemma 2.17, limt→+∞ m(t) t as in the proof of Theorem 3.25 show that l is continuous and strictly positive on [a1 , a2 − ] ∪ [a2 + , a(0)] and
0 2
2
√
f
0 mq
= DF q (a1 )m (1 + o(1)) as m → +∞ ,
f DF
√l (a1 ,a2 −) (a1 ,a2 −)
0 2
2
√
f
0 mq
= DF q (a1 )m (1 + o(1)) as m → +∞ .
f DF
√l (a2 +,a(0)) (a2 +,a(0))
The end of the proof is now obvious.
Remark 4.14. Like for Theorem 3.25 the assumptions of Theorem 4.13 are stronger than needed in order to obtain simpler results. A. Appendix Lemma A. Let Σ ⊂ R2 be a domain (i.e. nonempty, open and connected) such that for each y ∈ R, J y := {x ∈ R; (x, y) ∈ Σ} and for each x ∈ R, Jx := {y ∈ R; (x, y) ∈ Σ} are intervals of R. Let K1 := {x ∈ R; Σ ∩ ({x} × R) 6= ∅}, K2 := {y ∈ R; Σ ∩ (R × {y}) 6= ∅}. Then K1 and K2 are nonempty open intervals of R. 1 (Σ) and assume that ϕxy = 0 in D0 (Σ). Then there exist f ∈ Let ϕ ∈ Hloc 1 1 Hloc (K1 ), g ∈ Hloc (K2 ) such that ϕ(x, y) = f (x) + g(y)
a.e. in Σ .
Proof. Since Σ is nonempty, open and connected, the same holds for K1 and K2 ; so K1 and K2 are nonempty open intervals. 1 (Σ), ϕx ∈ L2loc (Σ) ⊂ L1loc (Σ). Further ϕxy = 0 in D0 (Σ) implies Since ϕ ∈ Hloc 1 that ∂y (ϕx ) ∈ L (Σ). Consequently (e.g. Kufner, John and Fuˇc´ık [24, Theorem 5.6.3, p. 274]), there exists a function u ∈ L1loc(Σ) such that: (i) u = ϕx ˆ 1 ⊂ K1 with m(K1 \ K ˆ 1 ) = 0 such that for all x ∈ K ˆ 1, a.e. in Σ, (ii) there exists K Jx 3 y 7→ u(x, y) is absolutely continuous and (iii) uy = ϕxy = 0 a.e. in Σ. ˜ 1) = 0 ˜ 1 ⊂ K1 with m(K1 \ K By Fubini’s theorem, (iii) ⇒ (iv): there exists K 0 ˆ1 ∩ K ˜ 1; ˜ 1 , uy (x, y) = 0 for almost every y ∈ Jx . Let K1 := K such that for all x ∈ K 0 0 m(K1 \ K1 ) = 0. The statements (ii) and (iv) imply that for all x ∈ K1 , Jx 3 y 7→ u(x, y) is constant. Let φ : K1 → R be the function defined by φ(x) := u(x, y) if x ∈ K10 and y ∈ Jx (φ is arbitrary on K1 \ K10 which is of zero measure). Let us show that φ ∈ L2loc(K1 ). It is sufficient to prove that for all x ∈ K1 , there exists r > 0 such that φ ∈ L2 ((x − r, x + r)). Let x0 ∈ K1 . Thus Jx0 6= ∅ and there exists r > 0 such that B ∞ ((x0 , y0 ), r) ⊂ Σ for some y0 ∈ Jx with B ∞ ((x0 , y0 ), r) := {(x, y) ∈ R2 ; max{|x − x0 |, |y − y0 |} < r}. Then for all x ∈ (x0 − r, x0 + r) ∩ K10 and for all y ∈ (y0 − r, y0 + r), one has u(x, y) = φ(x). The statement (i) implies
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that for almost every y ∈ (y0 − r, y0 + r), one has u(·, y) ∈ L2 ((x0 − r, x0 + r)). Let y1 ∈ (y0 − r, y0 + r) satisfying this assertion. Then for all x ∈ (x0 − r, x0 + r) ∩ K10 , one has φ(x) = u(x, y1 ); thus φ ∈ L2 ((x0 − r, x0 + r)). Since ϕ, ϕx ∈ L1loc(Σ), we use again the theorem in Kufner, John and Fuˇc´ık [24, Theorem 5.6.3, p. 274]: there exists a function v ∈ L1loc(Σ) such that: (i’) v = ϕ ˆ 2 ) = 0 such that for all y ∈ K ˆ 2, ˆ 2 ⊂ K2 with m(K2 \ K a.e. in Σ, (ii’) there exists K y J 3 x 7→ v(x, y) is absolutely continuous and (iii’) vx = ϕx = u a.e. in Σ. ˜ 2 ⊂ K2 with m(K2 \ Again by Fubini’s theorem, (iii’) ⇒ (iv’): there exists K ˜ ˜ K2 ) = 0 such that for all y ∈ K2 , vx (x, y) = u(x, y) for almost every x ∈ J y . By (ii’), with ay ∈ J y arbitrary: Z x ˆ 2 , ∀x ∈ J y , v(x, y) = v(ay , y) + vx (z, y) dz . (55) ∀y ∈ K ay
Using (iv’), (55) implies ˜ 2 =: K 0 , ˆ2 ∩ K ∀y ∈ K 2
Z ∀x ∈ J y ,
x
v(x, y) = v(ay , y) +
u(z, y) dz ay
Z
x
= v(ay , y) +
φ(z) dz . ay
Let a0 ∈ K1 and define: Z ∀x ∈ K1 ,
x
f (x) :=
φ(z) dz , a0
∀y ∈ K2 ,
Z
a0
g(y) := v(ay , y) +
φ(z) dz . ay
Since φ ∈ L1loc(K1 ), f is absolutely continuous on K1 and f 0 = φ a.e. on K1 : 1 (K1 ). We know that g : K2 → R is a function. Thus it follows that f ∈ Hloc ϕ(x, y) = v(x, y) = f (x) + g(y) a.e. on Σ. By changing the role of f and g, there 1 (K2 ) and a function f1 : K1 → R such that ϕ(x, y) = f1 (x) + g1 (y) exist g1 ∈ Hloc a.e. in Σ. Hence f1 = f − c a.e. and g1 = g + c a.e. for some c ∈ R. This yields ϕ(x, y) = f (x) + g(y) a.e. in Σ , 1 1 (K1 ), g ∈ Hloc (K2 ). with f ∈ Hloc
1 (Ω) satisfies Lemma B. Let a ∈ Lip(R), L(a) ∈ [0, 1) and a > 0. If ϕ ∈ Hloc 0 1 ϕtt − ϕxx = 0 in D (Ω), then there exist f, g ∈ Hloc (R) such that
ϕ(x, t) = f (t + x) + g(t − x)
a.e. in Ω .
¯ the traces of ϕ on each line t = Moreover ϕ can be continuously extended to Ω, 1 constant are in H ((0, a(constant))) and the traces of ϕ on the boundary ∂Ω of Ω 1 (∂Ω). are in Hloc
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Proof. The first statement is obvious by a change of variables and by using Lemma A. Then the second statement follows from the fact that f and g are in 1 (R) (and in particular f and g are continuous). Hloc Proof of Theorem 2.13. First step: by Lemma B, if a weak solution ϕ exists, then ϕ(x, t) = f (t + x) + 1 (R). g(t − x) a.e. in Ω with f , g ∈ Hloc Second step: the function ϕx has a trace on x = 0 by Lemma B and (6) implies that f 0 = g 0 a.e. Then g = f + c, where c ∈ R. Denote again by f the function f + 2c . Then ϕ(x, t) = f (t + x) + f (t − x) a.e. in Ω. Third step: the functions ϕ and ϕt have a trace on t = 0. Then f is given on I0 by the initial conditions ϕ0 and ϕ1 and by formulas similar to (19)–(20); thus f ∈ H 1 (I0 ). Fourth step: the function ϕx has a trace on x = a(t), thus (7) implies that f0 ◦ F = f0
a.e. in R .
Fifth step: by successive iterations, for all n ∈ Z, f 0 ◦ F n = f 0 a.e., thus f 0 is known a.e. on R. Up to a constant (depending on n) f is known on In by integration of f 0 . Using the third step the constants are determined by continuity 1 1 (R) and ϕ ∈ Hloc (Ω). of f . Obviously f ∈ Hloc Sixth step: by the third and fifth steps, f is unique on R. Thus ϕ is unique. B. Glossary a : R → R?+ , the law of the moving boundary, An := a ∈ C n (T); a > 0, a ∈ Lip(T), L(a) ∈ [0, 1)} a.e. almost everywhere BV(X), the space of functions on X of bounded variations, C k (X), the space of k-times continuously differentiable functions on X, C ω (X), the space of real analytic functions on X, C β (X) := Lipβ (X), whenever β ∈ (0, 1), Diffk (R), the C k -diffeomorphisms of R, Dk (T) the set of lifts to R of orientation-preserving C k diffeomorphisms of T, D0 (X) := (C0∞ (X))0 , the space of bounded linear functionals on the space of smooth functions with compact support in X, Diophantine number, a real number in a sense badly approximated by rationals, see Definition 3.10, E : R → R?+ , the energy of the field ϕ, F := k ◦ h−1 , F q , F composed by itself q times, F (k) or Dk F , the kth derivative of the function F , Fn , the set of functions F related to a ∈ An , i.e. Lipschitz homeomorphisms F ∈ Dn (T) such that F > Id, see Lemma 3.2.
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H 1 (X), Sobolev space of square integrable functions with square integrable generalized derivatives H01 (X) the closure of the space of smooth functions with compact support in H 1 (X); the fuctions satisfying Dirichlet boundary condition 1 1 (X), H0,loc (X), the spaces of functions the multiple of which by any smooth Hloc function of compact support are in H 1 (X) or H01 (X) h := Id − a, k := Id + a Id, the identity on R, In := [F n (−a(0)), F n (a(0))), n ∈ Z, ˙ the interior of a set X, Int X := X, (y) |, the Lipschitz constant of F ∈ Lip(X), L(F ) := supx,y∈X,x6=y | F (x)−F x−y Lipβ (X), the space of H¨older continuous functions with exponent β ∈ (0, 1], Lip(X) := Lip1 (X), the set of Lipschitz continuous functions, Morse–Smale diffeomorphism, a function from D1 (T) having nonzero finite number of periodic points which are all hyperbolic, m(B), the Lebesgue measure of a set B, P , function of class P , see Definition 2.6, periodic point of F ∈ D0 (T), a point in R with the projection in T invariant under F¯ q for some q ∈ N? , hyperbolic periodic point, a periodic point x with Dq F (x) 6= 1 ϕ, the real scalar field satistying the d’Alembert equation, ϕ0 , ϕ1 the initial value and initial velocity (the time derivative) of the field, π : R → T, x 7→ x + Z, the canonical projection, Q, function of class Q, see Definition 2.7, Rρ(F ) := Id + ρ(F ), the translation by ρ(F ) in R, i.e. the lift of rotation by the angle ρ(F ) in T, ρ(F ), the rotation number of F , whenever F ∈ D0 (T); a mean rotation angle of points in T under the projection F¯ ; see Eq. (9), T, the one dimensional torus (the circle of unit length), u.c.s., up to a countable set, d ) := inf{Var(b); b : T → R, b(x) = f (x) u.c.s. in T}, Var(f W (F ), the set of wandering point of F , see Definition 3.18, w.r.t., with respect to ? := {x ∈ X; x > 0} whenever X is a X ? := X \ {0}, X+ := {x ∈ X; x ≥ 0}, X+ subset of the real numbers, [x] the integer part of a real number x, kF k0 := kF kL∞ (X) := supx∈X |F (x)|, for a function F defined on X, kF kk := max1≤i≤k kF (i) k0 , kF kX , the L2 -norm of F on the set X, Fmin , Fmax , the essentiel infimum, resp. supremum of a real function F , F¯ , the projection of F on T whenever F is one-periodic, ¯ the closure of a set Ω, Ω, ∂Ω, the boundary of a set Ω
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Acknowledgments We wish to thank S. Vaienti for fruitful discussions. This work is partly supported by GACR grant No. 202/96/0218 (J. D.) and by a grant of the French Ministry of Foreign Affairs and the Czech Ministry of Education (N. G.). J. D. thanks the Centre de Physique Th´eorique of Marseille and the PhyMat and P. D., N. G. thank the Nuclear Physics Institute for hospitality. References [1] E. Fermi, “On the origin of the cosmic radiation”, Phys. Rev. 75 (1949) 1169–1174. [2] S. Ulam, “On some statistical properties of dynamical systems”, in Proceedings of the Fourth Berkeley Symposium on Mathematical and Statistical Problems, Vol. 3, Univ. of California Press, Berkeley, 1961, pp. 315–320. [3] L. D. Pustyl’nikov, “On Ulam’s problem”, Theoret. Math. Phys. 57 (1983) 1035–1038. [4] L. D. Pustyl’nikov, “Poincar´e models, rigorous justification of the second element of thermodynamics on the basis of mechanics, and the Fermi acceleration mechanism”, Russian Math. Surveys 50 (1995) 145–189. [5] L. D. Pustyl’nikov, “A new mechanism for particle acceleration and a relativistic analogue of the Fermi–Ulam model”, Theoret. Math. Phys. 77 (1988) 1110–1115. [6] L. D. Pustyl’nikov, “A new mechanism of particle acceleration and rotation numbers”, Theoret. Math. Phys. 82 (1990) 180–187. [7] G. Karner, “On the quantum Fermi accelerator and its relevance to “quantum chaos””, Lett. Math. Phys. 17 (1989) 329–339. ˇ [8] P. Seba, “Quantum chaos in the Fermi-accelerator model”, Phys. Rev. A41 (1990) 2306–2310. [9] S. T. Dembi´ nski, A. J. Makowski and P. Peplowski, “Quantum bouncer with chaos”, Phys. Rev. Lett. 70 (1993) 1093–1096. [10] V. V. Dodonov, A. B. Klimov and D. E. Nikonov, “Quantum particle in a box with moving walls”, J. Math. Phys. 34 (1993) 3391–3404. ˇˇtov´ıˇcek, “Floquet hamiltonians with pure point spectrum”, Com[11] P. Duclos and P. S mun. Math. Phys. 177 (1996) 327–347. [12] N. Balazs, “On the solution of the wave equation with moving boundaries”, J. Math. Anal. Appl. 3 (1961) 472–484. [13] J. Cooper, “Asymptotic behavior for the vibrating string with a moving boundary”, J. Math. Anal. Appl. 174 (1993) 67–87. ˇ [14] J. Dittrich, P. Duclos and P. Seba, “Instability in a classical periodically driven string”, Phys. Rev. E49 (1994) 3535–3538. [15] J. Cooper and H. Koch, “The spectrum of a hyperbolic evolution operator”, J. Funct. Anal. 133 (1995) 301-328. [16] O. M´eplan and C. Gignoux, “Exponential growth of the energy of a wave in a 1D vibrating cavity: Application to the quantum vacuum”, Phys. Rev. Lett. 76 (1996) 408–410. [17] J. Cooper, “Long time behavior and energy growth for electromagnetic waves reflected by a moving boundary”, IEEE Transac. Ant. Prop. 41 (1993) 1365–1370. [18] G. T. Moore, “Quantum field theory of the electromagnetic field in a variable-length one-dimensional cavity”, J. Math. Phys. 11 (1970) 2679–2691. [19] G. Calucci, “Casimir effect for moving bodies”, J. Phys. A: Math. Gen. 25 (1992) 3873–3882 and corrigenda: J. Phys. A: Math. Gen. 26 (1993) 5636. [20] V. V. Dodonov, A. B. Klimov and D. E. Nikonov, “Quantum phenomena in resonators with moving walls”, J. Math. Phys. 34 (1993) 2742–2756. [21] H. Johnston and S. Sarkar, “A re-examination of the quantum field theory of optical cavities with moving mirrors”, J. Phys. A: Math. Gen. 29 (1996) 1741–1746.
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[22] M. Herman, “Sur la conjugaison diff´ erentiable des diff´eomorphismes du cercle ` a des rotations”, Publ. Math. I.H.E.S. 49 (1979) 5–234 (in French). [23] A. Katok and B. Hasselblatt, “Introduction to the Modern Theory of Dynamical Systems”, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995. [24] A. Kufner, O. John and S. Fuˇc´ık, Function Spaces, Noordhoff Int. Publ., Leyden; Academia, Prague, 1977. [25] O. A. Ladyzhenskaya, “The Boundary Value Problems of Mathematical Physics”, Appl. Math. Sci. 49, Springer-Verlag, New York-Berlin, 1985. [26] J. L. Lions and E. Magenes, “Probl`emes aux Limites Non Homog`enes et Applications. Vol. I”, Travaux et Recherches Math´ematiques 17, Dunod, Paris, 1968 (in French). [27] S. Lang, Introduction to Diophantine Approximations, Second Ed., Springer-Verlag, New York, 1995. [28] Y. Katznelson and D. Ornstein, “The differentiability of the conjugation of certain diffeomorphisms of the circle”, Erg. Theory Dynam. Systems 9 (1989) 643–680. [29] N. Gonzalez, “L’equation des ondes dans un domaine d´ependant du temps”, Doctoral Thesis, UTV Toulon and CTU Prague, 1997 (in French). [30] V. I. Arnol’d, “Small denominators I. Mappings of the circumference onto itself”, Translations A.M.S. Series 2 46 (1965) 213–284. [31] Z. Nitecki, Differentiable Dynamics, M.I.T. Press, Cambridge-Massachusetts-London, 1971.
THE UNIQUENESS OF THE SOLUTION OF ¨ THE SCHRODINGER EQUATION WITH DISCONTINUOUS COEFFICIENTS ¨ WILLI JAGER University of Heidelberg Institute for Applied Mathematics D-69120 Heidelberg Germany
¯ YOSHIMI SAITO Department of Mathematics University of Alabama at Birmingham Birmingham, Alabama 35294 USA Received 15 March 1997 Revised 20 October 1997
1. Introduction Let us consider the reduced wave equation −∆u(x) + q(x)u(x) = 0
(1.1)
Ω ⊃ ER0 = {x ∈ RN : |x| > R0 } ,
(1.2)
on the domain Ω such that
where R0 > 0 and N ≥ 2. Suppose that q(x) has the form q(x) = −`(x) + s(x) ,
(1.3)
where `(x) a positive function, and |s(x)| is supposed to be dominated by `(x). The Eq. (1.1) has been studied extensively especially in the relation to the operator H1 = −∆ + V (x)
(1.4)
in L2 (Ω), or 1 ∆ (1.5) µ(x) in the weighted Hilbert space L2 (Ω; µ(x)dx) with boundary conditions on the boundary ∂Ω and at infinity. In this work we are concerned with the asymptotic behavior of the solution u of Eq. (1.1) at infinity. One of the important conclusions of the study is that we can establish the nonexistence of a class of (nontrivial) solutions of (1.1) which includes the L2 -solutions. And this result plays an important role in the attempt (the limiting absorption method, see, e.g. [1, 4]) to prove the H2 = −
963 Reviews in Mathematical Physics, Vol. 10, No. 7 (1998) 963–987 c World Scientific Publishing Company
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existence of the boundary value of the resolvent (H1 − z)−1 or (H2 − z)−1 of the operator H1 or H2 when the complex parameter z approaches the real axis. Consider the equation −∆u(x) + (−k 2 + s(x))u(x) = 0 ,
(1.6)
with k > 0, i.e., `(x) = k 2 in (1.3). In the celebrated work Kato [9] he showed, among others, that, under the condition τ ≡ (2k)−1 lim |x||s(x)| < 1 , |x|→∞
a nontrivial solution u of Eq. (1.1) satisfies Z (|∇u(x)|2 + |u(x)|2 ) dS = ∞ lim r2τ + r→∞
(1.7)
(1.8)
|x|=r
for any > 0. One of the important features of the work [9] is that the coefficient s(x) does not need to be spherically symmetric which makes the scope of application much wider than the preceding works (cf., e.g., M¨ uller [10], Rellich [11]). Another important feature of [9] is that the method is based on differential inequalities satisfied by several functionals of the solution u so that the problem was successfully treated as a local problem at infinity. As a result we do not need to use any boundary conditions at the boundary ∂Ω of Ω or at infinity such as radiation condition (cf., e.g., Wienholtz [15]). As is well-known this result has many applications. In Ikebe [3], in which the spectral theory and scattering theory for the Schr¨ odinger operator −∆ + V (x) in R3 was developed under the condition that V (x) = O(|x|−γ ) (|x| → ∞, γ > 2) ,
(1.9)
the result of Kato [9] was used to prove the existence of the boundary value of the Green function on the positive real axis as well as the nonexistence of the positive eigenvalues. After the work [9] various extensions and modifications were presented as many efforts were made to treat more general operators in a similar method. See e.g., Ikebe–Uchiyama [5] for Schr¨ odinger operators with magnetic potentials, J¨ ager [6] for the second order elliptic operators, Weidmann [14] for the many body Schr¨ odinger operators, and Ikebe–Sait¯ o [4], Sait¯ o [13] for Schr¨ odinger operators with long-range potentials. Now let us consider the case that `(x) is a positive function which may be discontinuous. One of the motivations to consider such `(x) comes from the study of the reduced wave equations in layered media. Consider the equation −µ(x)−1 ∆u − λu = 0 (x ∈ RN )
(1.10)
in layered media, where µ(x) is a positive function on RN . Suppose that the function µ(x) is a simple function with surfaces of discontinuity (separating surfaces) which may extend to infinity. Roach and Zhang [12] proved the nonexistence of the solution of Eq. (1.10) under a geometric condition (“cone-like” discontinuity on the
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separating surface, see also [2]). Then J¨ager and Sait¯ o [7] proved a similar results under another geometric condition (“cylindrical” discontinuity) on the separating surfaces. In these works the method is not local at infinity, but some global integral identity of the solution u are used. And the method seems to need some modifications in the case where µ is a perturbation such as µ(x) = µ0 (x) + µ` (x) + µs (x) ,
(1.11)
µ0 being a simple function, and µ` (x) and µs (x) behaving like a long-range and short-range potentials at infinity, respectively (cf. [8]). In this work we are going to obtain an extension of the result (1.8) by Kato [9] which can be applied the reduced wave Eq. (1.10) with µ(x) satisfying (1.11) as well as Eq. (1.6) where s(x) is the sum of a short-range potential and a long-range potential. Under the several assumptions (Assumptions 2.1, 4.2, 5.5 and 5.8) on the coefficient q(x) the following (Theorem 5.10 in Sec. 5) will be proved: Suppose that a solution u of Eq. (1.1) satisfies Z ∂u 2 2 lim − Re (q(x))|u| dS = 0 . ∂r r→∞ |x|=r
(1.12)
Then u has a compact support. Our method is a local method at infinity which is similar to the method of Kato [9]. As in [9], some type of differential inequalities on functionals of the solution u will play important roles. However, we shall first establish the differential inequalities not in the ordinary sense but in the sense of distributions, and then they will be interpreted in the ordinary sense. In Sec. 2 we define our reduced wave equation and give the main assumption (Assumption 2.1) on the coefficients. In Sec. 3 we introduce and evaluate the first functional M + (v, r). In order to complete the evaluation of M + (v, r), another functional N (v, m, r) is introduced and evaluated in Sec. 4. Section 5 is devoted for proving the main theorem (Theorem 5.9). Some examples are discussed in Sec. 6. In Sec. 7 we shall discuss how our result can be applied to some reduced wave operators which were studied in [8]. A lemma on distributional derivative is given in Appendix. 2. Schr¨ odinger-Type Homogeneous Equation Consider the homogeneous Schr¨ odinger equation −∆u(x) + q(x)u(x) = 0 (x ∈ ER0 ) ,
(2.1)
ER = {x ∈ RN : |x| > R} .
(2.2)
where R0 > 0, and Let S N −1 be the unit sphere of RN . We set X = L2 (S N −1 ) and the inner product and norm of X is denoted by ( , ) and | |, respectively.
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Assumption 2.1. (i) Let N be an integer such that N ≥ 2. Let u ∈ H 2 (ER0 )loc , R0 > 0, be a solution of Eq. (2.1), where q(x) is a complex-valued, measurable, locally bounded function on ER0 . (ii) Set (N − 1)(N − 3) . (2.3) Q(x) = q(x) + 4r2 (a) Then Q(x) is decomposed as Q(x) = Q0 (x) + Q1 (x) ,
(2.4)
where Q0 (x) is a real-valued, measurable, locally bounded function on ER0 such that (2.5) Q0 (x) ≤ 0 , and Q1 (x) is a complex-valued, measurable, locally bounded function on ER0 . (b) For any x ∈ X = L2 (S N −1 ), (Q0 (r·)x, x) has the right limit for all r > R0 as a function of r = |x|. (c) There exist h0 > 0 and, for 0 < h < h0 , a real-valued, measurable function Q0r (x; h) on ER0 such that sup {|Q0r (x; h)|/x ∈ G ,
0 < h < h0 } < ∞
(2.6)
for any compact set G ⊂ ER0 , 1 ({Q0 ((r + h)·) − Q0 (r·)}φ, φ) h ≤ (Q0r (r·; h)φ, φ) (φ ∈ X, r > R0 , 0 < h ≤ h0 ) ,
(2.7)
and the limit lim(Q0r (r·; h)φ, φ) = (Q0r (r·)φ, φ) h↓0
(φ ∈ X)
(2.8)
exists with a real-valued, measurable, locally bounded function Q0r (x) on ER0 . (iii) There exists a positive, measurable function h(r) defined on (R0 , ∞) such that 2 (a) (r > R0 ) , (2.9) h(r) ≤ r (b) and, setting a(r) = h−1 (r) sup |Q1 (x)| , |x|=r (2.10) b(r) = inf [−(Q0 (x) + h−1 (r)Q0r (x))] , |x|=r
where h−1 (r) = 1/h(r), we have a(r)2 ≤ b(r)
(r > R0 ) .
(2.11)
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In order to transform Eq. (2.1) into a differential equation on (R0 , ∞) with operator-valued coefficients, we give the following: Definition 2.2. (i) For r > R0 define a selfadjoint operator B(r) in X by ( D(B(r)) = D(ΛN ) , (2.12) B(r) = −r−2 ΛN , where D(T ) is the domain of T , and ΛN is the (selfadjoint realization of) Laplace– Beltrami operator on S N −1 . (ii) tit For r > R0 define a bounded operators C0 (r), C0r (r; h), C0r (r) and C1 (r) on X by C0 (r) = Q0 (r·)× , C0r (r; h) = Q0r (r·; h)× , (2.13) C0r (r) = Q0r (r·)× , C1 (r) = Q1 (r·) × . Proposition 2.3. Let u be a solution of Eq. (2.1) and let v be as in Assumption 2.1, (ii). Let J = (R0 , ∞). Then, (i) v ∈ C 1 (J, X). (ii) v(r) ∈ D((−ΛN )1/2 ) for r ∈ J. (iii) We have Z s {|v 0 (r)|2 + |B 1/2 (r)v(r)|2 } dr < ∞
(R0 < r < s < ∞) ,
(2.14)
r
(iv) (v)
(vi) (vii)
where v 0 (r) = dv(r)/dr and B 1/2 (r) = B(r)1/2 . v(r) ∈ D(ΛN ) for almost all r ∈ J, and Bv ∈ L2 ((r, s), X) for R0 < r < s < ∞. v 0 (r) ∈ Cac ([r, s], X) for R0 < r < s < ∞, where Cac ([r, s], X) is all X-valued absolutely continuous functions on [r, s]. There exists the weak derivative v 00 (r) of v 0 (r) for r ∈ J. v 0 (r) ∈ D((−ΛN )1/2 ) for almost all r ∈ J, and B 1/2 v 0 ∈ L2 ((r, s), X) for R0 < r < s < ∞. B 1/2 v ∈ Cac ([r, s], X) for R0 < r < s < ∞, and we have 2 d (B 1/2 (r)v(r), B 1/2 (r)v(r)) = − (B 1/2 (r)v(r), B 1/2 (r)v(r)) dr r +2 Re (B 1/2 (r)v 0 (r), B 1/2 (r)v(r))
for almost all r ∈ J. (viii) We have − v 00 (r) + B(r)v(r) + C0 (r)v(r) + C1 (r)v(r) = 0 in X for almost all r ∈ J.
(2.15)
(2.16)
968
¨ ¯ W. JAGER and Y. SAITO
Proof. See [13], Proposition 1.3.
Proposition 2.4. Suppose that Q0r (x) satisfies Assumption 2.1, (ii-b) and (ii-c). Let η ∈ C 1 (J, X). Let g(r) =
d (C0 (r)η(r), η(r)) dr
(2.17)
be the derivative of f (r) = (C0 (r)η(r), η(r)) in the sense of distributions on (R0 , ∞). Then we have g(r) ≤ (C0r (r)η(r), η(r)) + 2 Re (C0 (r)η(r), η 0 (r)) ,
(2.18)
where Ineq. (2.18) should be taken in the sense of distributions on (R0 , ∞) again. Proof. (I) Let ϕ be a nonnegative C0∞ ((R0 , ∞)) function. Then, by definition hg, ϕi = −hf, ϕ0 i Z ∞ ϕ(r + h) − ϕ(r) dr f (r) = − lim h↑0 R0 h Z 1 ∞ (f (r + h) − f (r))ϕ(r) dr, = lim h↓0 h R0
(2.19)
where h , i denotes the dual pair bracketing. (II) Here we have f (r + h) − f (r) = (C0 (r + h)η(r + h), η(r + h)) − (C0 (r)η(r), η(r)) = ({C0 (r + h) − C0 (r)}η(r + h), η(r + h)) +(C0 (r)η(r + h), η(r + h) − η(r)) +(η(r + h) − η(r), C0 (r)η(r)) ,
(2.20)
and hence, using (2.7) in (ii-b) of Assumption 2.1, we obtain 1 (f (r + h) − f (r)) ≤ (C0r (r; h)η(r + h), η(r + h)) h 1 + C0 (r)η(r + h), (η(r + h) − η(r)) h 1 (η(r + h) − η(r)), C0 (r)η(r) . + h
(2.21)
(III) It is easy to see from (2.8) in (ii-c) of Assumption 2.1 and (i) of Proposition 2.3 that the right-hand side of (2.21) converges to g0 (r) ≡ (C0r (r)η(r), η(r)) + 2 Re (C0 (r)η(r), η 0 (r))
(2.22)
¨ THE UNIQUENESS OF THE SOLUTION OF THE SCHRODINGER EQUATION WITH
...
969
boundedly on any compact interval in (R0 , ∞) as h ↓ 0. Therefore, noting that ϕ ≥ 0, we have Z ∞ g0 (r)ϕ(r) dr = hg0 , ϕi , (2.23) hg, ϕi ≤ R0
which completes the proof. 3. The Evaluation of the Functional M + (v, r)
Let v = v(r·) be as in (2.5). Then we are going to define the functional M + (v, r) by Definition 3.1. Let v be as in (ii-b) of Assumption 2.1. Then set M + (v, r) = |v 0 (r)|2 − (C0 (r)v(r), v(r)) − |B 1/2 (r)v(r)|2
(3.1)
for r > R0 . Proposition 3.2. Suppose that Assumption 2.1 is satisfied. Let M + (v, r) be as in Definition 3.1. (i) Then M + (v, r) is a real-valued, locally bounded function on J = (R0 , ∞). Further M + (v, r) is right continuous with its left limit for r ∈ J. (ii) We have d + M (v, r) ≥ −h(r)M + (v, r) (r > R0 ) , (3.2) dr where Ineq. (3.2) should be taken in the sense of distributions on (R0 , ∞). Proof. (i) follows from Assumption 2.1, (ii) and Proposition 2.3. From Propositions 2.3 and 2.4 we see that d + M (v, r) ≥ 2 Re (v 00 (r), v 0 (r)) dr −(C0r (r)v(r), v(r)) − 2 Re (C0 (r)v(r), v 0 (r)) 2 + (B(r)v(r), v(r)) − 2 Re (B(r)v(r), v 0 (r)) r
(3.3)
in the sense of distributions on (R0 , ∞). Using (2.16), we have from (3.3) d + M (v, r) ≥ −(C0r (r)v(r), v(r)) + 2 Re (C1 (r)v(r), v 0 (r)) dr 2 + (B(r)v(r), v(r)) r = −h(r)M + (v, r) +h(r) |v 0 (r)|2 − (C0 (r)v(r), v(r)) − (B(r)v(r), v(r)) −(C0r (r)v(r), v(r)) + 2 Re (C1 (r)v(r), v 0 (r)) 2 + (B(r)v(r), v(r)) r
(3.4)
¨ ¯ W. JAGER and Y. SAITO
970
Thus, using a(r) and b(r) defined by (2.10), and taking note of (2.9) in Assumption 2.1, we have d + M (v, r) ≥ −h(r)M + (v, r) dr +h(r)[|v 0 (r)|2 − 2a(r)|v 0 (r)||v(r)| + b(r)|v(r)|2 ] .
(3.5)
It follows from (2.11) in Assumption 2.1, that is, a(r)2 ≤ b(r), that (3.5) implies (3.2), which completes the proof. Proposition 3.3. Suppose that Assumption 2.1 is satisfied. For R1 > R0 we have Z r + h(t) dt M + (v, R1 ) (r ≥ R1 ) . (3.6) M (v, r) ≥ exp − R1
Proof. It follows from Proposition 3.2 that Z r d + M (v, r) h(t) dt exp dr R1 Z r h(t) dt M + (v, r) ≥ 0 (r > R1 ) , +h(r) exp
(3.7)
R1
and hence
Z r d + h(t) dt M (v, r) ≥ 0 exp dr R1
(r > R1 )
(3.8)
in the sense of distributions on (R1 , ∞). The Ineq. (3.6) follows from (3.8) and Lemma A of Appendix. 4. The Evaluation of the Functional N (v, m, r) Using M + (v, r), we are going to define another functional which will be used to evaluate M + (v, r) in Sec. 5. Definition 4.1. (i) Set N (v, m, r) = M + (w, r) + (m(m + 1) − F (r))r−2 |w|2
(w = rm v) ,
(4.1)
where m is a positive number and F (r) is a positive C 1 function on (R0 , ∞). (ii) For r > R0 define a bounded operators CR (r) on X by CR (r) = Re (Q(r·))× = (Q0 (r·) + Re (Q1 (r·)) × .
(4.2)
M (v, r) = |v 0 (r)|2 − (CR (r)v(r), v(r)) .
(4.3)
Set (iii) For r > R0 we set p(r) = inf [−(2Q0 (x) + rQ0r (x))] . |x|=r
(4.4)
¨ THE UNIQUENESS OF THE SOLUTION OF THE SCHRODINGER EQUATION WITH
...
971
Assumption 4.2. The function F (r) introduced in Definition 4.1 satisfies the following (i)–(iii): (i) There exists a positive constant c0 such that F 2 (r) ≤ c0 r4 h2 (r)b(r)
(r > R0 ) ,
(4.5)
where b(r) is given in (2.10), F 2 (r) = F (r)2 , and h2 (r) = h(r)2 . (ii) We have F (r) → ∞ as r → ∞. (iii) There exists a positive constant c1 such that Fr (r) ≡
d F (r) ≤ c1 r−1 dr
(r > R0 ) .
(4.6)
Proposition 4.3. (i) Let b(r) be given by (2.10) and assume that h(r) satisfies Ineq. (2.9) and that Q0 (x) is nonpositive. Then r2 h2 (r)b(r) ≤ 2p(r)
(r > R0 ) .
(4.7)
(ii) Assume that Ineq. (2.11) holds. Then !2 r sup |Q1 (x)|
≤ 2p(r)
|x|=r
(r > R0 ) .
(4.8)
(iii) Suppose that Ineq. (4.5) holds. Then r−2 F 2 (r) ≤ 2c0 p(r)
(r > R0 ) .
(4.9)
Proof. Since 0 < rh(r) ≤ 2 and −Q0 (x) ≥ 0, we have r2 h2 (r)[−(Q0 (x) + h−1 (r)Q0r (x))] = rh(r)[rh(r)(−Q0 (x)) + r(−Q0r (x))] ≤ 2[2(−Q0 (x)) + r(−Q0r (x))] = 2[−(2Q0 (x) + rQ0r (x))] ,
(4.10)
which implies (4.7). It follows from (2.11) and (4.7) that !2 r sup |Q1 (x)|
!2 2 2
= r h (r) h
−1
|x|=r
(r) sup |Q1 (x)| |x|=r
≤ r h (r)b(r) 2 2
≤ 2p(r) .
(4.11)
From (4.5) and (4.7) we obtain F 2 (r) ≤ c0 r4 h2 (r)b(r) ≤ c0 r2 (2p(r)) = r2 (2c0 p(r)) for r > R0 , which implies (4.9).
(4.12)
¨ ¯ W. JAGER and Y. SAITO
972
Proposition 4.4. Suppose that Assumptions 2.1 and 4.2 hold. Then there exist m0 > 0 and r0 > R0 such that d 2 (r N (v, m, r)) ≥ 0 dr in the sense of distributions on (r0 , ∞).
(m ≥ m0 )
Proof. (I) By definition w = rm v satisfies 0 w = rm v 0 + mrm−1 v = rm v 0 + mr−1 w , 00 m 00 m−1 0 v + m(m − 1)rm−2 v w = r v + 2mr m 00 −1 w0 − mr−1 w + m(m − 1)r−2 w = r v + 2mr = rm v00 + 2mr−1 w0 − m(m + 1)r−2 w = rm (Bv + C0 v + C1 v) + 2mr−1 w0 − m(m + 1)r−2 w ,
(4.13)
(4.14)
and hence we have −w00 + 2mr−1 w0 + (B + C0 + C1 − m(m + 1)r−2 )w = 0 .
(4.15)
g(r, m) = (m(m + 1) − F (r))r−2 .
(4.16)
(II) Set Then, using (4.15) and Proposition 2.4, we have r−2
d 2 (r N (v, m, r)) ≥ 2r−1 (|w0 |2 − (C0 w, w) − (Bw, w) + (gw, w)) dr +2 Re (w00 − C0 w − Bw + gw, w0 ) 2 + |B 1/2 w|2 − (C0r w, w) + (gr w, w) r = 2r−1 (|w0 |2 − (C0 w, w) − (Bw, w) + (gw, w)) +2 Re (2mr−1 w0 + C1 w − m(m + 1)r−2 w + gw, w0 ) 2 + |B 1/2 w|2 − (C0r w, w) + (gr w, w) r = 2(1 + 2m)r−1 |w0 |2 + (2r−1 g + gr )|w|2 +r−1 ([− 2C0 − rC0r ]w, w) +2 Re (C1 w − m(m + 1)r−2 w + gw, w0 ) ,
where gr = dg/dr. Note that ( −1 2r g(r, m) + gr (r, m) = −Fr (r)r−2 , g(r, m) − m(m + 1)r−2 = −F (r)r−2 .
(4.17)
(4.18)
Then the above inequality (4.17) can be rewritten as r−2
d 2 (r N (v, m, r)) ≥ 2(1 + 2m)r−1 |w0 |2 + r−1 (p(r) − r−1 Fr (r))|w|2 dr −2 Re ((r−2 F (r) − C1 )w, w0 ) ,
where p(r) is as in (4.4).
(4.19)
¨ THE UNIQUENESS OF THE SOLUTION OF THE SCHRODINGER EQUATION WITH
...
973
(III) It follows from Assumption 4.2, (iii) and Proposition 4.3, (ii), (iii) that
p(r) − r−1 Fr (r) ≥ p(r) − c1 r−2 = r−2 (r2 p(r) − c1 ) , |r−2 F (r) − Q1 (x)| ≤ r−1 (2c0 p(r))1/2 + r−1 (2p(r))1/2 √ √ = c2 r−1 p1/2 (r) (c2 = 2c0 + 2) .
(4.20)
Further, from (iii) of Proposition 4.3 and (ii) of Assumption 4.2 we see that r2 p(r) ≥ (2c0 )−1 F 2 (r) → ∞ (r → ∞) , and hence there exists r0 > R0 such that −2 2 r (r p(r) − c1 ) = p(r) 1 −
c1 r2 p(r)
≥ 2−1 p(r)
(4.21)
(4.22)
for r ≥ r0 . Thus, we obtain from (5.19) r−2
d 2 (r N (v, m, r)) dr ≥ 2(1 + 2m)r−1 |w0 |2 + 2−1 r−1 p(r)|w|2 − 2c2 r−1 p1/2 (r)|w0 ||w| = r−1 [2(1 + 2m)|w0 |2 + 2−1 p(r)|w|2 − 2c2 p1/2 (r)|w0 ||w|]
(4.23)
for r ≥ r0 . Therefore, there exists a sufficiently large m0 > 0 such that r2
d 2 (r N (v, m, r)) ≥ 0 dr
for r ≥ r0 and m ≥ m0 , which completes the proof.
(4.24)
5. Uniqueness Theorem We are going to prove our main theorem (Theorem 5.10) which shows, under Assumptions 2.1 and 4.1, and some additional conditions (Assumptions 5.5 and 5.8), that the solution u has compact support if u satisfies ∂u 2 − Re (q(x))|u|2 dS = 0 . ∂r |x|=r
Z lim r→∞
(5.1)
Proposition 5.1. Suppose that Assumptions 2.1 and 4.2 hold. Suppose that the support of u is unbounded. Let r0 and m0 be as in Proposition 4.4. Then there exist m1 ≥ m0 and r1 ≥ r0 such that N (v, m1 , r) > 0
(r ≥ r1 ) .
(5.2)
¨ ¯ W. JAGER and Y. SAITO
974
Proof. Since the support of u is assumed to be unbounded, there exists r1 ≥ r0 such that |v(r1 )| > 0. Since r1−2m N (v, m, r1 ) = r1−2m {|w0 (r1 )|2 − (C0 (r1 )w(r1 ), w(r1 )) −|B 1/2 (r1 )w(r1 )|2 + (m(m + 1) − F (r1 ))|w(r1 )|2 } ≥ −(C0 (r1 )v(r1 ), v(r1 )) − |B 1/2 (r1 )v(r1 )|2 +(m(m + 1) − F (r1 ))|v(r1 )|2 ,
(5.3)
we can choose a sufficiently large m1 ≥ m0 so that r1−2m1 N (v, m1 , r1 ) > 0 ,
or r12 N (v, m1 , r1 ) > 0 .
(5.4)
Note that, by (ii)-2 of Assumption 2.1, N (r, m, v) is right-continuous. Then Ineq. (5.4) is combined with (4.13) and Lemma A in Appendix to see that r2 N (r, m1 , v) > 0 on [r1 , ∞), which completes the proof. Definition 5.2. Suppose that Assumptions 2.1 and 4.2 hold. Suppose that the support of u is unbounded. Let F (r) and m1 be given in Definition 4.1 and Proposition 5.1, respectively. Then we introduce the following two alternative cases: Case I : There exists an infinite sequence {r`0 } such that R0 < r`0 , r`0 → ∞ as ` → ∞, and (5.5) 2 Re (v 0 (r`0 ), v(r`0 )) ≤ (2m1 r`0 )−1 F (r`0 )|v(r`0 )|2 for all ` = 1, 2, . . . . Case II : There exists r2 > r1 such that 2 Re (v 0 (r), v(r)) > (2m1 r)−1 F (r)|v(r)|2
(r ≥ r2 ) ,
(5.6)
where r1 is as in Proposition 5.1. Proposition 5.3. Suppose that Assumptions 2.1 and 4.2 hold. Suppose that the support of u is unbounded. Suppose that Case I in Definition 5.2 holds. Then there exists an infinite sequence {r`00 } such that R0 < r`00 , r`00 → ∞ as ` → ∞, and M + (v, r`00 ) > 0
(` = 1, 2, . . .) .
(5.7)
Proof. Let {r`0 } be as in Case I of Definition 5.2. Let w = rm1 v, where m1 is as in Proposition 5.1. Then we have for r = r`0 r−2m1 |w0 |2 = |v 0 + m1 r−1 v|2 = |v 0 |2 + 2m1 r−1 Re (v 0 , v) + m21 r−2 |v|2 ≤ |v 0 |2 + m1 r−1 (2m1 r)−1 F (r)|v|2 + m21 r−2 |v|2 = |v 0 |2 + (2−1 F (r) + m21 )r−2 |v|2 .
(5.8)
¨ THE UNIQUENESS OF THE SOLUTION OF THE SCHRODINGER EQUATION WITH
...
975
Let r1 be as in Proposition 5.1. For r = r`0 such that r`0 ≥ r1 , it follows that 0 < N (v, m1 , r) = M + (w, r) + (m1 (m1 + 1) − F (r))r−2 |w|2 ≤ r2m1 {|v 0 |2 + (2−1 F (r) + m21 )r−2 |v|2 } −r2m1 {(C0 v, v) + (Bv, v)} +r2m1 (m1 (m1 + 1) − F (r))r−2 |v|2 = r2m1 {M + (v, r) + (m1 (2m1 + 1) − 2−1 F (r))r−2 |v|2 } .
(5.9)
Since F (r) → ∞ as r → ∞, there exists a positive integer `0 such that m1 (2m1 + 1) − 2−1 F (r`0 )) < 0 (` ≥ `0 ) .
(5.10)
Therefore we have only to define r`00 by r`00 = r`0 0 +`
(` = 1, 2, . . .) ,
(5.11)
which completes the proof.
Proposition 5.4. Suppose that Assumptions 2.1 and 4.2 hold. Suppose that the support of u is unbounded. Suppose that Case II in Definition 5.2 holds. Suppose, in addition, that (5.12) Re Q(x) ≤ 0 (x ∈ ER0 ) . Then there exist r3 > R0 and a positive constant c2 such that M (v, r) ≥ c2
(r ≥ r3 ) ,
(5.13)
where M (v, r) is given by (4.3). Proof. Since F (r) → ∞ as r → ∞, there exists r4 > R0 such that F (r) ≥2 2m1
(r ≥ r4 ) .
(5.14)
Then it follows from (5.6) that d |v(r)|2 ≥ 2r−1 |v(r)|2 dr
(r ≥ r4 ) .
Let r3 be such that r3 ≥ r4 and |v(r3 )| > 0. Then, since d d −2 2 −2 2 −1 2 (r |v(r)| ) = r |v(r)| − 2r |v(r)| ≥ 0 dr dr we have
r−2 |v(r)|2 ≥ r3−2 |v(r3 )|2 > 0
(5.15)
(r ≥ r4 ) ,
(r ≥ r3 ) .
(5.16)
(5.17)
Also, using (5.6) and (5.14) again, we see that 2r−1 |v(r)|2 ≤ (2m1 r)−1 F (r)|v(r)|2 ≤ 2|v(r)||v 0 (r)| ,
(5.18)
¨ ¯ W. JAGER and Y. SAITO
976
or r−1 |v(r)| ≤ |v 0 (r)|
(5.19)
for r ≥ r4 . Thus, it follows from (5.17) and (5.19) that |v 0 (r)|2 ≥ r−2 |v(r)|2 ≥ r2−2 |v(r3 )|2 > 0 for r ≥ r3 , which is combined with (5.12) to obtain (5.13).
(5.20)
Assumption 5.5. (i) Let h(r) be as above. Then h ∈ L1 ((R0 , ∞)). (ii) There exists a constant β ∈ (0, 1) such that 0 ≥ βQ0 (x) ≥ Re (Q(x))
(x ∈ ER0 ) .
(5.21)
Theorem 5.6. Suppose that Assumptions 2.1, 4.2 and 5.5 hold. Suppose that the support of u is unbounded. Then there exist a positive constant c3 and R2 > R0 such that (5.22) M (v, r) ≥ c3 (r ≥ R2 ) . Proof. Note that all the assumptions that are necessary for the conclusions of Propositions 3.2, 3.3, 4.3, 4.4, 5.1, 5.3 and 5.4 are satisfied. Suppose that Case I of Definition 5.2 is satisfied. Then, by Proposition 5.3 there exists R20 > R0 such that M + (v, R20 ) > 0. Therefore, setting R1 = R20 in Proposition 3.3, we have for r ≥ R20 , ! Z r
M + (v, r) ≥ exp −
R03
Z ≥ exp −
h(t) dt M + (v, R20 )
∞
R03
! h(t) dt M + (v, R20 ) .
(5.23)
Since we have from (5.21) M (v, r) = |v 0 (r)|2 − (CR (r)v(r), v(r)) ≥ |v 0 (r)|2 − β(C0 (r)v(r), v(r)) ≥ β|v 0 (r)|2 − β(C0 (r)v(r), v(r)) − β|B 1/2 (r)v(r)|2 = βM + (v, r) ,
(5.24)
it follows from (5.23) that M (v, r) ≥ c03 with c03
Z = β exp −
∞
R03
(r ≥ R20 )
(5.25)
! h(t) dt M + (v, R20 ) .
(5.26)
¨ THE UNIQUENESS OF THE SOLUTION OF THE SCHRODINGER EQUATION WITH
...
977
Suppose that Case II of Definition 5.2 is satisfied. Then from Proposition 5.4 we have (5.27) M (v, r) ≥ c2 (r ≥ r3 ) , where c2 and r3 are as in Proposition 5.4. Now set ( c3 = min{c03 , c2 } ,
(5.28)
R2 = max{R20 , r3 } .
Then (5.22) follows, which completes the proof.
Corollary 5.7. Suppose that Assumptions 2.1, 4.2 and 5.8 hold. Suppose that lim M (v, r) = 0 .
(5.29)
r→∞
Then u has compact support. In order to show our main theorem (Theorem 5.10) we need one more assumption. Assumption 5.8. We have 2 lim r inf Re (−q(x)) = ∞ . r→∞
|x|=r
(5.30)
Before we state and prove Theorem 5.10, we are going to unify Assumptions 2.1, 4.2, 5.5 and 5.8 in more organized form: Assumption 5.9. (1) Let N be an integer such that N ≥ 2. Let u ∈ H 2 (ER0 )loc , R0 > 0, be a solution of the homogeneous Schr¨ odinger Eq. (2.1), where ER0 is given by (2.2) (with R = R0 ). Here q(x) is a complex-valued, measurable, locally bounded function on ER0 which satisfies (5.30). (2) Set (N − 1)(N − 3) . (5.31) Q(x) = q(x) + 4r2 (a) Then Q(x) is decomposed as Q(x) = Q0 (x) + Q1 (x) ,
(5.32)
where Q0 (x) is a non-positive, measurable, locally bounded function on ER0 and Q1 (x) is a complex-valued, measurable, locally bounded function on ER0 such that 0 ≥ βQ0 (x) ≥ Re (Q(x)) with a constant β ∈ (0, 1).
(x ∈ ER0 )
(5.33)
¨ ¯ W. JAGER and Y. SAITO
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(b) For any φ ∈ X = L2 (S N −1 ), (Q0 (r·)φ, φ) has the right limit for all r > R0 as a function of r = |x|, where ( , ) is the inner product of X. (c) There exist h0 > 0 and, for 0 < h < h0 , a real-valued, measurable function Q0r (x; h) on ER0 such that (2.6), (2.7) and (2.8) hold. (d) There exists h(r) ∈ L1 ((R0 , ∞)) such that 0 < h(r) ≤
2 r
(r > R0 ) ,
(5.34)
and, setting a(r) = h−1 (r) sup |Q1 (x)| , |x|=r b(r) = inf [−(Q0 (x) + h−1 (r)Q0r (x))] , |x|=r (h−1 (r) = 1/h(r)) ,
(5.35)
we have a(r)2 ≤ b(r)
(r > R0 ) .
(5.36)
(3) The function F (r) introduced in Definition 4.1 satisfies Assumption 4.2. Theorem 5.10. Suppose that Assumptions 5.9 hold. Suppose that the solution u satisfies (5.1). Then u has compact support. Proof. Note that 0 ≤ |v 0 (r)|2 − (CR (r)v(r), v(r)) = |(r(N −1)/2 u(r·))0 |2 − rN −1 (Re (Q(r·))u(r·), u(r·)) = rN −1 |∂r u(r·) + 2−1 (N − 1)r−1 u(r·)|2 −rN −1 (Re (Q(r·))u(r·), u(r·)) ≤ 2rN −1 |∂r u(r·)|2 + 2−1 (N − 1)2 rN −3 |u(r·)|2 (N − 1)(N − 3) −rN −1 Re (q(r·)) + u(r·), u(r·) , 4r2 ≤ 2rN −1 |∂r u(r·)|2 +
N 2 − 1 N −3 r |u(r·)|2 4
−rN −1 (Re (q(r·))u(r·), u(r·)) ,
(5.37)
where ∂r = ∂/∂r and we have used
Q(x) = q(x) +
(N − 1)(N − 3) , 4r2
2 2 (N − 1) − (N − 1)(N − 3) = N − 1 . 2 4 4
(5.38)
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Therefore we have 0 ≤ |v 0 (r)|2 − (CR (r)v(r), v(r)) ≤ 2rN −1 {|∂r u(r·)|2 − (Re (q(r·))u(r·), u(r·))} −rN −3 ([−r2 Re (q(r·)) − 4−1 (N 2 − 1)]u(r·), u(r·)) .
(5.39)
It follows from (1) of Assumption 5.9 (5.30) that there exists R3 > R0 such that −r2 Re (q(x)) − 4−1 (N 2 − 1) > 0
(|x| = r, r ≥ R3 ) ,
(5.40)
and hence, for |x| ≥ R3 , 0 ≤ |v 0 (r)|2 − (CR (r)v(r), v(r)) ≤ 2
∂u 2 − Re (q(x))|u|2 dS , ∂r |x|=r
Z
(5.41)
which, together with (5.1), implies (5.29). Thus Corollary 5.7 can be applied to see that u has compact support, which completes the proof. 6. Examples In this section we are going to give some applications of Theorem 5.10. Example 6.1. Let R > 0 and let u ∈ H 2 (ER )loc be a solution of the equation (−∆ + V` (x) + Vs (x) − λ(x))u = 0
(x ∈ ER ) .
(6.1)
Here λ(x) is a real-valued, measurable, locally bounded function on ER satisfying the following (i) and (ii): (i) There exists m0 > 0 such that λ(x) ≥ m0
(x ∈ ER ) .
(6.2)
(ii) For any φ ∈ X the function fφ (r) = (λ(r·)φ, φ)
(6.3)
is a right continuous, nondecreasing function on (R, ∞). The functions V` (x) and Vs (x) are real-valued and complex-valued functions, respectively, satisfying the following (iii) and (iv): (iii) The long-range potential V` (x) is assumed to be C 1 function on ER such that lim sup |V` (x)| = 0 , r→∞ |x|=r (6.4) 1+ ∂V` < ∞ |x| sup ∂|x| r>R,|x|=r with ∈ (0, 2).
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(iv) The short-range potential Vs (x) is assumed to be measurable such that {r1+ |Vs (x)|} < ∞ ,
sup
(6.5)
r>R,|x|=r
where is as above. Suppose, in addition, that ∂u 2 + λ(x)|u|2 dS = 0 . ∂r |x|=r
Z lim r→∞
(6.6)
Then u is identically zero in ER . In fact, set Q0 (x) = −λ(x) + V` (x) , Z r+h ∂V` −1 (sω) ds Q0r (rω; h) = h ∂r r
Q0r (x) =
(ω ∈ S N −1 , h > 0) ,
∂V` , ∂r
Q1 (x) = Vs (x) +
(6.7)
(N − 1)(N − 3) . 4r2
Then, since Re (−q(x)) = λ(x) − V` (x) − Re (Vs (x)) ≥ m0 − V` (x) − Re (Vs (x))
(6.8)
(1) of Assumption 5.9 is satisfied for sufficiently large r. For φ ∈ X, we have 1 1 ([Q0 ((r + h)·) − Q0 (r·)]φ, φ) = − ([λ((r + h)·) − λ(r·)]φ, φ) h h 1 + ([V` ((r + h)·) − V` (r·)]φ, φ) h ≤ (Q0r (r·; h)φ, φ) → (Q0r (r·)φ, φ)
(6.9)
as h → 0 with h > 0. Thus 2(c) of Assumption 5.9 is satisfied. Set h(r) = r−1−/2
(r > R) .
(6.10)
Then h(r) ∈ L1 ((R, ∞)) and Ineq. (5.34) is satisfied for sufficiently large r. Also we have a(r) → 0 as r → ∞ and b(r) ≥ m0 /2 for sufficiently large r, and hence 2(d) of Assumption 5.9 is now satisfied. Noting that βQ0 (x) − Re (Q(x)) = (1 − β)λ(x) + (β − 1)V` (x) − Re (Vs (x)) ,
(6.11)
and that λ(x) ≥ m0 (6.2), we see that 2(a) of Assumption 5.9 holds for sufficiently large r with any β ∈ (0, 1). The condition 2(b) of Assumption 5.9 is verified by (ii) of Example 6.1 and the smoothness of V` (x). Define F (r) by F (r) = log r. Obviously (ii) and (iii) of Assumption 4.2 are satisfied by definition. Since r4 h2 (r)b(r) = r2− (λ(x) + o(1))
(6.12)
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as r → ∞, (4.5) in Assumption 4.2 holds for sufficiently large r. Therefore, by setting R0 sufficiently large, all the conditions of Assumption 5.9 are satisfied, which implies that the solution u has compact support in ER . Therefore it follows from the unique continuation theorem that u is identically zero in ER . We remark here that, if λ(x) is assumed to be bounded from above, too, then the condition (6.6) is equivalent to Z ∂u 2 2 (6.13) lim + |u| dS = 0 . ∂r r→∞ |x|=r Another remark is that, if Vs (x) is real-valued, then the condition (6.6) is implied by the generalized radiation condition p ∂u − i λ(x)u ∈ L2 (ER ) (6.14) rδ−1 ∂r with δ > 1/2 and R > R. Example 6.2. Let R > 0 and let u ∈ H 2 (ER )loc be a solution of the equation 1 ∆ − λ u = 0 (x ∈ ER ) (6.15) − µ(x) Here λ > 0 and the real-valued function µ(x) on ER is decomposed as µ(x) = µ0 (x) + µ` (x) + µs (x)
(x ∈ ER ) ,
(6.16)
where µ0 (x), µ` (x) and µs (x) satisfy the following (i)–(iv): f0 > 0 such that (i) µ0 (x) is real-valued and measurable and there exists m f0 µ0 (x) ≥ m
(x ∈ ER ) .
(6.17)
(ii) For any φ ∈ X the function gφ (r) = (µ0 (r·)φ, φ)
(6.18)
is a right continuous, nondecreasing function on (R, ∞). (iii) The real-valued function µ` satisfies Example 6.1(iii) with V` (x) replaced by µ` . (iv) The complex-valued function µs (x) satisfies Example 6.1(iv) with Vs (x) replaced by µs . Set
λ(x) = λµ0 (x) , V` (x) = λµ` (x) , Vs (x) = λµs (x) .
(6.19)
Then u satisfy Eq. (6.1) in Example 6.1, where λ(x), V` (x) and Vs (x) satisfy (i)–(iv) in Example 6.1. Thus the condition Z ∂u 2 (6.20) lim + λµ0 (x)|u|2 dS = 0 ∂r r→∞ |x|=r implies that u is identically zero.
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7. Reduced Wave Operator in Layered Media In [8] we considered the reduced wave operator H=−
1 ∆ in H = L2 (RN , µ(x) dx) , µ(x)
(7.1)
where µ(x) is a real-valued function such that 0 < inf µ(x) ≤ sup µ(x) < ∞ . x
(7.2)
x
By defining the domain D(H) of H by D(H) = H 2 (RN ), where H 2 (RN ) is the second order Sobolev space on RN , H becomes a self-adjoint operator on H. In this section we shall show that the nonexistence of the eigenvalues of H can be proved in some cases discussed in [8] by using the result of Sec. 6 (Example 6.2). Suppose that µ(x) has the decomposition (6.16) with a position function µ0 , a long-range perturbation µ` and a short-range perturbation µs . The functions µ0 , µ` , µs are assumed to satisfy (i)–(iv) of Example 6.2. In [8], for the sake of simplicity, we assumed that only one of a long-range perturbation or short-range perturbation appeared with the main term µ0 (x), but we can easily modify the arguments in [8] so that we can treat µ(x) of the form (6.16). Let K− be a nonpositive integer or K− = −∞ and let K+ be a nonnegative integer or K+ = ∞. Let K be a set of integers given by (7.3) K = {k/K− ≤ k ≤ K+ } . Let {Ωk }k∈K be a sequence of open sets of RN such that Ωk ∩ Ω` = ∅ (k 6= `) , [ Ωk = R N ,
(7.4)
k∈K
where A is the closure of A. Further we assume that the boundary ∂Ωk of Ωk has the form (−) (+) (7.5) ∂Ωk = Sk ∪ Sk , (−)
(+)
(−)
(+)
where Sk ∩ Sk = ∅, and each of Sk and Sk is a continuous surface which is a finite union of smooth surfaces. We also assume that (+) (−) (k ∈ K) , Sk = Sk+1 (+) (−) SK+ = SK+ +1 = ∅ (if K+ 6= ∞) , (7.6) S (−) = ∅ (if K− 6= −∞) . K− Now the function µ0 (x) is assumed to be a simple function which takes a constant value νk on each Ωk such that {νk }k∈K ia a bounded, positive sequence. We assume that the origin 0 of the coordinates is in Ω0 , and µ0 (x) satisfies the condition (+) (−) (7.7) (νk+1 − νk )(n(k) (x) · x) ≥ 0 x ∈ Sk = Sk+1 , k ∈ K ,
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where n(k) (x) is the unit outward normal of Ωk at x ∈ ∂Ωk and n(k) (x)·x is the inner product of n(k) (x) and x in RN . Then the following theorem has been obtained in [8] ([8, Theorem 4.6]): Theorem 7.1. Let H be as above. Suppose, in addition, that µ takes the form of either µ = µ0 + µs or µ = µ0 + µ` . Let σp (H) be the set of the point spectrum of H. Then the multiplicity of each λ ∈ σp (H) is finite, σp (H) does not have any accumulation points except at 0 and ∞. It is not difficult to extend this result to the general case that µ = µ0 + µs + µ` . Using the Example 6.2, we can show a sufficient condition for the nonexistence of the point spectrum of the operator H. Theorem 7.2. Let H be as above. Suppose that, for almost all ω ∈ S N −1 , µ0 (rω) is a nondecreasing function of r ∈ [0, ∞). Then σp (H) = 0. Proof. The condition (ii) of Example 6.2 is now satisfied since µ0 (rω) is nondecreasing. Here we are going to give some examples. N such that Example 7.3. Let {Uk }∞ k=0 be a sequence of open sets of R
Uk ⊂ Uk+1 (k ≥ 0) , ∞ [ Uk = RN ,
(7.8)
k=0
where the boundary ∂Uk of Uk is a continuous surface which is a finite union of smooth surfaces. Suppose that n ˜ (k) (x) · x ≥ 0 (k = 0, 1, 2, . . .) ,
(7.9)
where n ˜ (k) (x) is the unit outward normal of Uk at x ∈ ∂Uk . Set Ω0 = U 0 , Ωk = Uk \Uk−1 (k ≥ 1) , (+) Sk = ∂Uk (−) Sk = ∂Uk−1
(k ≥ 0) ,
(7.10)
(k ≥ 1) .
This is the case that K− = 0 and K+ = ∞. Let µ0 (x) be given by µ0 (x) = νk
(x ∈ Ωk ) ,
(7.11)
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where {νk }∞ k=0 ia a bounded, positive, increasing sequence. Then we see that not only the condition (7.7) is satisfied but also µ0 (rω) is a nondecreasing function of r ∈ [0, ∞) for almost all ω ∈ S N −1 . Thus Theorem 7.2 can be applied to see that there is no point spectrum of H. Therefore the limiting absorption principle holds on the whole positive interval (0, ∞) (see Sec. 5 of [8]). Example 7.4. Let {ck /k = ±1, ±2, . . .} be an increasing sequence of real numbers such that c−1 < 0 < c1 , (7.12) lim ck = ±∞ . k→±∞
Let xN be the N th coordinate of x = (x1 , x2 , . . . , xN ), and set {x ∈ RN /c−1 < xN < c1 } (k = 0) , N Ωk = {x ∈ R /ck−1 < xN < ck } (k = −1, −2, . . .) , {x ∈ RN /c < x < c } (k = 1, 2, . . .) . k
N
(7.13)
k+1
We also set (±)
S0
= {x ∈ RN /xN = c±1 } , (
(+) Sk
=
(−) Sk
{x ∈ RN /xN = ck }
(k = −1, −2, . . .) ,
{x ∈ R /xN = ck+1 }
(k = 1, 2, . . .) ,
{x ∈ RN /xN = ck−1 }
(k = −1, −2, . . .) ,
{x ∈ RN /xN = ck }
(k = 1, 2, . . .) ,
N
(
and =
(7.14) (7.15)
(7.16)
(+)
Note that, for x ∈ Sk ,
( n(k) (x) · x
≥0
(k ≥ 0) ,
≤0
(k < 0) .
(7.17)
Define a simple function µ0 (x) by (7.11), where the sequence {νk }∞ k=−∞ is assumed ∞ is decreasing and {ν to be bounded and positive such that {νk }−1 k }k=1 is increask=−∞ ing. Then, as in Example 7.3, Theorem 7.2 can be applied to show that σp (H) = ∅. The planes {x ∈ RN /xN = ck } can be perturbed as far as the condition (7.17) is satisfied. Appendix Here we are going to prove a lemma on distributions on a half interval (a, ∞) which was used when we evaluate the functionals M + (v, r) and N (v, m, r). Lemma A. Let f (r) be a real-valued function on I = (a, ∞) such that f is locally L1 and right continuous on I. Suppose that f 0 ≥ 0, where f 0 is the distributional derivative of f and the inequality should be taken in the sense of distributions. Then f is nondecreasing on I.
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Proof. Here we are giving a rather elementary proof. (I) Let r ∈ I and h > 0. Then, for φ ∈ C0∞ (I), we have Z Z [f (r + h) − f (r)]φ(r) dr = − f (r)[φ(r) − φ(r − h)] dr I
I
Z
Z =−
r
f (r) I
φ0 (s) ds dr ,
(A.1)
r−h
where φ is supposed to be extended on the whole line (−∞, ∞) by setting φ(r) = 0 for r ≤ a. Since Z h Z r 0 φ (s) ds = φ0 (t + r − h) dt , (A.2) r−h
0
it follows that Z Z [f (r + h) − f (r)]φ(r) dr = I
0
h
Z − f (r)φ0 (t + r − h) dr dt.
(A.3)
I
(II) Let φ ∈ C0∞ (I) and φ ≥ 0. Then, since Z − f (r)φ0 (t + r − h) dr = hf 0 , φ(· + t − h)i ≥ 0
(A.4)
I
for h > 0 and 0 ≤ t ≤ h, where hF, Gi denotes the value of the distribution F for the test function G, it follows from (A.3) that Z [f (r + h) − f (r)]φ(r) dr ≥ 0 (A.5) I
for any φ ∈ C0∞ (I) with φ ≥ 0. (III) Suppose that there exist r0 ∈ I, h0 > 0 and η0 > 0 such that f (r0 + h0 ) − f (r0 ) = −η0 . Since f is right continuous, there exists r1 > r0 such that ( |f (r0 ) − f (r)| < η0 /3 , |f (r0 + h0 ) − f (r + h0 )| < η0 /3
(A.6)
(A.7)
for r0 ≤ r ≤ r1 . Then, for r0 ≤ r ≤ r1 , we have f (r + h0 ) − f (r) = f (r0 + h0 ) − f (r0 ) + {f (r + h0 ) − f (r0 + h0 )} + {f (r0 ) − f (r)} ≤ f (r0 + h0 ) − f (r0 ) + |f (r + h0 ) − f (r0 + h0 )| + |f (r0 ) − f (r)| < −η0 /3 .
(A.8)
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Let φ ∈ C0∞ (I) such that
supp φ ⊂ [r0 , r1 ] , φ ≥ 0, Z r1 φ(r) dr = 1 .
(A.9)
r0
Then, it follows that Z Z [f (r + h) − f (r)]φ(r) dr = I
r1
r0
≤− =−
η0 3
[f (r + h) − f (r)]φ(r) dr Z
r1
φ(r) dr r0
η0 3
< 0, which contradicts (A.5). This completes the proof.
(A.10)
Acknowledgements This work was finished when the second author was visiting the University of Heidelberg for February 1997. Here he would like to thank Deutsche Forschungs Gemeinschaft for its support through SFB 359. Also the second author is thankful to Professor Willi J¨ ager for his kind hospitality during this period. References [1] D. Eidus, “The principle of limiting absorption”, Amer. Math. Soc. Translations 47 (1965) 157–191 (Mat. Sb. 57 (1962)). [2] D. Eidus, “The limiting absorption and amplitude problems for the diffraction problem with two unbounded media”, Commun Math. Phys. 107 (1986) 29–38. [3] T. Ikebe, “Eigenfunction expansions associated with the Schr¨ odinger operators and their applications to scattering theory”, Arch. Rational Mech. Anal. 5 (1960) 1–34. [4] T. Ikebe and Y. Sait¯ o, “Limiting absorption method and absolute continuity for the Schr¨ odinger operator”, J. Math. Kyoto Univ. 12 (1972) 513–612. [5] T. Ikebe and J. Uchiyama, “On the asymptotic behavior of eigenfunctions of secondorder elliptic differential operators”, J. Math. Kyoto Univ. 11 (1971) 425–448. [6] W. J¨ ager, “Zur Theorie der Schwingugsgleichung mit variablen Koeffizienten in Aussengebieten”, Math. Z. 102 (1969) 62–88. [7] W. J¨ ager and Y. Sait¯ o, “On the Spectrum of the Reduced Wave Operator with Cylindrical Discontinuity”, Forum Mathematicum 9 (1997) 29–60. [8] W. J¨ ager and Y. Sait¯ o, “The reduced wave equation in layered materials”, to appear in Osaka J. Math. [9] T. Kato, “Growth properties of solutions of the reduced wave equation with a variable coefficients”, Commun. Pure Appl. Math. 12 (1959) 403–425. [10] C. M¨ uller, Grundprobleme der Mathematischen Theorie Elektromagnetischer Schwingungen, Springer, Berlin 1957. ¨ [11] F. Rellich, “Uber das asymptotische Verhalten des L¨ osungen von ∆u + λu = 0”, Jber. Deutsche. Math. Verein. 53 (1943) 57–65.
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[12] G. Roach and B. Zhang, “On Sommerfeld radiation conditions for the diffraction problem with two unbounded media”, Proc. Royal Soc. Edinburgh 121A (1992) 149– 161. [13] Y. Sait¯ o, Spectral Representations for Schr¨ odinger Operators with Long-Range Potentials, Lecture Notes in Mathematics, Vol. 727, Springer, Berlin, 1979. [14] J. Weidmann, “On the Continuous spectrum of Schr¨ odinger operators”, Commun. Pure and Appl. Math. 19 (1966) 107–110. [15] E. Wienholtz, “Halbbeschr¨ anke partielle Differentialoperatoren zweiter Ordnung vom elliptischen Typus”, Math. Ann. 135 (1958) 50–80.
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD ERIK SKIBSTED Institut for Matematiske Fag Aarhus Universitet Ny Munkegade 8000 Aarhus C Denmark E-mail : [email protected] Received 16 April 1997 We develop an extension of the abstract Mourre theory which consecutively is used to prove spectral properties of various systems coupled to a massless bosonic field. Our models include the spin-boson model and the standard model of quantum electrodynamics for a non-relativistic atom considered recently in [8] and [3] respectively.
1. Introduction It is well known that Mourre’s commutator method [9] is a powerful machinery for proving basic spectral properties of N -body Schr¨ odinger operators H: Letting F (H), σpp (H) and σsc (H) denote the set of thresholds (i.e. eigenvalues of sub-Hamiltonians), the set of eigenvalues and the continuous singular spectrum, respectively, these properties include: (1) F (H) is closed and countable. (2) The eigenvalues (counted with multiplicity) can only a‘ccumulate at F (H). (3) There is a limiting absorption principle away from F (H) ∪ σpp (H). In particular σsc (H) = ∅. The purpose of the paper is to prove these or similar properties for various systems coupled weakly to a massless bosonic field. We discuss N -body as well as finite number of states systems motivated by the recent papers [3] and [8], respectively. For the spin-boson Hamiltonians of [8] the above properties should be replaced by: (2)0 σpp (H) is finite. (3)0 There is a limiting absorption principle away from σpp (H). In particular σsc (H) = ∅. To explain a basic idea of [8] let us consider bosons with positive mass m which p 2 means that the 1-boson energy function is given by ω(k) = |k| + m2 . We introduce B = 12 (F · p + p · F ); F (k) = −ω(k)|k|−2 k, p = −i∇k , as an operator on the 1-boson space L2 (Rνk ), and compute i[ω, B] = I. Upon lifting to the symmetric 989 Reviews in Mathematical Physics, Vol. 10, No. 7 (1998) 989–1026 c World Scientific Publishing Company
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E. SKIBSTED
Fock space F (by second quantization) this gives i[dΓ(ω), dΓ(B)] = N , where N denotes the “number operator”. Of course it is non-negative which in the applications of [8] is used to establish a Mourre estimate for spin-boson Hamiltonians. To get the consequences (2)0 and (3)0 listed above the authors developed an extended abstract Mourre theory since their “conjugate operator” is not selfadjoint and consequently does not meet the requirements of [9]. (Notice that the above conjugate operators B and A = dΓ(B) do not have self-adjoint realizations.) It is noticed in [8] that the massless case m = 0 is worse. As a matter of fact the authors did not obtain results in this case (except for a certain compressed model in which N is bounded). The basic obstacle for m = 0 is that N is not bounded relatively to H = dΓ(ω). More general for all nonzero real-valued f ∈ C0∞ (σ(H)), f (H)i[H, A]f (H) is unbounded.
(1.1)
In this paper we develop a Mourre theory that overcomes the above difficulty. We notice that (1.1) does not fit into the existing refinements of Mourre’s paper (see for example [10, 4, 1, 12]). Of course our refinement of the Mourre theory involves new conditions. One of these is that the H-unbounded piece M of a decomposition i[H, A] = M + G has that property that i[H, M ] is H-bounded. (Notice that for the above example the latter condition is trivial: i[H, N ] = 0.) Another complication is the one mentioned above: A might not be self-adjoint. The procedure of [8] essentially consists in mimicking [9] using the semigroup generated by −iA (assumed to exist). One ingredient of our approach is the introduction of a sequence of self-adjoint operators (An ) which (by assumption) in some sense converges to A. Also each An is assumed to have the usual good commutator property i[H, An ] being H-bounded. Hence our procedure is facilitated by the basic technical results of [9] that are directly applicable for the approximating operators. For the example discussed above (1.1) and the lack of self-adjointness are due to the singularity at k = 0 of the vector field F , and therefore the approximating self-adjoint operators may naturally be constructed by suitably smoothing our the vector field. Our examples concern Hamiltonians defined on a Hilbert space of the form L2 (X) ⊗ F . The spin-boson model corresponds to X being finite. For N -body systems X is a finite dimensional vector space. In [3] the authors propose the conjugate operator A = Ael ⊗ I − I ⊗ Af , where Ael is the generator of the dilations and Af is the second quantized generator of the dilations lifted from the 1-photon space. The basic idea is to invoke the well-known Mourre estimate for “the N -body part” Hel of the Hamiltonian (given in terms of Ael ) and then show local positivity away from the thresholds of Hel for small perturbations of Hel ⊗ I + I ⊗ dΓ(ω) .
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This procedure does not yield to the properties (1)–(3) listed above even for perturbations for which there is a natural notion of thresholds for the perturbed Hamiltonians. Explicitly the authors do not prove spectral properties in neighbourhoods of thresholds of Hel . Moreover to get spectral information in neighbourhoods of eigenvalues of Hel they need implicit conditions (cf. Fermi’s Golden Rule) and an analyticity assumption on the perturbation. In this paper we shall consider perturbations of the above tensor sum for which there is a natural notion of thresholds for the full Hamiltonian. We shall obtain all of the properties (1)–(3) for some classes of perturbations. Our basic idea is to consider the (non-self-adjoint) conjugate operators A = CBR ⊗ I + I ⊗ dΓ(B) ,
(1.2)
where C and R are adjustable positive parameters, B is given as in the beginning of this section, and BR is somewhat similar to Ael . More precisely BR is given in terms of a scaled version of the vector field invented by Graf [7]. We notice that by a previous result of the author [14, Theorem B2] this operator provides another Mourre estimate for “the N -body part” Hel . In [14] the latter estimate had obvious applications in studying propagation properties with locally singular potentials. In the present context the use of BR is convenient not only for dealing with local singularities but for another reason too. (As a matter of fact we don’t see how to obtain (1)–(3) for non-trivial perturbations say by using (1.2) with BR replaced by Ael .) We emphasize that for all our examples we only have results for weak coupling. It is an open problem to go beyond this restriction for the classes of perturbations we consider. For the spin-boson Hamiltonians and our first model of an N -body system coupled to the bosonic field, the latter phrased as the electron-boson model, we give a very explicit bound on the perturbation (in fact similar to the one of [8] obtained for positive mass). For the more complicated model, the so-called standard model of quantum electrodynamics for a non-relativistic atom, we do not derive an explicit bound. (In that case our methods would probably not yield realistic bounds.) Although we shall not discuss it in this paper our limiting absorption principle may be used to show absence of embedded eigenvalues (in a weak coupling regime) in accordance to Fermi’s Golden Rule and to treat related issues. This does not require the notion of resonance as defined in [3] or any other such notion. We notice that in the theory of N -body Schr¨ odinger operators the analyticity assumption of [13] was similarly removed in the treatment of instability of embedded eigenvalues of [2]. In Sec. 2 we formulate and derive the extended abstract Mourre theory. In Sec. 3 we discuss in detail the applications to the spin-boson model. We prove (2)0 and (3)0 . In the next section we introduce “the electron-boson model” and do some preliminaries. As an easy corollary of the main result Proposition 4.4 we recover the Mourre estimate [14, Theorem B2]. In Sec. 5 we use the proposition to complete “the electron-boson model”, that is to establish (1)–(3). Finally in Sec. 6 we discuss the standard model of quantum electrodynamics for a non-relativistic atom. For a
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certain class of perturbations we obtain again the above properties. In that section we skip many details referring the reader to various procedures and preliminary results for the previously discussed models. 2. Abstract Theory For a (densely defined) operator T on a Hilbert space we denote by D(T ) and ρ(T ) its domain and resolvent set, respectively. Assumptions 2.1. Let H, M, An , n ∈ N, be self-adjoint operators on a Hilbert space H. Suppose M ≥ δI for some positive number δ, and that there exists a closed symmetric operator A on H with the properties −i ∈ ρ(A) and for some core C of A with C ⊆ D(An ) the identity limn→∞ An φ = Aφ holds for all φ ∈ C. Suppose (1) (Compatibility) The set D(H)∩D(M ) is dense in D(H) as well as in D(M ). The form i[H, M ] defined on this set extends to an H-bounded operator i[H, M ]0 , and sup kHeitM (H − i)−1 k < ∞ . |t| 12 , where we use the notation eitA for the semigroup generated by −iA. In order to prove Theorem 2.4 we need some preliminary results. Lemma 2.6. Suppose Assumption 2.1 (1). Consider for ∈ R \ {0}H() = H − iM with the domain D = D(H) ∩ D(M ). The adjoint operator is given by H()? = H(−) . In particular z ∈ ρ(H()) for either Im z + δ and both positive or both negative. (Here δ refers to the delta of Assumptions 2.1.) Moreover in these cases the resolvent Rz0 () = (H() − z)−1 obeys the bound kRz0 ()k ≤ |Im z + δ|−1 .
(2.1)
Proof. Obviously H()? ⊇ H(−). Therefore for given ψ ∈ D(H()? ) we just need to show that ψ ∈ D. To that end we shall use that with C = kH()? ψk |hH()φ, ψi| ≤ Ckφk for all φ ∈ D .
(2.2)
For any non-real complex η we can compute (cf. [9]) M (H − η)−1 M −1 = (H − η)−1 − i(H − η)−1 i[H, M ]0 (H − η)−1 M −1 .
(2.3)
Since obviously the right-hand side is bounded we get that D is preserved by (H − η)−1 . Therefore we may replace φ in (2.2) by (H − η¯)−1 φ. Commuting the resolvent by using (2.3) we thus obtain |hM φ, (H − η)−1 ψi| ≤ Cψ,η kφk for all φ ∈ D .
(2.4)
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
995
Since (by assumption) D is dense in D(M ) we conclude that (H − η)−1 ψ ∈ D(M ). By repeating the argument we obtain that φκ = Dκ ψ ∈ D where for κ > 0, Dκ = H(κH − i)−1 (κH + i)−1 = κ−1 (I + i(κH − i)−1 )(κH + i)−1 . Inserting in (2.2) yields |Re hH()φκ , ψi| ≤ Ckφκ k ≤ CkH(κH + i)−1 ψk .
(2.5)
On the other hand, |Re hH()φκ , ψi| ≥ kH(κH + i)−1 ψk2 − | Im hM φκ , ψi| .
(2.6)
To estimate the last term we notice that (by using a technique of [9] cf. Lemma 2.2) Im hM φκ , ψi = lim
λ→∞
=
i h[M iλ(M + iλ)−1 , Dκ ]iψ 2
1 + 0 + hR H R + iR− H 0 R− R+ + iR− R+ H 0 R+ iψ ; 2
R+ = (κH + i)−1 ,
R− = (κH − i)−1 ,
H 0 = i[H, M ]0 .
In particular, 3 | Im hM φκ ψi| ≤ || kψk kH 0 (H − i)−1 k k(H − i)(κH + i)−1 ψk 2 ≤
1 kH(κH + i)−1 ψk2 + C1 kψk2 , 2
(2.7)
where C1 is independent of κ (and ψ). Combining (2.5), (2.6) and (2.7) yields (by subtraction) 1 kH(κH + i)−1 ψk2 − C1 kψk2 ≤ kH()? ψk kH(κH + i)−1 ψk , 2
(2.8)
from which we obtain the bound kH(κH + i)−1 ψk ≤ Cψ ,
(2.9)
the constant Cψ being independent of κ. Letting κ → 0 we obtain that ψ ∈ D(H). This fact combined with (2.2) implies that also ψ ∈ D(M ). The second part of the lemma follows from the first part just proven and the fact that the numerical range of H() is a subset of {η|Im η ≤ −δ}. Lemma 2.7. Suppose the assumption of Lemma 2.6 and that f ∈ C0∞ (R) is equal to one on a neighbourhood of E. Then there exist constants C, 0 > 0 and neighbourhood V of E such that k(H − i)(I − f (H))Rz0 ()k ≤ C provided || ≤ 0 , Im z > 0 and Re z ∈ V.
996
E. SKIBSTED
Proof. We pick an almost analytic extension f˜ ∈ C0∞ (C) of f so that we can represent Z 1 (∂¯f˜)(η)(H − η)−1 dudv , η = u + iv . (2.10) f (H) = π C In conjunction with (2.3) we obtain that M f (H)M −1 and [f (H), M ] are bounded.
(2.11)
In particular, for any φ ∈ H, ψ = (I − f (H))Rz0 ()φ ∈ D = D(H) ∩ D(M ) . We commute (H() − z)ψ = (I − f (H))φ − i[f (H), M ]Rz0 ()φ , yielding together with (2.11) and Lemma 2.6 k(H() − z)ψk ≤ k(I − f (H))φk + ||C1 kRz0 ()φk ≤ C2 kφk
(2.12)
for constants C1 and C2 independent of and z. We can also compute k(H() − z)ψk2 = k(H − Re z)ψk2 + k(M + Im z)ψk2 − hi[H, M ]iψ .
(2.13)
Clearly the last term on the right-hand may be estimated by |hi[H, M ]iψ | ≤ ||C3 k(H − i)ψk2 .
(2.14)
Combining (2.12), (2.13) and (2.14) gives the bound k(H − Re z)ψk2 + k(M + Im z)ψk2 ≤ C22 kφk2 + ||C3 k(H − i)ψk2 .
(2.15)
Clearly we can find a neighbourhood V of E and a positive number κ such that for all t ∈ R and s ∈ V, {(t − s)2 − 2κ|t − i|2 }|1 − f (t)|2 ≥ 0 .
(2.16)
For ||C3 ≤ κ and Re z(= s) ∈ V, (2.15) and (2.16) give the bound (by subtraction) (2.17) k(H − i)ψk2 ≤ κ−1 C22 kφk2 . In the rest of this section we impose Assumptions 2.1 with E not being an eigenvalue of H. We pick a real-valued f ∈ C0∞ (R) equal to one on a neighbourhood of E such that the form inequality M + f (H)Gf (H) ≥
α f (H)2 − (I − f (H))L(I − f (H)) 2
(2.18)
997
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD 1
holds on D(H) ∩ D(M 2 ). We would like to prove Lemmas 2.6 and 2.7 for the perturbed operator H() − if (H)Gf (H). Introducing the notation Rz () = (H − i(M + f (H)Gf (H)) − z)−1 for its resolvent, we have: Lemma 2.8. There exist constants C, 0 > 0 and a neighbourhood V of E such that kRz ()k ≤
C ||
,
(2.19)
k(H − i)(I − f (H))Rz ()k ≤ C
(2.20)
provided || ≤ 0 , Im z > 0 and Re z ∈ V. Proof. It is not obvious that the resolvent exists for all z in question. Clearly a perturbation argument based on (2.1) gives the existence for large vales of |Im z|. Below we shall prove (2.19) in a domain of the desired form assuming that the resolvent exists. Then by a simple connectedness argument it follows that it exists in the whole domain. So suppose z is given such that Rz () exists. Then from Rz () = Rz0 ()(I + if (H)Gf (H)Rz ()) and Lemma 2.7 we obtain (with 0 and V given as in Lemma 2.7) k(H − i)(I − f (H))Rz ()k ≤ C1 (1 + || kRz ()k) .
(2.21)
By (2.18) and Lemma 2.6, 2 Im Rz () 2 + Rz¯(−)(I − f (H))L(I − f (H))Rz () , Rz¯(−)f (H) Rz () ≤ α (2.22) yielding
kf (H)Rz ()k2 ≤ C2
kRz ()k + k(H − i)(I − f (H))Rz ()k2 ||
.
(2.23)
Combining (2.21) and (2.23) we obtain kRz ()k2 ≤ 2(k(I − f (H))Rz ()k2 + kf (H)Rz ()k2 ) kRz ()k 2 2 2 2 + C1 (1 + || kRz ()k) ≤ 2C1 (1 + || kRz ()k) + 2C2 .(2.24) || We may assume that 2C12 (1 + C2 )20 < 1. Then (by subtraction) (2.24) implies the bound (2.19). Upon combining with (2.21) we get (2.20). With Rz () given as in Lemma 2.8, we introduce the notation Fz () = (A + i)−1? Rz ()(A + i)−1 .
998
E. SKIBSTED
Lemma 2.9. In addition to the bounds of Lemma 2.8 we have (with a possibly larger constant C) kRz ()(A + i)−1 k ≤ ||− 2 C(1 + kFz ()k) 2 ,
(2.25)
kM Rz ()(A + i)−1 k ≤ C(1 + kFz ()k) 2
(2.26)
1
1
1
for all and z given as in the lemma. Proof. For any φ ∈ H, we put ψ = Rz ()(A + i)−1 φ. The expectation of (2.22) in the state (A + i)−1 φ gives 2 kFz ()k kφk2 + C1 k(H − i)(I − f (H))ψk2 . kf (H)ψk2 ≤ α || Upon combining with Lemma 2.8 we thus obtain 4 kFz ()k 2 2 2 + C2 kφk2 , kψk ≤ 2kf (H)ψk + 2k(I − f (H))ψk ≤ α ||
(2.27)
which clearly gives (2.25). As for (2.26) we compute (A + i)−1 φ + (H() − z)ψ − if (H)Gf (H)ψ , which in conjunction with (2.13), (2.14) and Lemma 2.8 leads to k(A + i)−1 φk2 ≥
1 k(H() − z)ψk2 − 2 C3 kψk2 2
≥
1 k(M + Im z)ψk2 − (||C4 + 2 C3 )k(H − i)ψk2 2
≥
1 kM ψk2 − 2(||C4 + 2 C3 )k(H − i)f (H)k2 kψk2 − C5 kφk2 . 2
So by (2.27), kM ψk2 ≤ C6 (|| kψk2 + kφk2 ≤ C6
4 kFz ()k + ||C2 + 1 kφk2 . α
which clearly gives (2.26).
Proof of Theorem 2.4. We shall use Lemmas 2.8 and 2.9 to prove the differential inequality
d
Fz () ≤ ||− 12 C(1 + kFz ()k) . (2.28)
d In conjunction with (2.19) this will give Theorem 2.4 (1) by three integrations (iterations) with respect to using the fact that for any φ ∈ H, lim Rz ()φ = (H − z)−1 φ ,
→0
which in turn is readily obtained from Assumption 2.1 (1) and (2.1).
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
999
To prove (2.28) we compute (for 6= 0), d Fz () = (A + i)−1? Rz ()i(M + f (H)Gf (H))Rz ()(A + i)−1 . d The middle term is rewritten as M + f (H)Gf (H) = (M + G) − (I − f )Gf − G(I − f ) .
(2.29)
(2.30)
Upon substituting (2.30) into the right-hand side of (2.29) we obtain three terms. The second and third terms are bounded by ||− 2 C(1 + kFz ()k) 2 (≤ ||− 2 c(1 + kFz ()k)) 1
1
1
(2.31)
by (2.20) and (2.25). It remains to bound the operator Bz () = (A + i)−1? Rz ()i(M + G)Rz ()(A + i)−1 .
(2.32)
For that we rewrite hφ1 , Bz ()φ2 i for φ1 , φ2 ∈ C1 = (A + i)C as follows. We insert (cf. Assumption 2.1 (2) and Lemma 2.2 (1)) M + G = limn→∞ limλ→∞ i[H, An (λ)] = T1 + · · · + T5 ; T1 = limn→∞ limλ→∞ i[H − i(M + f Gf ), An (λ)] , T2 = − limn→∞ limλ→∞ [M, An (λ)] , T3 = − limn→∞ limλ→∞ [f, An (λ)]Gf ,
(2.33)
T4 = − limn→∞ limλ→∞ f [G, An (λ)]f , T5 = − limn→∞ limλ→∞ f G[f, An (λ)] . The contribution from T1 is given by lim lim hφ1 , {(A + i)−1? i[Rz (), λAn (An + iλ)−1 ](A + i)−1 }φ2 i
n→∞ λ→∞
= hRz¯(−)(A + i)−1 φ1 , A(A + i)−1 φ2 i − hA(A + i)−1 φ1 , Rz ()(A + i)−1 φ2 i . (2.34) Here we used that for φ ∈ C1 , lim lim iλAn (An + iλ)−1 (A + i)−1 φ = lim An (A + i)−1 φ = A(A + i)−1 φ .
n→∞ λ→∞
n→∞
Since C1 is dense in H we obtain from (2.34), the Cauchy Schwarz inequality and (2.25) that the contribution from T1 to (2.32) is bounded by the constant (2.31). We are left with bounding the contributions from T2 , . . . , T5 to (2.32). For that we compute as forms on D(M ) using Assumptions 2.1, Lemmas 2.2 and 2.6, (2.10) and (2.3) T2 = iM∞ , Z 1 (∂¯f˜)(η)(H − η)−1 (M + G)(H − η)−1 dudv Gf , T3 = −i π C T4 = if G∞ f , Z 1 (∂¯f˜)(η)(H − η)−1 (M + G)(H − η)−1 dudv . T5 = −if G π C
(2.35)
1000
E. SKIBSTED
By (2.3) the terms T3 and T5 are of the form: T3 = M B and T5 = −B ? M ;
B bounded.
Hence by (2.25), (2.26) and the Cauchy Schwarz inequality all terms of (2.35) contribute to (2.32) by operators with the bounding constant ||− 2 C(1 + kFz ()k) . 1
We have proved (2.28). The second statement Theorem 2.4 (2) may be proved using (2.25) and (2.28) (without the last factor) exactly as in [10]. 3. The Spin-Boson Model Let F denoteR the symmetric Fock space over L2 (Rν ), ν ∈ N. Let m ∈ N. On H = Cm ⊗ F = X ⊗Fdx, X = {1, . . . , m}, we consider a Hamiltonian of the form: Z ⊕{a(λx ) + a? (λx )}dx , (3.1) H = S ⊗ I + I ⊗ Hf + V, V = X
where S is self-adjoint on C , Hf is the second quantization dΓ(w) on F of the operator of multiplication by the function ω(k) = |k| on L2 (Rν ), and a and a? are the operator of annihilation and creation, respectively, of a function λx (·) ∈ L2ω := L2 (Rν , (1 + ω(k)−1 )dk). We use the notation kλx kω for the corresponding weighted L2 -norm. It is well known (cf. [3] and [8]) that for any φ in the form domain of I ⊗ Hf ,
Z
Z
1
⊕a(λx )dxφ , ⊕a? (λx )dxφ ≤ sup kλx kω hI ⊗ Hf + Ii 2 . (3.2) φ
m
X
X
x∈X
Hence by the Kato–Rellich theorem [11, Theorem X.12] the Hamiltonian (3.1) with domain D(H) = Cm ⊗ D(Hf ) is self-adjoint. We shall impose the following stronger condition. Let for i, j ∈ {0, 1, 2}, i + j ≤ 2, x ∈ X i ∂ (i,j) −j λx (k) λx (k) = |k| ∂|k| considered as distributions on C0∞ (Rν \ {0}). We demand λx(i,j) (·) ∈ L2ω .
(3.3)
We can now verify Assumptions 2.1 (1)–(3). Up to this point we have only specified H. Consider the semigroup U (t) acting on L2 (Rν ) given by ν−1 2 k t φ (|k| − t) ,t ≥ 0. U (t)φ(k) = F (|k| > t) 1 − |k| |k| It is generated by −iB (that is U (t) = eitB ), where B is the closure on C0∞ (Rν \{0}) of k ∂ ∂ 1 k ·p+p· ,..., , p = −i . B=− 2 |k| |k| ∂k1 ∂kν
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1001
The A of Assumptions 2.1 is given by A = I ⊗ dΓ(B), where −idΓ(B) is the generator of the second quantized semigroup Γ(U (t)). Generating a contraction semigroup −1 ∈ ρ(−iA). Consequently, −i ∈ ρ(A). We specify (the last product consisting of all finite symmetric linear combinations of products of C0∞ -functions) C=
p [M
Cm ⊗ (C0∞ (Rν \ {0})⊗sq )(⊆ H) .
(3.4)
p∈N q=0
It follows immediately from [11, Theorem X.49] that C is a core of A. By the formal computations on F [dΓ(B), a? (λx )] = a? (Bλx ) , [dΓ(B), a(λx )] = −a(Bλx ) ,
(3.5)
we can (on a formal level) understand (3.3) as conditions guaranteeing H-boundedness of i[V, A] and i[i[V, A, A]. As for the commutator with I ⊗ Hf we have (still formally) i[I ⊗ Hf , A] = I ⊗ i[dΓ(ω), dΓ(B)] = I ⊗ dΓ(i[ω, B]) = I ⊗ dΓ(I) = I ⊗N,
(3.6)
where N is the “number operator”. Motivated by the above computations we define with PΩ given as the projection in F onto the vacuum M = I ⊗ (N + PΩ )(≥ I) , An = I ⊗ dΓ(Bn ) ; Bn =
(3.7)
1 −k (Fn · p + p · Fn ), Fn (k) = p . 2 |k|2 + n−2
The flow generated by the smooth vector field Fn is given by d θn (k, t) = Fn (θn (k, t)), θn (k, 0) = k , dt and satisfies the bound |θn (k, t)| ≤ en . |k| |t| 0, dbc (E + ) ≤ dbc (E) + ,
(4.9)
and similarly for the function defined in (4.7). For R ∈ R, the notation F (· < R) stands for the characteristic function of the interval (−∞, R). Let F (· ≥ R) = 1 − F (· < R). For δ > 0 the notation ηδ stands for any function η ∈ C0∞ (R) obeying 0 ≤ η ≤ 1, η(t) = 1 for |t| ≤ δ and η(t) = 0 for |t| > 2δ. We shall prove the following result expressed in terms of the definition (4.8). Lemma 4.1. Let b, c ∈ B, c ⊂ b, E0 ∈ R, > 0, Kf be compact on F and finally hc ∈ L∞ (X c ) be compactly supported. Then sup kBcb (δ, E, )k → 0 f or δ → 0 ;
E≤E0
Bcb (δ, E, ) = ηδ (H b − E)(B b ⊗ Kf ) , B b = hc (xc )F ((pbc )2 < dbc (E + ) − 2) . (For b = bmin B b is the number hc (0)F (0 < dbc (E + ) − 2).) Proof. We consider the statement of the lemma as a collection of statements labelled by b ∈ B and holding for all other quantities (including the constant C of (4.8)). Then we shall proceed by induction using the ordering of B. Thus the first step will be to verify the statement for b = bmin. For that we may proceed more general by showing it for c = b in which case we may actually assume that b = bmax (since bmin = bmax is not excluded at this point). To do that we notice that for δ < 12 and E ≤ E0 , Bc (δ, E, ) = ηδ (H − E)B1 KB2 ; B1 = F (H < E0 + 1)((p2 + 1) ⊗ I) , K = ((p2 + 1)−1 hc ) ⊗ Kf ,
(4.10)
B2 = B2 (E, ) = F ((pc )2 < dc (E + ) − 2) ⊗ I . Here we have omitted the notation b = bmax at various places while we have kept the notation c for a latter purpose. (If b = bmin we may interpret B1 = I and K = hc (0)Kf .) Clearly B1 is bounded and K is compact. If E ∈ σpp (H) then B2 = 0. On the other hand if E ∈ / σpp (H) then by the compactness kBc (δ, E, )k → 0 for δ → 0. To obtain this convergence uniformly with respect to E ≤ E0 we proceed
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1007
by the way of contradiction. So suppose for some κ > 0 that kBc (δn , En , )k > κ
(4.11)
for sequences δn → 0 and En → E. We notice that (4.9) implies the bound − ; |E − En | ≤ , dc (En + ) − 2 ≤ dc E + 2 2 yielding for large enough n that
B2 (En , ) . Bc (δn , En , ) = ηδn (H − En )Bc δ, E, 2
(4.12)
Here δ > 0 may be chosen arbitrarily. Choosing it so small that kBc (δ, E, 2 )k ≤ κ we obviously have a contradiction to (4.11). Next we conduct the induction step assuming again b = bmax , so that we can now use the statement of the lemma for all b 6= bmax . Suppose c 6= bmax . Since the last argument above involving (4.12) did not use the property c = bmax , it suffices to show convergence for fixed E. For that purpose we pick a family {jb }, b 6= bmax , of functions on X each one being smooth and homogeneous of degree 0 outside a compact set. We assume that X jb = 1 0 ≤ jb ≤ 1, b 0
/b and for any b ⊂ 0
|x|jb (x) ≤ C|xb |jb (x) ; We insert this partition ηδ (H − E)B ⊗ Kf =
X
C = C(b, b0 ) < ∞ .
(4.13)
ηδ (H − E)jb B ⊗ Kf .
(4.14)
b
Clearly we may assume that the constant C entering in the definition of B through max it is readily seen that (4.8) is large. Assuming C ≥ E0 + − inf Pbbmin 00
0
dbc00 (E 0 ) ≤ dbc0 (E 0 ); c00 ⊂ c0 ⊂ b0 ⊂ b00 ,
E 0 ≤ E0 + .
(4.15)
In particular, Fc = Fbmax Fc ;
Fc0 = Fc0 ((p0c )2 < dc0 (E + ) − 2) ,
and thus we can write for b with c ⊂ /b ηδ (H − E)jb B ⊗ Kf = ηδ (H − E)B1 KB2 B3 ; B1 = F (H < E0 + 1)((p2 + 1) ⊗ I) , K = ((p2 + 1)−1 jb hc ) ⊗ Kf , B2 = Fbmax ⊗ I ,
B3 = Fc ⊗ I .
By (4.13) K is compact and hence we recognize the form (4.10) treated above.
1008
E. SKIBSTED
Thus we only need to bound those terms with c ⊂ b. To do that we split ηδ (H − E)jb = ηδ (H − E)jb ηδ0 (Hb − E) + ηδ (H − E)T (δ 0 ) ; T (δ 0 ) = ηδ0 (H − E)jb − jb ηδ0 (Hb − E), δ 0 ≥ 2δ .
(4.16)
We may write ηδ (H − E)T (δ 0 )B ⊗ Kf = ηδ (H − E)KB2 B3 ; K = T (δ 0 )(hc ⊗ Kf ) , B2 = Fbmax ⊗ I, B3 = Fc ⊗ I . By a computation using (2.10) and (4.13) we see that kT (δ 0 )(F (|x| ≥ R) ⊗ I)k → 0 for R → ∞ , implying that K is compact. Hence for fixed δ 0 > 0 we can again proceed as above to treat the contribution from the second term on the right-hand side of (4.16). It remains to show that the contribution from the first term on the right-hand side of (4.16) can be estimated arbitrarily small by choosing δ 0 > 0 small enough. For that purpose we use (4.5) to reduce to a statement for the sub-Hamiltonian H b . We decompose (pc )2 = (pbc )2 + (pb )2 and estimate (cf. (4.9) and 4.15)) dc (E + ) − (pb )2 ≤ dc (E − (pb )2 + ) ≤ dbc (E − (pb )2 + ) ,
(4.17)
yielding kηδ0 (Hb − e)B ⊗ Kf k ≤ kηδ0 (H b − (E − (pb )2 )) · · · (hc (xc )F ((pbc )2 < dbc (E − (pb )2 + ) − 2)) ⊗ Kf k ≤ sup kηδ0 (H b − E 0 )(hc (xc )F ((pbc )2 E 0 <E0
< dbc (E 0 + ) − 2)) ⊗ Kf k .
(4.18)
Clearly by the induction hypothesis the right-hand side of (4.18) → 0 for δ 0 → 0 completing the proof. In the rest of this section bmin 6= bmax . We can now prove a similar result expressed in terms of the definition (4.7). Corollary 4.2. Let c ∈ B, E ∈ R, > 0, Kf be compact on F and finally hc ∈ L∞ (X c ) be compactly supported. Then for any given ε > 0, there exists for all small enough δ > 0 a compact operator K = K(ε, δ) on H such that kBc − Kk ≤ ε ; Bc = ηδ (H − E)(B ⊗ Kf ) ,
B = hc (xc )F ((pc )2 < d(E + ) − 2) .
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1009
Proof. We use the partition of unity in the proof of Lemma 4.1 to obtain (4.14) (with the present definition of B). The terms with c ⊂ / b are compact. As for the terms with c ⊂ b we use (4.16) and the discussion following (4.16) to conclude that we only need to find a δ 0 > 0 such that X kηδ0 (Hb − E)B ⊗ Kf k ≤ ε . (4.19) c⊂b6=bmax
For that we pick C = max(1, E + − inf F (H)) to be used as input in the definition (4.8) for any b and c such that c ⊂ b 6= bmax . Then it is readily seen (cf. (4.17)) that (4.20) d(E + ) − (pb )2 ≤ d(E − (pb )2 + ) ≤ dbc (E − (pb )2 + ) , which implies the bound (4.18). Using Lemma 4.1 to the right-hand side of (4.18) we clearly obtain (4.19) for some small δ 0 > 0. In order to apply Corollary 4.2 we introduce a certain vector field on X invented by Graf [7]. We have enlisted a collection of its characteristics in the following lemma. These may be derived by mimicking [7] or a simplified procedure due to Derezinski [5]. The property (5) was derived in [14, Appendix A]. Lemma 4.3. There exist on X a smooth vector field ω with symmetric derivative qb } indexed by b ∈ B and consisting of smooth functions, ω∗ and a partition of unity {˜ 0 ≤ q˜b ≤ 1, such that for some positive constants r1 and r2 P (1) ω∗ (x) ≥ b Πb q˜b (x). (2) ω b (x) = 0 if |xb | < r1 . qb ) if c ⊂ / b. (3) |xc | > r1 on supp (˜ b qb ). (4) |x | < r2 on supp (˜ (5) For all multi-indicies α and n ∈ N ∪ {0} there exists C ∈ R: |∂xα q˜b (x)| + |∂xα (x · ∆)n (ω(x) − x)| ≤ C ;
x∈X.
We state some consequences. Combining (3) and (4) yields to qb (kx) = 0 if c ⊂ / b; q˜c (x)˜
k=
r1 . r2
(4.21)
We define for any b ∈ B, X
qb (x) = q˜b (kx)
!− 12 q˜c (kx)2
.
c
Then by (1) and (4.21) ω? (x) ≥
X
Πb qb (x)2 .
(4.22)
b
Finally using (2) and the fact that ω is a gradient field, we obtain ω(x) − ω(xb ) if |xb | < r1 .
(4.23)
1010
E. SKIBSTED
We shall now consider the operator BR on L2 (X) given by BR =
x x 1 Rω ·p+p·ω R ; 2 R R
R > 0.
More precisely, we shall consider the formal commutator x x X 1 (∆(∇ · ω)) Vb0 ; p− ⊗ I + i[H, BR ⊗ I] := 2pω? R 2R2 R b∈B x b Z · ⊕{a(∇b λbxb ) + a? (∇b λbxb ) + (∇b v b )(xb )I}dxb . Vb0 = −Rω R Xb
(4.24)
(4.25)
Here R should be chosen so large that the right-hand side makes sense as a symmetric operator on D(H). Notice that this is possible due to our conditions (4.2) and (4.3), (3.2) and Lemma 4.3 (2) and (5). In fact we observe that kVb0 (H − i)−1 k → 0 for R → ∞ .
(4.26)
To state the main result of this section we notice (cf. [11, Theorem X.41]) that P ˜ b obeying (4.2) is any potential V˜ = b∈B V˜b of the form (4.1) with v˜b = 0 and λ essentially self-adjoint on C=
p [O
C0∞ (X) ⊗ (C0∞ (Rν \ {0})⊗s q )(⊆ H) .
(4.27)
p∈N q=0
In the next section C will play a similar role as the space defined by (3.4) did in Sec. 3. Proposition 4.4. Let V˜ be given as above, E ∈ R and > 0. Then there exists R0 > 0 so that for any R ≥ R0 there exist an open neighbourhood U = U(R) of E and a compact operator K = K(R) on H such that with T = exp(iV˜ ) f (H)i[H, BR ⊗ I]f (H) ≥ f (H){2(d(E + ) − 3)T I ⊗ PΩ T −1 − K − I}f (H)
(4.28)
for all real-valued f ∈ C0∞ (U). Proof. Let in the following the notation o(1) stands for an R-depending bounded operator obeying ko(1)k → 0 for R → ∞. Operators or forms on the form o(1)(H − i) or (H + i)o(1)(H − i) are denoted by oH (1) and oHH (1), respectively. By (4.22), Lemma 4.3 (5) and (4.26) x x X (pb )2 qb , (4.29) Qb (R) ⊗ I + oH (1) ; Qb (R) = qb i[H, BR ⊗ I] ≥ 2 R R b
as a form inequality on D(H).
1011
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
Pdim Xb 2 For each of the terms Qb (R) ⊗ I we may write (pb )2 = pj , (p1 , . . . , j=1 pdim Xb ) being the moment operator with respect to some orthonormal basis in Xb , and then estimate Qb (R) ⊗ I ≥
dim Xb X
Pj? T I ⊗ PΩ T −1 Pj = T
j=1
dim XXb
Pj? I ⊗ PΩ Pj T −1 + oHH (1)
j=1
= T Qb (R) ⊗ PΩ T −1 + oHH (1) ; x ⊗I. Pj = pj qb R
(4.30)
The commutations of the middle step require justification: We introduce T (t) = exp(itV˜ ); −1 ≤ t ≤ 1, and obtain by a differentiation on the core C ⊂ D(H) XZ 1 T (t)i[V˜c , Pj ]T (−t)dt . (4.31) T Pj T −1 − Pj = c
0
Here by assumption Z V˜c = Xc
˜ c c ) + a? (λ ˜ c c )}dxc , ⊕{a(λ x x
˜ b satisfies (4.2). We compute where λ Z ˜c c ) + a? (∂j λ ˜ c c )}qb x dxc . ˜ ⊕{a(∂j λ i[Vc , Pj ] = − x x R Xc
(4.32)
(4.33)
Clearly this operator is zero for c ⊂ b and in general of the form oH (1) (cf. Lemma 4.3 (3)). By iterating the formula (4.31), we obtain X XXZ 1 Z t i[V˜c , Pj ] = dt dsT (s)i2 [V˜d , [V˜c , Pj ]]T (−s) T Pj T −1 − Pj − d
c⊂ /b
=
c⊂ /b
0
XX 1 d
c⊂ /b
2
0
fc,d ⊗ I ,
(4.34)
where by the canonical commutation relations and (4.33) we have computed Z x 2 ˜ d c ˜ ˜ ˜ Im λxd (k)∂j λxc (k)dk ⊗ I =: fc,d (x) ⊗ I . i [Vd , [Vc , Pj ]] = −2qb R Rν (4.38) Since fc,d ⊗ I = o(1) we can write (4.34) on the form T −1 Pj = Pj T −1 + oH (1)
(4.35)
as an operator on C and thus (by density) on D(H). Clearly we can justify the middle step of (4.30) by using (4.35) and the adjoint formula.
1012
E. SKIBSTED
Using Qb (R) ≥ (d(E + ) − 3)qb
x R
F ((pb )2 ≥ d(E + ) − 3)qb
x R
to the right-hand side of (4.30) and substituting in (4.20) yields to i[H, BR ⊗ I] ≥ 2(d(E + ) − 3)T I ⊗ PΩ T −1 X − 2d(E + ) T Pb (R) ⊗ PΩ T −1 + oHH (1) ;
(4.36)
b
Pb (R) = qb
x R
F ((pb )2 < d(E + ) − 3)qb
x R
.
For each of the terms T Pb (R) ⊗ PΩ T −1 , we shall now use the following analogue of (4.31). XZ 1 −1 T (t1 )iad1c1 (Pb (R) ⊗ PΩ )T (−t1 )dt1 , T Pb (R) ⊗ PΩ T − Pb (R) ⊗ PΩ = c1
0
(4.37) utilizing here and below the notation m−1 ˜ adm c1 ,...,cm (B) = [Vcm , adc1 ,...,cm−1 (B)] ,
ad0 (B) = B; c1 , . . . , cm ∈ B . The commutator in (4.37) is given by (1)
ad1c1 (Pb (R) ⊗ PΩ ) = Bc1 + Pb (R) ⊗ Iad1c1 (I ⊗ PΩ ) ; (m) (I ⊗ PΩ ) . Bc1 ,...,cm = [V˜cm , Pb (R) ⊗ I]adcm−1 1 ,...,cm−1 (m)
(4.38)
(m)
For cm ⊂ bBc1 ,...,cm = 0. In general Bc1 ,...,cm = o(1) (cf. (4.42) below and Lemma 4.3 (3)). To treat the contribution from the second term on the right-hand side of (4.38) to (4.37), we iterate by inserting T (t1 )Pb (R) ⊗ Ii ad1c1 (I ⊗ PΩ )T (−t1 ) − Pb (R) ⊗ Ii ad1c1 (I ⊗ PΩ ) X Z t1 T (t2 )(Pb (R) ⊗ I)i2 ad2c1 ,c2 (I ⊗ PΩ )T (−t2 )dt2 + o(1) , =
(4.39)
0
c2
(2)
where we used that Bc1 ,c2 = o(1). By iterating the above procedure n(∈ N) times we obtain the formula T Pb (R) ⊗ PΩ T −1 = Tn + Rn + o(1) ; Tn =
n−1 X
X
(m!)−1 Pb (R) ⊗ Iim adm c1 ,...,cm (I ⊗ PΩ ) ,
(4.40)
m=0 c1 ,...,cm
Rn =
X Z c1 ,...,cn
0
Z
1
dt1 · · · 0
tn−1
dtn T (tn )Pb (R) ⊗ Iin adnc1 ,...,cn (I ⊗ PΩ )T (−tn ) .
1013
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
We claim that kRn k → 0 for n → ∞ ,
(4.41)
uniformly with respect to R. To show (4.41) we notice the bound (cf. the proof of [11, Theorem X.41]) kadnc1 ,...,cn (I
⊗ PΩ )k ≤ 4
n
√
Z ˜ c c (k)|2 dk |λ x
n! sup c,xc
Rν
n2 ,
(4.42)
which upon computing the integrals implies kRn k ≤ C n (n!)− 2 ; 1
Z C = 4(#B) sup c,xc
Rν
˜ c c (k)|2 dk |λ x
12 ,
and therefore clearly (4.41). 1 We fix n such that 2d(E + )(#B)C n (n!)− 2 ≤ 4 bounding the contribution from all Rn (one for each b) to the second term on the right-hand side of (4.36) by 4 . Next we look at the term Tn . Clearly Z (I ⊗ P ) = ⊕Kf (xc )dxc =: Dc adm Ω c1 ,...,cm Xc
with X c = X c1 + · · · + X cm and Kf (·) being a bounded weakly measurable compact operator-valued function with kKf (xc )k → 0 for |xc | → ∞. Up to a small bounded operator on Hc = L2 (X c ) ⊗ F Dc ((pc )2 + 1)−1 ⊗ I ≈ Dc K c ⊗ I , where K c is on the form K c = g c (xc )((pc )2 + 1)−1 , g c ∈ L∞ (X c ) being compactly supported. Again up to a small bounded operator we can write Z ⊕QKf (xc )Qdxc K c ⊗ I , Dc K c ⊗ I ≈ Xc
where Q is a finite dimensional projection on F (by the Lebesgue dominated convergence theorem). Finally using that (pc )2 is H-bounded we conclude that − 2d(E + )(m!)−1 Pb (R) ⊗ Iim adm c1 ,...,cm (I ⊗ PΩ )η1 (H − E) X (Pb (R)hc (xc )) ⊗ Kf η1 (H − E) , ≈
(4.43)
hc ,Kf
where the summation is finite and ranges over compactly supported hc ∈ L∞ (X c ) and compact operators Kf on F . Of course the approximation in (4.43) is uniform with respect to R. We fix for each term an approximation such that the total contribution from the errors to the second term on the right-hand side of (4.36) is bounded by 4 (uniformly in R).
1014
E. SKIBSTED
x x Since we can write qb ( R )hc (xc ) = qb ( R )hdR (xd ) with X d = X b + X c and hdR ∈ d L (X ) being compactly support (cf. Lemma 4.3 (4)), and for this d ∞
Pb (R) = Pb (R)F ((pd )2 < d(E + ) − 2) + o(1) (cf. (2.10), (4.16) and Lemma 4.3 (5)), we conclude (Pb (R)hc (xc )) ⊗ Kf = (Pb (R)F ((pd )2 < d(E + ) − 2)hdR (xd )) ⊗ Kf + o(1) .
(4.44)
Now we fix R0 such that the contributions from the terms o(1) on the right-hand side of (4.44) and (4.40) plus the one from η1 (H −E)oHH (1)η1 (H −E) (with oHH (1) given in (4.36)) altogether are bounded by 4 for R ≥ R0 . Finally by Corollary 4.2 we can find a compact operator K = K(R) on H such that for a small δ > 0, X Pb,d,Kf ⊗ Kf ηb (H − E)k ≤ ; kK − 4 b,d,Kf
Pb,d,Kf = Pb (R)F ((pd )2 < d(E + ) − 2)hdR (xd ) . Upon symmetrizing all previously encountered terms we readily obtain the result using that the four 4 -bounds appeared up to this point adds to . The proof of Proposition 4.4 was involved due to the appearance of the operator T . Partly as a warm up for the next section we close Sec. 4 by specializing to the case T = I and λb = 0 (for all b). We shall derive spectral properties for the generalized Schr¨ odinger operator Hel := −∆ + V acting (as a factor) on L2 (X). Notice that in this case H = Hel ⊗ I + I ⊗ Hf and that the set of thresholds of H as defined by (4.6) coincides with the set Fel of thresholds of Hel (defined similarly in terms of sub-Hamiltonians on the Hilbert space L2 (X)). Corollary 4.5 (A Mourre estimate). Under the condition (4.3) there exists for any given E ∈ R and > 0, a positive number R0 so that for any R ≥ R0 we can find an open neighbourhood U = U(R) of E and a compact operator K = K(R) on L2 (X) such that f (Hel i[Hel , BR ]f (Hel ) ≥ f (Hel ){(2d(E) − )I − K}f (Hel )
(4.45)
for all real-valued f ∈ C0∞ (U). Moreover the eigenvalues of Hel (counted with multiplicity) can only accumulate at Fel , the latter set being closed and countable. Proof. We proceed by induction with respect to the ordering of B (cf. the proof of Lemma 4.1). So suppose we know the statements of the corollary for subHamiltonians (leaving the start of induction to the reader), then we need to verify them for Hel . Since (by assumption) the eigenvalues of any sub-Hamiltonian can
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1015
only accumulate at the set of thresholds of this operator we obtain that Fel is closed and countable. Now we can verify the other statements without further reference to the induction hypothesis. We apply (4.28) to the invariant subspace L2 (X) = (I ⊗ PΩ )H yielding f (Hel )i[Hel , BR ]f (Hel ) ≥ f (Hel {(2d(E + ) − 7)I − K}f (Hel ) .
(4.46)
If E ∈ Fel then clearly (4.45) follows from (4.46). Using that Fel is closed then also for E ∈ / Fel (4.46) implies (4.45). The remaining statement that the eigenvalues of Hel can only accumulate at Fel follows readily from (4.45) and a virial theorem (cf. Corollary 2.3). 5. The Electron-Boson Model, Continued We proceed somewhat similar to Sec. 3 skipping some arguments given there. The basic set-up is the same as the one in Sec. 4, but we shall need the following stronger conditions compared to (4.2) and (4.3). We require b(i,j)
λ(·) b(i,j) λxb (k)
−j
= |k|
∂ ∂|k|
(·) ∈ L∞ (X b , L2ω ) ;
i
b(i,j)
k∂yα (|y|α λy
λbxb (k) ,
i, j ∈ {0, 1, 2}, i + j ≤ 2 ,
(5.1)
(·))kω → 0 for |y| → ∞ ;
i + j + |α| ≤ 2 , and v b (−∆b + 1)−1 compact, v b real-valued, |∂yα (|y||α| v b (y))| → 0 for |y| → ∞; |α| ≤ 2 .
(5.2)
As in Sec. 3 the function λbxb (·) in (5.1) (for fixed xb ) is considered as a distribution on C0∞ (Rν \ {0}). Moreover all the above derivatives with respect to xb are only required to exist outside a compact subset Kb ⊂ X b in the sense of distribution. More precisely we consider in the case of (5.1) λb(i,j) as a distribution on C0∞ (X b \ Kb ) ⊗ C0∞ (Rν \ {0}) and in the case of (5.2) v b as a distribution on C0∞ (X b \ Kb ). We shall need the following analogue of (3.12). Assume that for all b ∈ B, 2 Z X c(1) λxc (k) dk < 1 ; β b := sup b b ν x ∈X R c⊂b (5.3) ν − 1 c(0,1) c(1) c(1,0) λxc . + λxc = λxc 2 For later convenience, we abbreviate X X b(1) λbxb , λ(1) λxb , β = β bmax . (5.4) λx = x = b⊂B
b∈B
1016
E. SKIBSTED
We shall verify Assumptions 2.1 for some fixed energy E ∈ / F(H) in Assumption 2.1 (4) assuming that F (H) is closed and countable. The latter condition can be verified inductively, cf. the proof of Corollary 4.5. For that let BR be defined as the closure in L2 (X) of the expression (4.24) on ∞ C0 (X) and Bf = dΓ(B) as in Sec. 3. We shall consider A = CBR ⊗ I + I ⊗ Bf
(5.5)
defined as the closure on C, the latter given by (4.27). Here C is an arbitrary constant chosen such that (with d(E) given by (4.7)) C2d(E) ≥ 1 .
(5.6)
It is readily proved (cf. [11, Theorem X.49]) that −iA generates a contraction semigroup. With Bn given by (3.7) we define An = CBR ⊗ I + I ⊗ dΓ(Bn ) as the closure on C, which constitutes a family of self-adjoint operators approaching A as n → ∞. Let (5.7) M = I ⊗ (N + PΩ )(≥ I) . We can now verify Assumptions 2.1 for R large enough. We shall mainly focus on Assumption 2.1 (4) referring to Sec. 3 for a more detailed account. (Many arguments there may readily be modified.) We notice that the invariance of D(H) and D(M ) under eitAn follows from the formula eitAn = (eitCBR ⊗ I)(I ⊗ eitdΓ(Bn ) )
(5.8)
since by explicit computation these properties hold for both factors on the righthand side. Moreover (5.8) implies invariance of C, and therefore it suffices to calculate commutators as forms on C (cf. Sec. 3). We obtain Z ? (1) ⊕{a(λ(1) (5.9) G= x ) + a (λx )}dx − I ⊗ PΩ + Ci[H, BR ⊗ I] . X
Clearly by (5.1) and (4.25) G is bounded relative to H. (1) To verify Assumption 2.1 (4) we let again T be given by (3.13) (with λx given in (5.4)). Then using (5.7) and (5.9) we split for any real-valued f ∈ C0∞ (R) (cf. (3.15)) M + f (H)Gf (H) = P − L + f (H)(L + Q)f (H) ; Z ? (1) −1 ⊕{a(λ(1) , P = I ⊗N + x ) + a (λx )}dx + T I ⊗ PΩ T Z L=
X ? (1) −1 ⊕{a(λ(1) , x ) + a (λx )}dx − I ⊗ PΩ + T I ⊗ PΩ T
X
Q = Ci[H, BR ⊗ I] − T I ⊗ PΩ T −1 .
(5.10)
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1017
Due to (5.3) and (3.14) P ≥ (1 − β) =: 2α > 0. By (4.28) and (5.6) f (H)Qf (H) ≥ −Cf (H){K + I}f (H) ,
(5.11)
for > 0 small, K = K(, R), R large, and finally for all f supported sufficiently close to E. We fix = α2 C −1 and a corresponding large R. Then we may proceed exactly as in Sec. 3 after the formula (3.15). Using (3.16) with the there chosen such that 2 kBk2 ≤ α2 we obtain now from (5.10) and (5.11) the inequality of Assumption 2.1 (4) (with the above α and K replaced by CK). We have now outlined a verification of Assumptions 2.1 for our electron-boson Hamiltonian. Leaving it to the reader to use Corollary 2.3 (see the first part of the proof of Theorem 3.1) and to perform an induction argument (cf. the proof of Corollary 4.5, the start of induction using Theorem 3.1 with m = 1) we summarize our results as follows. Theorem 5.1. Suppose (5.1), (2.5) and (5.3). Then any eigenstate of H belongs 1 to D(M 2 ), with M given by (5.7), and the eigenvalues of H (counted with multiplicity) can only accumulate at the set of thresholds F (H), the latter being closed and countable. Moreover the bounds of Theorem 2.4 hold with A given by (5.5) and E not being an eigenvalue nor threshold energy of H. In particular Hsc , the continuous singular subspace of H, is empty. 6. Extended Models We shall now consider an extension of the model of the previous section. Up to inclusion of spin, polarization and various cut-offs the standard (Dirac) model of quantum electrodynamics (see [3]) can be put on this form as to be discussed at the end of this section. We consider again H = p2 ⊗ I + I ⊗ Hf + V˜ on H = L2 (X) ⊗ F , P where the “potential” V˜ = b∈B V˜b now is more general. We demand
(6.1)
¯ b · pb + pb · U ¯b + (W ¯ b )2 ; V˜b = Vb + U Z ⊕{a(λVxbb + a? (λVxbb ) + v Vb (xb )}dxb , Vb = Xb
Z ¯b )l = (U
Xb
bl bl ⊕{a(λU ) + a? (λU ) + v Ubl (xb )}dxb , xb xb
(6.2)
Z ¯ b )l = (W
Xb
bl bl ⊕{a(λW ) + a? (λW ) + v Wbl (xb )}dxb , xb xb
where we write (with respect to an orthonormal basis in X b ) ¯ b = (Wb1 , . . . , Wb dim X b ) , ¯b = (Ub1 , . . . , Ub dim X b ), W U
pb = (pb1 , . . . , pbdim X b ) .
We aim at an analogue of Theorem 5.1 for H on this form by verifying Assumptions 2.1 with the same inputs for A, An and M .
1018
E. SKIBSTED
We impose the conditions (5.1) and (5.2) for the first term Vb on the right-hand ¯ b , which ¯b and W side of (6.2) and need to specify conditions on the “vectors” U again will allow us to apply the Kato–Rellich theorem. A related problem is to find conditions such that our new G is bounded relatively to H. For the latter purpose we (formally) compute the following analogue of (5.9): G = T1 + T2 + T3 − I ⊗ PΩ + Ci[H, BR ⊗ I] ; Z ⊕{a(λVx (1) ) + a? (λVx (1) )}dx , T1 = X
T2 =
X
X
(6.3)
i{[Ubl , Bf ]pbl + pbl [Ubl , Bf ]} ,
b∈B 1≤l≤dim X b
T3 =
X
X
i{Wbl [Wbl , Bf ] + [Wbl , Bf ]Wbl } .
b∈B 1≤l≤dim X b
P V (1) V (1) = b∈B λxbb is given as in (5.3) and (5.4). With a similar notation Here λx the commutators are given by Z U (1) U (1) ⊕{a(λxbbl ) + a? (λxbbl )}dxb , i[Ubl , Bf ] = Xb
Z i[Wbl , Bf ] =
Xb
(6.4) W (1) ⊕{a(λxbbl )
+a
?
W (1) (λxbbl )}dxb
.
Moreover we may write ∂ Ubl (1) ∂ Ubl (1) ? + ⊕ a λ b λ b +a dxb . ∂xl x ∂xl x Xb Z
ipbl [Ubl , Bf ]
=
i[Ubl , Bf ]pbl
(6.5) Motivated by the above computations we can now specify conditions on the new terms. Clearly (6.4), (6.5) and (3.2) suggest the condition U (1)
∂xβb λxbbl
(·) ∈ L∞ (X b , L2ω ) ;
|β| ≤ 1 ,
(6.6)
since it assures that the term T1 on the right-hand side of (6.3) is H-bounded. 1 1 (Notice that (−∆b + 1) 2 ⊗ (Hf + 1) 2 is H-bounded.) Similarly the condition bl (·) ∈ L∞ (X b , L2ω ) ; ∂xβb λU xb
|β| ≤ 1 ,
assures relative boundedness of the contribution to H from the terms involving λUbl in the definition of V˜b . To obtain relative boundedness of the term T3 , we use the bounds
Z
− 32
(Hf + 1) ⊕a(λx )dx(Hf + 1) ≤ 2 sup k(1 + ω)λx kω ,
X
x∈X
Z
? − 32
(Hf + 1) ⊕a (λx )dx(Hf + 1) ≤ 2 sup k(1 + ω)λx kω ,
X
x∈X
(6.7)
1019
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
obtained from (3.2), (3.5) and commutations. By interpolating (3.2) and (6.7) we obtain
Z
1 −1
(Hf + 1) 21 2 sup k(1 + ω)λ k , ⊕a(λ )dx(H + 1) x f x ω
≤ 2 x∈X
X (6.8)
Z
1 1 ? −1
(Hf + 1) 2 2 ⊕a (λx )dx(Hf + 1)
≤ 2 sup k(1 + ω)λx kω ,
x∈X
X
We expand each term of T3 into a sum of products using (6.4). Those terms that are products of two factors each one either being of the form a(λx ) or a∗ (λx ) are handled by inserting 1 1 I = (Hf + 1)− 2 (Hf + 1) 2 between the factors and then invoke (3.2) and (6.8). This procedure leads to the conditions (6.9) (1 + ω)λWbl , (1 + ω)λWbl (1) ∈ L∞ (X b , L2ω ) . W (1)
To handle the contribution from the remaining terms a(λxbbl vxWbbl , we impose (v Wbl )2 (−∆b + 1)−1 bounded, Wbl
b
W (1)
)vxWbbl and a? (λxbbl
)
(6.10) − 12
which by interpolation implies boundedness of v (−∆ + 1) . Combining this property with (6.9) we readily treat the above terms. Moreover we remark that (6.9) and (6.10) assure the relative boundeness of the ¯ b in the definition of V˜b . contribution from the term involving W It remains to elaborate on the last term on the right-hand side of (6.3). Formally (cf. (4.25)) x x X 1 p− V˜b0 ; (∆(∇ · ω)) ⊗ I + i[H, BR ⊗ I] := 2pω∗ R 2R2 R b∈B
V˜b0
=
0 T1b
+
0 T2b
+
0 T3b
,
0 = i[Vb , BR ⊗ I] , T1b
(6.11)
0 ¯ b · pb + pb · U ¯b , BR ⊗ I] , = i[U T2b 0 ¯ b )2 , BR ⊗ I] , = i[(W T3b
where we may compute x b Z 0 = Vb0 = −Rω · ⊕{a(∇b λVxbb ) + a? (∇b λVxbb ) + (∇b v Vb )(xb )}dxb , T1b R Xb X 0 0 0 = {Ubl0 pbl + pbl Ubl0 + Ubl BRl + BRl Ubl } ; T2b 1≤l≤dim X b
x 1 0x ∂ ωl · p + p · ωl0 , ωl0 = ω, 2 R R ∂xl X = {Wbl Wbl0 + Wbl0 Wbl } .
0 = BRl 0 T3b
1≤l≤dim X b
1020
E. SKIBSTED
Here Ubl0 and Wbl0 are given as Vb0 by replacing Vb by Ubl and Wbl , respectively. The conditions bl (·)) ∈ L∞ (X b , L2ω ) , ∂xαb (|xb ||α| ∂xβb λU xb
∂xαb (|xb ||α| ∂xβb v Ubl (·)) ∈ L∞ (X b ) , bl (·)) ∈ L∞ (X b , L2ω ) , ∂xαb (|xb ||α| (1 + w)λW xb
∂xαb (|xb ||α| v Wbl (·)) ∈ L∞ (X b ) ; |β|, |α| ≤ 1 , 0 and and Lemma 4.3 assure H-boundedness of the contribution from the terms T2b 0 T3b . Actually due to Lemma 4.3 and (4.23) we here only need boundedness outside a compact set in X b . Since we shall verify more than just relative boundeness (6.6), (6.9), (6.10) and (6.12) would not suffice. For example, we shall need decay at infinity in (6.9) and (6.12). Also we shall need assumptions for |α| = 2. Explicitly (and as a conclusion of the previous discussion) we impose in addition to (5.1) and (5.2) for Vb the following conditions. Let for i, j ∈ {0, 1, 2}, i + j ≤ 2, i ∂ Ubl (i,j) −j bl (k) = |k| λU (k) . λxb xb ∂|k| W (i,j)
(k) be defined similarly. Let λxbbl ¯b , we demand As for U U (i,j)
∂xβb λxbbl
(·) ∈ L∞ (X b , L2ω ) ; U (i,j)
k∂yα (|y|α ∂yβ λy bl
i + j ≤ 2, |β| ≤ 1 ,
(·))kω → 0 for |y| → ∞ ;
i + j + |α| ≤ 2 ,
(6.12)
|β| ≤ 1 ,
and (∂xβb v Ubl (xb ))(−∆b + 1)−
1+β 2
compact;
|β| ≤ 1 ,
|∂yα (|y||α| ∂yβ v Ubl (y))| → 0 for |y| → ∞ ;
v Ubl real-valued,
|α| ≤ 2 ,
|β| ≤ 1 .
(6.13)
¯ b , we demand As for W W (i,j)
(1 + ω)λxbbl
∈ L∞ (X b , L2ω ) ;
i+j ≤ 2,
k∂yα (|y|α (1 + ω)λWbl (i,j) )kω → 0 for |y| → ∞ ;
(6.14)
i + j + |α| ≤ 2 , and (v Wbl )2 (−∆b + 1)−1 compact,
v Wbl real-valued,
|∂yα (|y||α| v Wbl (y))| → 0 for |y| → ∞; |α| ≤ 2 .
(6.15)
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1021
To verify the Kato–Rellich criterion, kV˜ b φk ≤ (1 − b )k((pb )2 ⊗ I + I ⊗ Hf )φk + Cb kφk
(6.16)
(for some b > 0; b ∈ B) one would need an additional smallness assumption. It may readily be shown that the following bound implies (6.17): X X 5 Ubl Wbl 2 2 sup kλxb kω + sup k(1 + ω)λxb kω < 1 . 2 b∈B 1≤l≤dim X b
xb ∈X b
xb ∈X b
With the above assumptions we may justify the previous computations. In particular G is given by the H-bounded expression(6.3). To verify Assumption 2.1 (4) we notice that the decay conditions of (5.1), (5.2) and (6.13)–(6.16), and the property (4.23), imply that the terms V˜b0 in (6.11) in the form sense vanish for R → ∞. Explicitly using the notation of the proof of Proposition 4.4 V˜b0 = oHH (1). The other terms on the right-hand side of (6.11) may be treated as in the proof of Proposition 4.4 (with T = I), and since we may generalize the previous results of Sec. 4 to the present more general class of Hamiltonians we conclude (4.28) with T = I under the present assumptions (5.1), (5.2) and (6.13)–(6.17). We shall need a lower bound for the form T1 + T2 + T3 on the right-hand side of (6.3) (substituting (5.3)). Due to the appearance of momentum operators in the definition of T2 we need at this point to restrict our attention to a bounded energy regime. So let I0 := (−∞, E0 ] for some fixed E0 . Similarly to Sec. 5 we assume that F (H) ∩ I0 is closed and countable. We claim that the latter property may be proved inductively under the following additional smallness condition replacing (5.3). We demand that for all b ∈ B, F0b (T1b + T2b + T3b )F0b ≥ −β b F0b ; F0b = F (H b < E0 + 1) ,
β b ∈ [0, 1) ,
(6.17)
where for b = bmax (T1b + T2b + T3b ) is given by T1 + T2 + T3 on the right-hand side of (6.3) and for b 6= bmax by similarly expressions in terms of the potentials V˜c with c ⊂ b (cf. (5.3)). We notice that (6.17) and (6.18) are smallness conditions in the sense that they are satisfied upon replacing the potential V˜ subjected to the previous conditions by eV˜ for any sufficiently small constant e > 0. Assuming (6.18) and that E ∈ I0 \ F (H) we verify Assumption 2.1 (4) by choosing the constant C (defining A and An ) such that (5.6) holds (with d(E) defined similarly). Then we decompose (with F0 = F0bmax ) M + f (H)Gf (H) = P − L + f (H)(L + Q)f (H) ; P = I ⊗ N + F0 (T1 + T2 + T3 )F0 + I ⊗ PΩ , L = F0 (T1 + T2 + T3 )F0 , Q = Ci[H, BR ⊗ I] − I ⊗ PΩ .
(6.18)
1022
E. SKIBSTED
Again P ≥ (1 − β) =: 2α (with β = β bmax ). Since (due to arguments given above) we can use (5.11) again we can from this point proceed as in Sec. 5. We conclude: Theorem 6.1. Let E0 ∈ R and I0 := (−∞, E0 ]. Suppose (5.1) and (5.2) for the ¯b and W ¯ b , respectively, and the smallness terms Vb , (6.13)–(6.16) for the “vectors” U 1 conditions (6.17) and (6.18). Then any eigenstate of H belongs to D(M 2 ), with M given by (5.7), and the eigenvalues of H in I0 (counted with multiplicity) can only accumulate at the set of thresholds in I0 , F (H) ∩ I0 , the latter being closed and countable. Moreover the bounds of Theorem 2.4 hold with A given by (5.5) and E ∈ I0 not being an eigenvalue nor threshold energy of H. In particular, the restriction of H to the spectral subspace F (H < E0 )H does not have continuous singular spectrum. For high energies we can obtain a similar result under some other assumptions on the potential. The idea is to verify Assumptions 2.1 with M = I and A = An =
1 (x · p + p · x) ⊗ I + I ⊗ dΓ(−(k · pk + pk · k)) 2
(6.19)
(cf. [3]). Formally i[H, A] = 2(H − V˜ ) + i[V˜ , A] .
(6.20)
If V˜ and i[V˜ , A] are “small” then the right-hand side of (6.21) is large for high energies and hence in particular positive. To implement this idea we need the following conditions on V˜ : (xb · ∇xb )m1 (k · ∇k )m2 λVxbb (k) ∈ L∞ (X b , L2ω ) ; m1 , m2 ∈ {0, 1, 2} ,
m1 + m2 ≤ 2 .
(6.21)
((xb · ∇xb )m v Vb (xb ))(−∆b + 1)−1 bounded; m ∈ {0, 1, 2} .
(6.22)
bl (k) ∈ L∞ (X b , L2ω ) ; (xb · ∇xb )m1 ∂xβb (k · ∇k )m2 λU xb
m1 , m2 ∈ {0, 1, 2} ,
m1 + m2 ≤ 2, |β| ≤ 1 .
((xb · ∇xb )m ∂xβb v Ubl (xb ))(−∆b + 1)− m ∈ {0, 1, 2} ,
1+β 2
(6.23)
bounded;
|β| ≤ 1 .
(6.24)
bl ∈ L∞ (X b , L2ω ) ; (xb · ∇xb )m1 (1 + ω)(k · ∇k )m2 λW xb
m1 , m2 ∈ {0, 1, 2} ,
m1 + m2 ≤ 2 .
(6.25)
((xb · ∇xb )m (v Wbl (xb ))j )(−∆b + 1)− 2 bounded; 1
m ∈ {0, 1, 2} ,
j ∈ {1, 2} .
(6.26)
Also we need (6.17) and the form inequality to be valid for some (large) E0 ∈ R, any b ∈ B and some αb ∈ (0, 2),
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
F¯0b (i[V˜ b , Ab ] − 2V˜ b )F¯0b ≥ −(2 − αb )H b F¯0b ; F¯ b = F (H b ≥ E0 − 2) .
1023
(6.27)
0
Here Ab is given by (6.20) for b = bmax and similarly for b 6= bmax . Similar to a previous discussion of the conditions (6.17) and (6.18) the conditions (6.22)–(6.27) imply (6.17) and (6.28) upon introducing a small enough coupling constant. With the above conditions it is now straightforward to justify the form identity (6.21) (on the domain D(H)) and the inequality (with α = αbmax ), η1 (H − E)i[H, A]η1 (H − E) ≥ α(E0 − 2)η1 (H − E)2 ;
E ≥ E0 .
(6.28)
Hence we have verified Assumption 2.1 (4). Moreover the strong form of (6.29) implies absence of eigenvalues ≥ E0 (by a virial theorem). We may argue similarly for sub-Hamiltonians. Therefore we have outlined a proof of: Theorem 6.2. Suppose (6.22)–(6.27) and the smallness conditions (6.17) and (6.28), the latter for some E0 > 2. Let I¯0 := [E0 , ∞). Then (σpp (H)∩F(H))∩I¯0 = ∅. Moreover the bounds of Theorem 2.4 hold for any E ∈ I¯0 and with A given by (6.20). In particular the restriction of H to the spectral subspace F (H ≥ E0 )H is purely absolutely continuous. Under all of the previous conditions (5.1), (5.2), (6.13)–(6.16), (6.22)–(6.27) we may combine Theorems 6.1 and 6.2 upon introducing a small coupling constant. This is done by using the E0 of Theorem 6.2 as input in Theorem 6.1 and would imply absence of continuous singular spectrum and also the same conclusion on the set of eigenvalues and the set of thresholds as in Theorem 5.1. We end this section essentially by proceeding in this way for an example in a slightly more general framework, namely the standard model of quantum electrodynamics (see [3] for a more detailed account). We consider the following model for an atom coupled to a photon field: Including spin and polarization the Hilbert space is given by H = Hel ⊗Hf , where Hel is the N times anti-symmetric tensor product of the 1-electron space L2 (R3x ) ⊗ C2 while Hf denotes the symmetric Fock space built from the 1-photon space L2 (R3k ) ⊗ C2 . (The first factor C2 accounts for spin while the latter factor accounts for polarization.) Let m and e denotes the electron mass and charge, respectively, and σj the triple consisting of the standard Pauli matrices (each one considered as acting on the spin space C2 of the jth electron). We shall use units in which h(bar), c = 1, and √ therefore e = α where the feinstructure constant α ' (137)−1 . We shall consider e as a small parameter. The Hamiltonian reads N X 1 [σj · (−i∇j − eA(xj ))]2 + I ⊗ Hf + e2 V (x) ⊗ I , (6.29) H= 2m j=1 where A(y) is a quantized vector potential to be elaborated on below and with Z being the charge of the nucleus
1024
E. SKIBSTED
V (x) =
N X j=1
−
Z + |xj |
X 1≤i≤j≤N
1 . |xi − xj |
(6.30)
The theory of Dirac amounts to putting X Z 1 p [µ (k)eiky aµ (k) + µ (k)e−iky a?µ (k)]dk , A(y) = 2π ω(k) µ=1,2
(6.31)
where the vectors 1 (k), 1 (k) and k|k|−1 constitute an orthonormal basis in R3 , and the operators a1 (k), a?1 (k), a2 (k) and a?2 (k) obey [a1 (k), a2 (k 0 )] = [a1 (k), a?2 (k 0 )] = 0 , [a1 (k), a?( k 0 )] = [a2 (k), a?2 (k 0 )] = δ(k − k 0 ) . Since the latter operators are only formally defined (cf. [11, Sec. X.7]) the expression (6.32) is a formal object. In order to have a well-defined operator one makes the so-called ultraviolet cut-off. Then the form is X ¯ yµ )+a? (λ ¯ ¯yµ (k) = (λyµ1 (k), λyµ2 (k), λyµ3 (k)) , [aµ (λ λ (6.32) A(y) = µ yµ )] ; µ=1,2
where ˜µl (k) λyµl = e−iky λ ˜ ∈ L2 (R3 ). for a suitable real-valued function λµl k The methods discussed previously in this section fail to handle λyµl of this form. The problem is the lack of decay as y → ∞ of the y-dependence which may be overcome by cutting off suitably at y = ∞. There is another problem at k = 0 ˜ µl indicated above which may be (the infrared region) for the concrete value of λ overcome by cutting off suitably in this region. In any case imposing similar conditions as before on each component λyµl (k) the methods may be generalized to the present extended model. Notice that we may rewrite the first term on the right-hand side of (6.30) as N N X X 1 [σj · (−i∇j − eA(xj ))]2 = −∆ ⊗ I + V˜j ; 2m j=1 j=1
∆=
N X 1 ∆j , 2m j=1
Vj = −
¯ j · pj + pj · U ¯j + (W ¯ j )2 , V˜j = Vj + U
e σj · curl A(xj ) , 2m
¯j = − e A(xj ) , U 2m
pj = −i∇j ,
(6.33)
¯ j = √ e A(xj ) . W 2m
Except for the additional spin and polarization structure the form of (6.30) and (6.34) fits into the framework given by (6.1) and (6.2). The conditions we need on λyµl (k) may be derived from (6.34) by comparing with (5.1), (5.2), (6.13)–(6.16), (6.22)–(6.27). Notice that the last term on the right side of (6.30) is in agreement
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1025
with (5.2) and (6.23). We consider the following simplified condition given in terms of some > 0: m+i |α|+ ∂ λyµl (k) ∈ L∞ (R3y , L2ω ) ; (1 + y 2 ) 2 ∂yα+β (1 + ω(k))1−|β| |k|m−j ∂|k| (6.34) |α| + m + i + j ≤ 2 , |β| ≤ 1 . For the above model our methods imply: Theorem 6.3. Consider the Hamiltonian H (6.30) with V (x) given by (6.31) and A(y) by (6.33), each component λyµl obeying (6.35). Then for small enough e > 0 the following statements hold: The eigenvalues of H (counted with multiplicity) can only accumulate at the set of thresholds F (H), the latter being closed and countable. Moreover there is a limiting absorption principle away from the eigenvalues and thresholds. In particular, the continuous singular subspace of H is empty. Remarks 6.4. (1) We have not attempted to prove bounds on e > 0 comparably 1 with the physically relevant value e ' (137)− 2 . It is an open problem whether the smallness of e is strictly needed for the conclusion of Theorem 6.3. (2) For molecules consisting of static nuclei there is a similar model (see [3]). In this case the Coulomb potentials describing the interaction between the electrons and the nuclei do not fulfil (6.23). On the other hand, the weaker analogue condition (5.2) of Theorem 6.1 is fulfilled. Consequently the conclusion of that theorem holds for the molecule-photon model too. References [1] W. O. Amrein, A. Boutet de Monvel and V. Georgescu, “C0 -groups, commutator methods and spectral theory of N -body Hamiltonians”, Progress in Math. Series, 135 Birkh¨ auser, Basel, 1996. [2] S. Agmon, I. Herbst and E. Skibsted, “Perturbation of embedded eigenvalues in the generalized N -body problem”, Commun. Math. Phys. 122 (1989), 411–438. [3] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Quantum electrodynamics of confined nonrelativistic particles, preprint July 1996, to appear in Adv. Math. [4] A. M. Boutet de Monvel-Bertheir, D. Manda and R. Purice, “The commutator method for form-relatively compact perturbations”, Lett. Math. Phys. 22 (1991) 211–223. [5] J. Derezi´ nski, “Asymptotic completeness for N -particle long-range quantum systems”, Ann. Math. 38 (1993) 427–476. [6] R. Froese and I. Herbst, “A new proof of the Mourre estimate”, Duke Math. J. 49 (4) (1982) 1075–1085. [7] G. M. Graf, “Asymptotic completeness for N -body short-range quantum system: a new proof”, Commun. Math. Phys. 132 (1990) 73–101. [8] M. H¨ ubner and H. Spohn, “Spectral properties of the spin-boson Hamiltonian”, Ann. Inst. Henri Poincar´e 62 (3) (1995) 289–323. [9] E. Mourre, “Absence of singular continuous spectrum for certain self-adjoint operators”, Commun. Math. Phys. 91 (1981) 391–408.
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E. SKIBSTED
[10] P. Perry, I. M. Sigal and B. Simon, “Spectral analysis of N -body Schr¨ odinger operators”, Ann. Math. 114 (1981) 519–567. [11] M. Reed and B. Simon, Fourier Analysis, Self-Adjointness. Methods of Modern Mathematical Physics II, New York, Academic Press, 1975. [12] J. Sahbani, “The conjugate operator method for locally regular Hamiltonians”, preprint June 1996, J. Operator Theory 38 (2) (1997) 297–322. [13] B. Simon, “Resonances in N -body quantum systems with dilation analytic potentials and the foundations of time-dependent perturbation theory”, Ann. Math. 97 (1973) 247–274. [14] E. Skibsted, “Propagation estimates for N -body Schr¨ odinger operators”, Commun. Math. Phys. 142 (1991) 67–98.
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION C. FRØNSDAL Physics Department, University of California Los Angeles CA 90024, USA
A. GALINDO Departamento de F´ısica Te´ orica Universidad Complutense 28040 Madrid, Spain Received 10 February 1998 1991 Mathematics Subject Classification: 81R50, 70G50 We study the vertex operators Φ(z) associated with standard quantum groups. The element Z = RRt is a “Casimir operator” for quantized Kac–Moody algebras and the quantum Knizhnik–Zamolodchikov (q-KZ) equation is interpreted as the statement :ZΦ(z) := Φ(z). We study the covariance of the q-KZ equation under twisting, first within the category of Hopf algebras, and then in the wider context of quasi Hopf algebras. We obtain the intertwining operators associated with the elliptic R-matrix and calculate the two-point correlation function for the eight-vertex model.
1. Introduction In this paper we study the quantum Knizhnik–Zamolodchikov equation [12] for quasi Hopf algebras, with its covariance properties with respect to twisting, and its relation to matrix elements of intertwining operators. The conclusions bear on the interpretation of the solutions of similar equations with exotic R-matrices. We calculate the correlation functions for the eight-vertex model. Correlation Functions for the Eight-Vertex Model d Baxter [2] introduced the trigonometric and elliptic quantum R-matrix for sl(2); this paper is mostly about the elliptic case, and about the generalization [4] to \). The trigonometric R-matrices found their interpretation in elliptic quantum sl(N terms of quantized Kac–Moody algebras, viewed as Hopf algebras; that is, quantum groups [7]. The elliptic R-matrices had, until recently, not found their place in an algebraic framework. Surprisingly the elliptic R-matrices also turned out to be related to quantized Kac–Moody algebras, but with a quasi Hopf structure [14, 15]. More precisely, the algebraic structure is the same as in the trigonometric case, while the coproduct ∆ of the trigonometric quantum group is replaced by a new, deformed coproduct ∆ (“elliptic coproduct”) that depends on a deformation parameter . It can be expressed as ∆ = (Ft )−1 ∆Ft ; the twistor F must satisfy a cocycle 1027 Reviews in Mathematical Physics, Vol. 10, No. 8 (1998) 1027–1059 c World Scientific Publishing Company
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C. FRØNSDAL and A. GALINDO
condition that has been solved to give an explicit expression for F as a power series in . The quotient of the elliptic quantum group, by the ideal generated by the center, is a Hopf algebra; it is the quantization, in the sense of Drinfel’d, of the classical, affine Lie bialgebra with elliptic r-matrix in the classification of Belavin and Drinfeld [5]. To understand the role of these elliptic quantum groups in the context of integrable models and conformal field theory, we calculate the correlation functions of the eight-vertex model. The premise is that Baxter’s vertex operators can be interpreted mathematically as intertwining operators for representations of quantized Kac–Moody algebras [17]; this is the interpretation that affords the most direct link between statistical models and conformal field theory. Here we define new intertwining operators in terms of the elliptic coproduct and calculate the correlation functions that are associated with them; that is, matrix elements of products of intertwining operators. We find that these functions satisfy equations similar to the quantum Knizhnik–Zamolodchikov equations of Frenkel and Reshetikhin [12], but that they can be described much more easily in terms of the familiar correlation functions that govern the six-vertex model. Twist Covariance The larger issue is the question of the covariance of the q-KZ equation under twisting in the category of quasi Hopf algebras. To begin with, we point out that the q-KZ of Frenkel and Reshetikhin [12] can be easily generalized to all simple, affine quantum groups endowed with what we call a “standard” R-matrix: a universal R-matrix (expressed as a series in Chevalley–Drinfeld generators, see Definition 2.1.) that commutes with the Cartan subalgebra. Reshetikhin [22] has described a highly specialized form of twisting under which a standard R-matrix remains of standard type. From now on, by the term “twisting” we always have in mind a more radical twist that transforms a standard R-matrix to a nonstandard or esoteric R-matrix. A quantum group in the sense of this paper is a quantized, affine Kac–Moody algebra ˆ g based on a simple Lie algebra g. The structure of coboundary Hopf algebra is given by a coproduct, an antipode and a counit, but only the coproduct plays a direct role in this paper. A coboundary Hopf algebra is a Hopf algebra ˆg with an invertible element R ∈ ˆ g ⊗ gˆ that satisfies the Yang–Baxter relation and that intertwines the coproduct ∆ with its opposite ∆0 : R∆0 = ∆R .
(1.1)
The q-KZ equation is a holonomic system of difference equations that are satisfied by certain intertwining operators, Φ, Ψ : Vµ,k → V (z) ⊗ Vν,k ,
(1.2)
where Vµ,k and Vν,k are irreducible, highest weight gˆ-modules of level k and V (z) is an evaluation module. The intertwining property of Φ and of Ψ is expressed as Φx = ∆(x)Φ ,
Ψx = ∆0 (x)Ψ ,
(1.3)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1029
for x ∈ ˆ g. When R is of standard type (Definition 2.1), then the q-KZ equation for Ψ takes the form (Z 0 − 1)Ψ = 1 , (1.4) where Z 0 is a Casimir operator (acting in V (z) ⊗ Vν,k ) for ˆg. To define this operator let us express R as R = Ri ⊗ Ri , where we use the summation convention for the index i; then formally, Z 0 = Rt R ,
Rt := Ri ⊗ Ri .
(1.5)
However, to make sense of an operator product such as Z 0 Ψ it is necessary to renormalize it. The correct form of the q-KZ equation is indeed (1.4), but with Z 0 Ψ replaced by the normal-ordered product ˆ
:Z 0 Ψ: = Rt (Ri q H ⊗ 1)ΨRi ,
(1.6)
ˆ ˆ. where the factor q H belongs to the Cartan subalgebra of g We study a deformation of the initial, standard quantum group, implemented by twisting with an invertible element F ∈ ˆg ⊗ ˆg that is a formal power series in a deformation parameter . The twisted quantities are:
R = (Ft )−1 RF , Ψ = F−1 Ψ ,
∆0 = F ∆0 F−1 , Z0 = F−1 Z 0 F ,
and the twisted KZ equation is :Z0 Ψ : = Ψ ; it has the same form as in the standard case. However, Eq. (1.6) is not covariant; we mean by that it cannot be generalized by simply replacing R by R , since the expression Rt (Ri ⊗ 1)Ψ Ri is not well defined. Instead, the correct expression for the normal-ordered product is ˆ :Z0 Ψ : = F−1 :Z 0 Ψ: = F−1 Rt (Ri q H ⊗ 1)ΨRi . Therefore, although there is a clear sense in which “the q-KZ equation” is covariant, the normal-ordered product (1.6) is not. This observation has analogous implications for correlation function. To illustrate this, consider the two-point correlation function g(z1 , z2 ) = hΨ(z1 )Ψ(z2 )i. In the standard case the q-KZ equation reduces to g(q −k−g z1 , z2 ) = q A1 R−1 (z1 , z2 )g(z1 , z2 ) .
(1.7)
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C. FRØNSDAL and A. GALINDO
The twisted correlation function obeys g (q −k−g z1 , z2 ) = (F−1 (z2 , q −k−g z1 )q A1 R−1 (z1 , z2 )F (z2 , z1 ))g (z1 , z2 ) , and this is not the same as Eq. (1.7) with R replaced by R . This conclusion casts some light on the proposed generalization of of the q-KZ equations for correlation functions. Integrability is assured by the Yang–Baxter relation for the R-matrix. It is natural to study the equations that result from replacing the trigonometric R-matrix in (1.7) and the rest, by more exotic R-matrices. Since this requires a knowledge of such R-matrices in finite dimensional representations only, it is possible, in particular, to use the elliptic R-matrix of Baxter in this connection. As long as the elliptic quasi Hopf algebra was not known, it was possible to speculate that the solutions of such “elliptic q-KZ equations” relate in some way to (unknown) elliptic intertwiners. Our conclusion is that this interpretation is not the correct one. Outline of the Paper Section 2 summarizes some facts about standard, universal R-matrices and sets our notation. Section 3 examines certain intertwining operators and draws some conclusions (Proposition 3.1) that are used later to determine the correct approach to regularizing operator products. Sections 4 and 5 present a view of the KZ and q-KZ equations. Both can be interpreted very simply as eigenvalue equations, ζΦ = 0 or (Z − 1)Φ = 0, for the Casimir operators ζ or Z of affine Kac–Moody or quantized, affine Kac– Moody algebras. Section 4 deals with the classical KZ equation ζΦ = 0; the effect of different polarizations is discussed, as well as the invariance of the operator ζ (Propositions 4.1 and 4.2). The quantum case is taken up in Sec. 5; the correct normal-ordered action of the Casimir elements Z and Z 0 on the intertwiners Φ and Ψ is established (Proposition 5.1), and the q-KZ equations are presented in Eqs. (5.6) and (5.8). Sections 6 and 7 explore the effect on intertwiners of twisting in the categories of Hopf and quasi Hopf algebras. In Sec. 6 we stress the distinction between “finite” and “elliptic” twisting. The twisted q-KZ equation is presented (Definition 6.3). In Sec. 7 quasi Hopf twisting is discussed and a recursion relation to actually calculate the elliptic twistor is given. Sections 8 and 9 apply the results to correlation functions. In Sec. 8 the classical and quantum q-KZ equations for correlation functions are given; the effect of twisting is exhibited and a certain lack of covariance is emphasized. In Sec. 9 the two-point correlation function for the eight-vertex model is calculated, as well as explicit expressions for the twisting matrix in the fundamental representation of d sl(2). Finally, some auxiliary material is relegated to an Appendix. Relation to Other Work (1) Our original goal was to discover the enigmatic “elliptic quantum groups” and to use it to define and calculate the correlation functions for the eight-vertex
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1031
model. This is precisely the problematics of a series of paper by Jimbo, Miwa and others; see especially the review [17] and the papers [11, 18]. These authors did not have available the universal, elliptic R-matrix and did not anticipate the fact that the algebraic structure of the elliptic quantum group would turn out to be the same as in the trigonometric case. (Only the coproduct is changed.) They postulated a new algebraic structure, but in the absence of a coproduct they could not define intertwiners. In spite of this they did succeed in calculating correlation functions that stand up to analysis and that reproduce some of Baxter’s results on the eightvertex model. Nevertheless, the correlation functions that we here propose for the eight-vertex model are quite different. (2) One of the most interesting aspects of the elliptic quantum group is its quasi Hopf nature. Quasi Hopf algebras, characterized by a modified quantum Yang– Baxter relation, are basic to the Knizhnik–Zamolodchikov–Bernard generalization of the KZ equation that was discovered by Bernard [6]. This equation also arises in connection with Felder’s elliptic quantum groups [10]. However, these developments are not concerned with highest weight matrix elements of intertwiner operators, and the quasi Hopf algebras of Felder et al. are not related to the elliptic R-matrices of Baxter and Belavin. The new r-matrices discovered by Enriquez and Rubtsov [9] and by Frenkel, Reshetikhin and Semenov–Tian–Shansky [13] are of a different sort. These interesting developments go beyond the classification of classical r-matrices by Belavin and Drinfel’d [5] and are outside the scope of this paper. 2. Standard, Affine, Universal, Quantum R-Matrices This section contains basic definitions and notation. Let M , N be two finite sets, ϕ, ψ two maps, ϕ : M × M → C , a, b 7→ ϕab , ψ : M × N → C,
a, β 7→ Ha (β) ,
and q a complex parameter. Let A or A(ϕ, ψ) be the universal, associative, unital algebra over C with generators {Ha }a∈M , {e±α }α∈N , and relations [Ha , Hb ] = 0 ,
[Ha , e±β ] = ±Ha (β)e±β ,
[eα , e−β ] = δαβ (q ϕ(α,·) − q −ϕ(·,α) ) , with ϕ(α, ·) = ϕab Ha (α)Hb , ϕ(·, α) = ϕab Ha Hb (α) and q ϕ(α,·)+ϕ(·,α) 6= 1, α ∈ N . The algebra of actual interest is a quotient A0 = A/I, where I is a certain ideal; in this paper we suppose that I is generated by a complete set of (quantized) Serre relations among the eα ’s and among the e−α ’s; then A0 is a quantized (generalized) Kac–Moody algebra. In the case when A0 is a quantized Kac–Moody algebra of affine type, based on a simple Lie algebra g, we sometimes write ˆg for A0 . The “Cartan subalgebra” A00 is generated by {Ha }a∈M , extended by the inclusion of exponentials.
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C. FRØNSDAL and A. GALINDO
Definition 2.1. The standard, universal R-matrix has the form R = qϕ T = qϕ where t0 = 1 ⊗ 1, t1 = and tn has the form
P
∞ X
tn ,
ϕ=
X
ϕab Ha ⊗ Hb ,
(2.1)
n=0
e−α ⊗ eα (the sum is over the Serre generators, α ∈ N ) (α0 )
tn = t(α) e−α1 . . . e−αn ⊗ eα01 . . . eα0n .
(2.2)
Sums over repeated indices are implied; the multi-index (α0 ) runs over the permutations of (α). (α0 )
The coefficients t(α) ∈ C are essentially determined (the elements tn are determined uniquely) by the imposition of the Yang–Baxter relation, R12 R13 R23 = R23 R13 R12 .
(2.3)
It has been shown that, for a universal R-matrix of the type (2.1), this relation is equivalent to the recursion relation [14] [eγ ⊗ 1, tn ] = tn−1 (q ϕ(γ,.) ⊗ eγ ) − (q −ϕ(.,γ) ⊗ eγ )tn−1 ,
(2.4)
with the initial condition t0 = 1. There is exactly one solution in A0 ⊗ A0 . ˆ is a quantized, affine Kac–Moody algebra based on We suppose now that A0 = g a simple Lie algebra g. The coproduct is then generated by the following formulas: ∆(eα ) = 1 ⊗ eα + eα ⊗ q ϕ(α,.) ,
∆(e−α ) = q −ϕ(.,α) ⊗ e−α + e−α ⊗ 1 ,
(2.5)
and ∆Ha = Ha ⊗ 1 + 1 ⊗ Ha . Let π1 , π2 be finite dimensional representations of g, and πi (zi ) the associated evaluation representations of gˆ with spectral parameters zi . Let (2.6) R(z1 , z2 ) := π1 (z1 ) ⊗ π2 (z2 )R . The spectral parameters are regarded as formal variables; R(z1 , z2 ) is a formal power series R12 (z2 /z1 ) in z2 /z1 . The effectiveness of the recursion relation (2.4) is illustrated in the Appendix. Finally, given A = ai ⊗ai ∈ A0 ⊗A0 , we shall write At := ai ⊗ai and mA := ai ai . 3. Highest Weight Modules and Intertwining Operators Let Vµ be an irreducible, finite dimensional, highest weight ˆg-module, and Vµ,k = ⊕n≥0 Vµ,k [−n] the associated level k, highest weight, irreducible, graded ˆg-module. The intertwining operators of greatest interest are imbeddings Φ = Φ(z) : Vµ,k → V (z) ⊗ Vν,k , where V (z) is an evaluation module over ˆg. The defining property of Φ is Φx = ∆(x)Φ , for all x in ˆ g.
(3.1)
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8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
We shall obtain some very essential information about the structure of the intertwining operators. Proposition 3.1. Let v be a homogeneous element of Vµ,k . Then Φv = bn , with bn ∈ Vν,k [−n], where the sum is not, in general, finite.
P n
an ⊗
Proof. It will be enough to verify that the sum is effectively infinite in one d with V the typical case. Thus consider the quantized Kac–Moody algebra sl(2) fundamental representation. In this case, for any v ∈ Vµ,k , Φv takes the form: ! A(z)v A Φv = v. (3.2) = B B(z)v Necessary conditions to be satisfied by the operators A, B : Vµ,k → Vν,k are A A eα v . v= ∆(eα ) (3.3) B B The first space is two-dimensional, 0 e1 ⊗ 1 = κ 0
with 1 , 0
e0 ⊗ 1 = κ
0 z
0 0
.
(3.4)
The parameter κ is related to q, κ2 = q − q −1 . In full detail, [e0 , A] = 0 , [e1 , A] = −κq ϕ(1,.)B , [e1 , B] = 0 , [e0 , B] = −κzq ϕ(0,.)A .
(3.5)
On the highest weight vector v0 in Vµ,k , we have e0 Av0 = 0 ,
e1 Bv0 = 0 ,
e0 Bv0 = −κzq ϕ(0,.)Av0 ,
e1 Av0 = −κq ϕ(1,.) Bv0 , (3.6)
with a unique solution of the form: Av0 =
∞ X
0 z n v2n ,
Bv0 =
n=0
∞ X
0 z n+1 v2n+1 ,
(3.7)
n=0
with v00 a highest weight vector in Vν,k and vectors vn0 ∈ Vν,k determined recursively by 0 0 = e1 v2n+1 = 0, e0 v2n
0 0 e1 v2n = −κq ϕ(1,.) v2n−1 ,
The solutions have the form: X 0 n 0 v2n = A2n σ σ(e−1 e−0 ) v0 , σ∈S2n
0 v2n+1 =
0 0 e0 v2n+1 = −κq ϕ(0,.) v2n . (3.8)
X
Bσ2n+1 σ(e−1 e−0 )n e−0 v00 ,
σ∈S2n+1
where the sum is over all permutations of the generators. It is clear that vn0 6= 0 for all n and the proposition is proved.
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ˆ is a quantized Kac–Moody algebra of We return to the general case, A0 = g affine type, based on a simple Lie algebra g, Vµ,k is a highest weight module over ˆ g, Vi (zi ) finite dimensional evaluation modules. Remark 3.2. The product Φ2 Φ1 is a compound map Φ
Φ
1 2 Φ2 Φ1 : Vµ,k −→V 1 ⊗ Vν,k −→V1 ⊗ V2 ⊗ Vλ,k .
(3.9)
It has the property Φ2 Φ1 x = Φ2 ∆(x)Φ1 = (id ⊗ ∆)∆(x)Φ2 Φ1 .
(3.10)
By coassociativity of ∆, Φ2 Φ1 is an intertwiner of the same type as Φ1 and Φ2 : Φ2 Φ1 : Vµ,k → (V1 (z1 ) ⊗ V2 (z2 )) ⊗ Vλ,k .
(3.11)
Consequently, universal statements about intertwiners apply to products of intertwiners as well. This observation will be of use in Sec. 8. Of course, it does not apply in the quasi Hopf case (Sec. 9). 4. The Classical KZ Equation The object Z = RRt ∈ A0 ⊗ A0 ,
(4.1) 0
if it exists, is invariant in the sense that it commutes with ∆(x), ∀x ∈ A . It plays the role of a Casimir element for the quantized Kac–Moody algebra. Since the intertwiner Φ projects on an irreducible representation, one expects that there is hZi ∈ C such that (Z − hZi)Φ = 0 . (4.2) We shall begin our study of this equation by considering its classical limit. The result is Propositions 4.2 and 4.3. The important concepts are normal ordering and “polarization”. Then we shall return to the quantum case to show that (4.2) is the q-KZ equation of Frenkel and Reshetikhin [12, Sec. 5]. The classical limit is defined by setting q = eη , expanding in powers of η, and retaining the first nonvanishing term. When A0 = ˆg is a quantized Kac–Moody algebra of finite type, one finds that X E−α ⊗ Eα , (4.3) R = 1 + ηr + O(η 2 ) , r = ϕ + α∈∆+
where the sum runs over the positive roots of g. For simple roots one has eα = √ η(Eα + O(η)); the others are normalized so that the Casimir element in g ⊗ g takes the form: C = r + rt . In the case of an untwisted affine loop algebra one gets X X E−α ⊗ Eα + (z2 /z1 )n C , R = 1 + ηr + O(η 2 ) , r = ϕ + α∈∆+
n≥1
(4.4)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1035
where ∆+ is the set of positive roots of the underlying Lie algebra and where z1 , z2 are the spectral parameters in the first, resp. second space. It is important to keep in mind that this expression is, until further development, nothing more than a formal power series in z2 /z1 . In terms of the basis n = z n E±α , E±α
Han = z n Ha ,
(4.5)
the expression for r becomes r = ϕ + E−α ⊗ Eα +
X
Cn ,
(4.6)
n≥1
with −n n ⊗ Eαn + Eα−n ⊗ E−α , C n = K ab Ha−n ⊗ Hbn + E−α
K ab = (ϕ + ϕt )ab .
(4.7)
Summation over a, b and α ∈ ∆+ will henceforth be taken for granted. Note that Eqs. (4.6) and (4.7) are valid in the case of twisted loop algebras as well. Returning to affine Kac–Moody algebras, it will be convenient to change our conventions just a little. Retain the above notation for the loop algebra, so that, in particular, (4.8) ϕ = ϕab Ha ⊗ Hb , where the sum runs over the basis of the Cartan subalgebra of a simple Lie algebra g. The form that characterizes the full, quantized affine Kac–Moody algebra ˆg is ϕˆ = ϕ + uc ⊗ d + (1 − u)d ⊗ c ,
(4.9)
where d is the degree operator, c is a basis for the central extension and u is a parameter. For the full quantized Kac–Moody algebra the limit is R = 1 + ηˆ r + O(η 2 ) ,
rˆ = r + uc ⊗ d + (1 − u)d ⊗ c .
(4.10)
The classical limit of Z is Z = 1 + ηζ + O(η 2 ) , Formally, ζ = rˆ + rˆt =
∞ X
ζ = rˆ + rˆt .
(4.11)
Cn + c ⊗ d + d ⊗ c .
−∞
When both spaces are evaluation modules, where c 7→ 0, ζ=
+∞ X
(z2 /z1 )n C .
(4.12)
n=−∞
This sum becomes zero when projected on a quotient algebra of meromorphic functions.
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C. FRØNSDAL and A. GALINDO
We try to make sense out of the classical limit of (4.2), namely (ζ − hζi)Φ = 0 . By abuse of notation we retain the notation Φ for the classical limit of the intertwiner. Now the first space is an evaluation module, where c vanishes, and if c 7→ k (k is the level) on the second space, then formally ζΦ(z) = kz
X X d Φ(z) + C −n Φ(z) + C n Φ(z) . dz n>0
(4.13)
n≥0
Let us introduce a uniform basis {La } for g, so that the Casimir element takes the form C = La ⊗ La (summation implied). Then (in the untwisted case) (4.13) takes the form X X d + La ⊗ z n L−n z −n Lna Φ(z) . ζΦ(z) = kz (4.14) a + La ⊗ dz n>0 n≥0
However, the significance of this formula is doubtful, as we shall see. This is the reason for the introduction of normal-ordered products in [12]. Polarization It is usual, at this point of the development, to replace the operator products by normal-ordered products. It is a step that merits comment. Normal-ordered operator products are introduced in field theory when ordinary operator products fail to make sense. The typical example is this product of destruction and creation operators: ! ! X X inω −imω ∗ e an e am . n
m
When it is applied to the vacuum one gets ! ! +∞ X X X inω −imω ∗ e an e am |0i = |0i , n
m
n=−∞
which is without meaning. The last term in (4.14) is of this kind; the degreedecreasing operators in Φ correspond to the creation operators, and the degreeincreasing operators Lna correspond to the destruction operators. Collecting all terms of the same degree in the product one gets a divergent series. We want to avoid having to interpret such infinite series, if it is possible. Using the (classical) intertwining property of the intertwiner, ! N N N X X X z −n (La ⊗ Lna )Φ = z −n (La ⊗ 1)ΦLna − La La ⊗ 1 Φ . (4.15) n=1
n=1
n=1
P Passing with N to infinity we encounter the meaningless expression n>0 La La , an exact analogue of the divergent sum that is thrown away when a field operator
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1037
product is replaced by the normal-ordered product. It is tempting to redefine the operator ζ, by dropping this offensive term, thus ζΦ = kz
d Φ + (La ⊗ 1) :Ja (z)Φ: (?) , dz
with Ja = Ja+ + Ja− =
X n≥0
z n L−n a +
X
z −n Lna ,
(4.16)
(4.17)
n>0
and :Ja Φ: := Ja+ Φ + ΦJa− .
(4.18)
This new operator is well defined on highest weight modules and it will serve if it has the property that the formal expression (4.1) was intended to assure; that is, if it is invariant. Actually it is, almost. Proposition 4.1. The covariant definition of the operator product ζΦ(z) is ζΦ(z) = (k + g)z
d Φ(z) + (La ⊗ 1):Ja (z)Φ(z): , dz
(4.19)
where g is the dual Coxeter number of g. This result of [12] is an analogue of Proposition 4.2 that we prove below. The replacement of the factor k by k + g, at first sight somewhat mysterious, is thus required by covariance. For sl(N ), g = N We calculate the value hζi. The operator J− annihilates the highest weight vector v0 ; therefore d + + La ⊗ Ja Φ(z)v0 . ζΦ(z)v0 = (k + g)z dz In terms of the contravariant bilinear form (.,.), with v00 the highest weight vector of Vν,k , one gets a V (z)-valued function: (v00 , Φ(z)v0 ) =: Φv00 v0 (z) ∈ V (z) , and (v00 , ζΦ(z)v0 ) = (k + g)z
d Φv0 v (z) + (v00 , La ⊗ Ja+ Φ(z)v0 ) . dz 0 0
In Ji+ only the zero mode contributes, and the second term reduces to const. × Φv00 v0 (z). The constant has the value 1 (C(µ) − C(ν) − C(π)) , 2
(4.20)
where C(µ) is the value of the Casimir operator C = m C in Vµ,k [0]. (Recall that if A = a ⊗ b ∈ A0 ⊗ A0 , then mA = ab ∈ A0 .)
1038
C. FRØNSDAL and A. GALINDO
We can reduce the value hζi of ζ to zero by choosing the grading of Φ according to Φ(z) =
X
Φ[n]z −n−(µ|vπ) ,
(µ|ν, π) :=
n∈Z
1 (C(µ) − C(ν) − C(π)) ; 2(k + g)
then for any weight vector w in V , (w ⊗ v00 , Φ(z)v0 ) = z −(µ|vπ) (w ⊗ v00 , Φ[0]v0 ) ,
(4.21)
and
d Φ(z) + :La ⊗ Ja Φ(z): = 0 . dz This is the “classical” Knizhnik–Zamolodchikov equation [20]. ζΦ(z) = (k + g)z
(4.22)
Alternative Polarizations The polarization defined by (4.17) and (4.18) is ad hoc. We have the freedom of shifting any finite set of summands from J + to J − , as in X X z n L−n z −n Lna ; Ja = Ja+ + Ja− = a + n>0
n≥0
the effect in this particular case is merely to change the sign of C(π) in (4.20). The result now agrees with [12]. Another polarization is suggested by (4.11), ζ = La ⊗ Ja = La ⊗ Ja+ + La ⊗ Ja− = rˆt + rˆ . Here we are dealing directly with the full Kac–Moody algebra, including the c, dterms in rˆ. Formally, the intertwining property gives ri rˆi ⊗ 1)Φ + (ˆ ri ⊗ 1)Φˆ ri , rˆΦ = (ˆ ri ⊗ rˆi )Φ = −(ˆ and
1 1 X n ri , rˆi ] . (4.23) C + c ⊗ d + d ⊗ c + [ˆ 2 2 The first term on the right-hand side of this last equation, although meaningless, looks like it may be a scalar, and thus ignorable. Proceeding heuristically up to ˆ Proposition 4.2, we begin by dropping this term. The other term is an element H of the Cartan subalgebra of ˆ g, rˆi rˆi =
ˆ = 1 [ˆ ri , rˆi ] ; H 2
(4.24)
it is determined up to an additive central element by ˆ eα ] = [eα , rˆi rˆi ] = m[eα ⊗ 1 + 1 ⊗ eα, rˆ] = m(ϕ(α, .) ∧ eα ) = ϕ(α, α)eα . [H,
(4.25)
If we restrict the relation (4.25) to the real simple roots, then it determines a unique element in the Cartan subalgebra of g, namely 1X [Eα , E−α ] . H= 2 α>0
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1039
Therefore, there is a unique element in the extended Cartan subalgebra, of the form: ˆ = H + gd , H
(4.26)
such that (4.25) holds for the affine root e0 as well. The integer g is the dual Coxeter number of g. The redefined operator is ˆ ⊗ 1)Φ(z) + :(ˆ ζΦ(z) := (H rt + rˆ)Φ(z): ˆ ⊗ 1)Φ(z) + rˆt (z)Φ(z) + (ˆ = (H ri (z) ⊗ 1)Φ(z)ˆ ri .
(4.27)
ˆ as the element of Notice that we did not actually use (4.24); instead we defined H ˆ g that has the same commutator with eα as (4.23). This makes it plausible that the term that was dropped is a scalar and that covariance is preserved. Indeed we have: Proposition 4.2. The operator product ζΦ(z) defined in (4.27) is covariant; that is, if Φ is an intertwiner then so is ζΦ. Proof. The coproduct is that of the classical limit, ∆(x) = x ⊗ 1 + 1 ⊗ x for x∈ˆ g. ˆ ⊗ 1)Φ(z) + [∆(x), rˆt (z)]Φ(z) ∆(x)ζΦ(z) − (ζΦ(z))x = ([x, H] ri + (ˆ ri (z) ⊗ 1)Φ[x, rˆi ] . + ([x, rˆi (z)] ⊗ 1)Φ(z)ˆ
(4.28)
Suppose first that x ∈ g; then in terms 2, 3, 4 only the zero modes contribute. The sums over the degree are now finite and ∆(x)ζΦ(z) − ζΦ(z)x = ([x, H] ⊗ 1)Φ(z) + ([x, ri (0)ri (0)] ⊗ 1)Φ(z) ,
(4.29)
which vanishes in view of (4.26). We shall verify that (4.28) holds for x = e0 . Besides (4.29) there are additional terms that arise from the extension term in the commutation relations, others that arise from the fact that e0 does not commute with the degree operator, and finally the more subtle contributions that come from the fact that the degree of [e0 , y] is shifted by 1 from that of y: ˆ ⊗ 1)Φ = (−ge0 ⊗ 1)Φ + ([e0 , H] ⊗ 1)Φ , ([e0 , H] X [∆(e0 ), rˆt (z)]Φ = [∆(e0 ), rˆt (0)]Φ + [∆(e0 ), Lna ⊗ L−n a ]Φ n>0
= [∆(e0 ), rˆ (0)]Φ + (L1a ⊗ [e0 , L−1 a ])Φ , X n ([e0 , rˆi (z)] ⊗ 1)Φˆ ri = ([e0 , rˆi (0)] ⊗ 1)Φˆ ri + ([e0 , L−n a ] ⊗ 1)ΦLa , t
n>0
(ˆ ri (z) ⊗ 1)Φ[e0 , rˆi ] = (ˆ ri (0) ⊗ 1)Φ[e0 , rˆi ] +
X
n (L−n a ⊗ 1)Φ[e0 , La ] .
n>0
1040
C. FRØNSDAL and A. GALINDO
The two infinite sums almost cancel, leaving only the first term of the second one. The sum of the last two expressions is 1 [∆(e0 ), rˆ(0)]Φ + ([e0 , ri (0)ri (0)] ⊗ 1)Φ + ([e0 , L−1 a ] ⊗ 1)ΦLa .
Adding the second expression we obtain 1 1 −1 [∆(e0 ), rˆ(0) + rˆt (0)]Φ + ([e0 , L−1 a ] ⊗ La )Φ + (La ⊗ [e0 , La ])Φ 1 + ([e0 , ri (0)ri (0)] ⊗ 1)Φ + ([e0 , L−1 a ]La ⊗ 1)Φ .
The first three terms cancel exactly and we have ∆(e0 )ζΦ(z) − ζΦ(z)e0 = −g(e0 ⊗ 1)Φ + ([e0 , H] ⊗ 1)Φ 1 + ([e0 , ri (0)ri (0)] ⊗ 1)Φ + ([e0 , L−1 a ]La ⊗ 1)Φ .
Terms two and three cancel as in (4.29) and the proposition is proved when we 1 verify that, in the evaluation module, [e0 , L−1 a ]La = [e0 , La ]La = ge0 , and repeat the calculation with e0 replaced by e−0 . Normalization Returning to (4.27) we put the degree operator into evidence: ζΦ(z) = (k + g)z
d Φ + (H ⊗ 1)Φ(z) + rt (z)Φ(z) + (ri (z) ⊗ 1)Φ(z)ri . dz
(4.30)
Again we fix the grading of the intertwiner as in (4.21), but now with (µ|ν, π) replaced by (µ|ν) =
A(w) , k+g
A := ϕ(., v0 ) + ϕ(v00 , .) + H =
1 (C(µ) − C(ν)) , 2
(4.31)
so that the operator form of the Knizhnik–Zamolodchikov equation takes the form: ζΦ(z) = 0 .
(4.32)
Note that this makes the grading of Φ independent of the choice of evaluation module; this grading/normalization is thus “universal”. Remark 4.3. In view of the interpretation of the quantum field Φ(z) as an intertwiner for highest weight affine Kac–Moody modules, the appearance of the rational r-matrix in the original KZ equation (4.22) has always seemed somewhat mysterious. The mystery is deepened by the discovery [19] that the monodromy associated with the solutions yields a representation of Uq (g). The alternative, to use the polarization based on the decompostion ζ = rˆ+ˆ rt , was first suggested in [12]; it seems to be more natural. However, Φ is defined as an intertwiner of Kac–Moody modules, with the classical coproduct; it knows nothing about r-matrices. Normal ordering is an example of additive renormalization, or “subtraction”, necessary
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1041
only if the ordinary product is ill defined. Any two polarizations that eliminate the divergent term by subtracting a scalar (that is; without compromising covariance) are equivalent, and one is not more natural than the other, in the present context at least. The fact that renormalization is required is revealed by the fact that the subtracted term, the last term in (4.15), is divergent. It is related to the fact that the classical r-matrix has a pole at z1 /z2 = 1. Remark 4.4. The appearance of the factor k + g as a coefficient of the degree operator in both versions is justified by covariance, as is indeed implied by the proof of Proposition 4.1 in [12]. The term (H ⊗ 1)Φ(z) has exactly the same origin. Perhaps it should be pointed out that the concept of “covariance” that is evoked in this section is quite distinct from the covariance under twisting that is alluded to in the title of the paper and in from Sec. 6 onward. 5. The Quantum KZ Equation Here we shall make sense of Eq. (4.2), (Z − hZi)Φ(z) = 0 ,
Z = RRt ,
in the quantized Kac–Moody algebra, to recover the q-KZ equation of Frenkel and Reshetikhin [12]. The action of Rt Φ on Vµ,k is well defined (in terms of formal series), since both Φ and Rt act by degree-decreasing operators in the second space (Proposition 3.1 and Eq. (3.9)), but the subsequent action of R is not. We therefore investigate the effect of normal ordering. Thus if R = Ri ⊗ Ri , we set (tentatively) :ZΦ: = (Ri ⊗ 1)ΨRi ,
Ψ := Rt Φ
(?) ,
and try to prove that the operator Z : Φ 7→ :ZΦ: is invariant; that is, that it commutes with the coproduct. In view of the intertwining property of Φ this is the same as ∆(x):ZΦ: = :ZΦ: x (?) . Attempts to verify this equation leads to: ˆ be the element in the Cartan subalgebra A0 of A0 with Proposition 5.1. Let H 0 the property ˆ ˆ (5.1) q H eα q −H = q ϕ(α,α) eα , and define the normal-ordered product :ZΦ: by ˆ i ⊗ 1)ΨRi , :ZΦ: := (R
Ψ := Rt Φ ,
ˆ i := Ri q Hˆ . R
(5.2)
Then ∆(x):ZΦ: = :ZΦ: x ,
∀x ∈ A0 .
(5.3)
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C. FRØNSDAL and A. GALINDO
Proof. We begin with R ∆0 (eα ) = ∆(eα )R; that is (Ri ⊗ Ri )(eα ⊗ 1 + q ϕ(α,.) ⊗ eα ) = (1 ⊗ eα + eα ⊗ q ϕ(α,.) )(Ri ⊗ Ri ) , and thus Ri ⊗ Ri eα = −Ri eα q −ϕ(α,.) ⊗ Ri + Ri q −ϕ(α,.) ⊗ eα Ri + eα Ri q −ϕ(α,.) ⊗ q ϕ(α,.) Ri , which gives us ˆ
:ZΦ: eα = (Ri q H ⊗ 1)ΨRi eα ˆ
= −(Ri eα q −ϕ(α,.) q H ⊗ 1)ΨRi ˆ
ˆ
+ (Ri q −ϕ(α,.) q H ⊗ 1)Ψeα Ri + (eα Ri q −ϕ(α,.) q H ⊗ 1)Ψq ϕ(α,.) Ri . Using the intertwining property of Ψ we convert the last two terms to (Ri q H ⊗ eα )ΨRi + (Ri q −ϕ(α,.) q H eα ⊗ 1)ΨRi + (eα Ri q H ⊗ q ϕ(α,.) )ΨRi . ˆ
ˆ
ˆ
As for the first term, we shift the operator eα to the right; since eα commutes with ˆ − ϕ(α, .) we get the required cancellation and the result is H ˆ i ⊗ eα )ΨRi + (eα R ˆ i ⊗ 1)ΨRi . ˆ i ⊗ q ϕ(α,.) )ΨRi = ∆(eα )(R :ZΦ: eα = (R ˆ In the classical limit the q-factor in (5.2) produces the H-term in Eq. (4.27). A similar calculation with e−α completes the proof of Proposition 5.1, and we have an independent confirmation of the covariance of (4.27). Normalization Our next task is to pull out the degree operator. Since the first space is an evaluation module, on which the central element c is zero, the degree operator d . We define L∓ by appears only in the first factors of R and Rt , as z dz ˆ i (z) ⊗ Ri = q (1−u)kd+gd (Ad(q −gd ) ⊗ 1)L− (z) , R
Rt (z) = q ukd L+ (z) ,
(5.4)
d acts in the evaluation module and Ad(x)y = xyx−1 . Objects denoted where d = z dz by the letter L (with ornamentation) do not contain d. We also need the expansions
L− (z) = L−i (z) ⊗ L− i ,
L+ (z) = L+i (z) ⊗ L+ i .
Now :ZΦ(z): = q (1−u)kd+gd (Ad(q −gd )L−i (z) ⊗ 1)(q ukd ⊗ 1)L+ (z)Φ(z)L− i = q (k+g)d (L−i (q −g−uk z) ⊗ 1)L+ (z)Φ(z)L− i =: q (k+g)d :L(z)Φ(z): .
(5.5)
Thus, the q-KZ equation for Φ: Φ(q −k−g z) = :L(z)Φ(z): = (L−i (q −g−uk z) ⊗ 1)L+ (z)Φ(z)L− i .
(5.6)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1043
Here we have fixed hZi = 1. It means that the grading of Φ is so chosen that, on any weight vector w ∈ V and the highest weight vectors v0 ∈ Vµ,k , v00 ∈ Vν,k , we have (w ⊗ v00 , Φ(q −k−g z)v0 ) = q (µ|ν) , (w ⊗ v00 , Φ(z)v0 ) with (µ|ν) as in (4.31). The other intertwiner, Ψ ∝ Rt Φ, satisfies Ψx = ∆0 (x)Ψ and :Z 0 Ψ: = Ψ where 0 Z := Rt R. We find ˆ i ⊗ 1)Ψ(z)Ri = q ukd L+ (z)q (k−uk+g)d (L−i (q −g z) ⊗ 1)Ψ(z)L− , :Z 0 Ψ(z): = Rt (R i (5.7) and thus, the q-KZ equation for Ψ: Ψ(q −k−g z) = :L0 (z)Ψ(z): = L+ (q −g−k+uk z)(L−i (q −g z) ⊗ 1)Ψ(z)L− i .
(5.8)
6. Hopf Twisting It is remarkable that the elliptic quantum group can be viewed as deformation of the trigonometric quantum group. The deformation does not affect the algebraic structure, which remains that of a quantized, affine Kac–Moody algebra. Only the coproduct distinguishes the elliptic case from the trigonometric one. The deformation is implemented by a twist in the category of Hopf algebras (this section) or quasi Hopf algebras (next section). The full elliptic quantum group is quasi Hopf; it becomes Hopf on the quotient by the ideal generated by the center. In this section we investigate the effect of twisting on the intertwiners and on the KZ equation, in the quantum case where the relationship between the intertwiner and the R-matrix is clearer. Definition 6.1. A formal Hopf deformation of a standard R-matrix R is a formal power series R = R + R1 + · · · , that satisfies the Yang–Baxter relation to each order in . It turns out [14] that the deformations of greatest interest have the form of a twist. Theorem 6.2. Let R be the R-matrix, ∆ the coproduct, of a coboundary Hopf algebra A0 , and F ∈ A0 ⊗ A0 , invertible, such that ((1 ⊗ ∆21 )F )F12 = ((∆13 ⊗ 1)F )F31 . Then ˜ := (F t )−1 RF R
(6.1)
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C. FRØNSDAL and A. GALINDO
(a) satisfies the Yang–Baxter relation and (b) defines a Hopf algebra A˜ with the same product and with coproduct ˜ = (F t )−1 ∆F t . ∆ This is a result of Drinfel’d [8]; a detailed proof was given in [14]. We say that a deformation R of a standard R-matrix R is implemented by a twistor F if there is a formal power series F = 1 + F1 + · · · that satisfies (6.1) to each order in and R = (Ft )−1 RF .
(6.2)
In this case the deformed R-matrix intertwines a deformed coproduct, R ∆0 = ∆ R ,
∆0 := F−1 ∆0 F .
(6.3)
Known solutions of (6.1) have the following structure [14]. We need a pair of ˆ i ⊂ {eα }α∈N , and a diagram subalgebras Γ1 , Γ2 of A0 = gˆ, generated by sets Γ ˆ ˆ isomorphism τ : Γ1 → Γ2 . A deformation exists when the parameters of A0 satisfy the following condition: ϕ(σ, .) + ϕ(., τ σ) = 0 ,
ˆ1 . σ∈Γ
ˆ 1 . Then there is a cocycle F of the form: Note that eτ σ is defined only if eσ ∈ Γ Y Fm := F1 F2 . . . Fm . . . , F = m≥1
Fm =
X
m(ρ)
mn F(σ) fσ1 . . . fσn ⊗ f−ρ1 . . . f−ρn ,
(6.4)
(σ)
fσ := q −ϕ(σ,.) eσ ,
f−ρ := e−ρ q ϕ(.,ρ) ,
where the sum is over all (σ) = σ1 , . . . , σn , and all permutations (σ 0 ) of (σ), such m(ρ) that ρi = τ m σi0 is defined. We take F(σ) = 1 when the set (σ) is empty. Note that the family of deformation of this type is large enough to contain the quantization of all the classical Lie bialgebras classified by Belavin and Drinfel’d, with r-matrices of constant, trigonometric and elliptic type. Two cases need to be distinguished. (a) Finite twisting is by definition the case when there is k such that for all σ, ˆ 1 ; then Γ ˆ1, Γ ˆ 2 are distinct and the product over m is finite. /Γ τ kσ ∈ (b) Elliptic twisting. The only other possibility (see [14, Sec. 16]) is that A0 = \) and Γ1 = Γ2 is generated by all the simple roots. This section deals sl(N with twisting in the category of Hopf algebras; elliptic twisting within the context of Hopf algebras implies [15] that we drop the central extension and descend to loop algebras. The full elliptic Kac–Moody algebra is quasi Hopf and will be discussed in the next section.
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1045
The deformed R-matrix and coproduct are R = (Ft )−1 RF ,
∆ = (Ft )−1 ∆Ft .
(6.5)
Here are some products that seem ill defined; thus R has degree-increasing operators in the second space, where F has degree-decreasing operators. This problem can be handled in a general way by adopting an interpretation that is quite natural in deformation theory. One notes that F is a formal power series in the deformation parameter . One interprets all the operators this way; then the problem reduces to making sure that the coefficients are well defined. Indeed, to any fixed order in , the product RF is, in the second space, a power series in the operators eα multiplied by a polynomial in the other generators. It is, nevertheless, of some interest to determine whether singularities arise as one assigns a value to and attempts to sum up the deformation series. In this respect cases (a) and (b) are quite different. (a) Finite twisting. The sum in (6.4) becomes finite when projected on a finite dimensional representation in either one of the two spaces. Infinite sums will appear if both representations are infinite, but there is a finite number of terms with fixed weight; therefore no infinite, purely numerical series will appear. Infinite sums with operator coefficients are beyond (our power of) analysis in the general case, and of no immediate concern to us. The value of is basis dependent; the only distinct possibilities are = 0, 1. (b) Elliptic twisting. We note that the range of is in this case || < 1. Here the situation is more delicate, and of some interest. Under twisting, the Casimir element Z suffers an equivalence transformation Z = (Ft )−1 ZFt ,
(6.6)
and one expects that an intertwiner Φ , satisfying ∆ (x)Φ = Φ x ,
x ∈ A0 ,
(6.7)
may be expressed as Φ = (Ft )−1 Φ. However, Ft has a structure similar to that of R, with degree-increasing operators in the second space, and we must consider the possibility that normal ordering may be required. In fact probably not, but since we have not proved this, we shall switch our attention to the other intertwiner. We consider instead the alternative intertwiner Ψ, and the alternative Casimir operator Z 0 that commutes with ∆0 (x), namely Z 0 = Rt R ,
Z 0 ∆0 (x) = ∆0 (x)Z 0 ,
(Z 0 − 1)Ψ = 0 .
(6.8)
We have Z0 = F−1 Z 0 F , F−1 Ψ
The operator product property of Ψ , namely
is therefore in the clear.
Ψ = F−1 Ψ .
(6.9)
is well defined as an operator on Vµ,k . The intertwining ∆0 (x)Ψ = Ψ x ,
(6.10)
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C. FRØNSDAL and A. GALINDO
We define the operator product Z0 Ψ (z). Formally, Z0 Ψ = Rt R Ψ = F−1 Z 0 Ψ , and this too is well defined, provided we define the untwisted product as in (5.7); that is ˆ Z 0 Ψ → :Z 0 Ψ: = Rt (Ri q H ⊗ 1)ΨRi . The equation satisfied by the twisted correlation function is (Z0 − 1)Ψ = 0 or more precisely: Definition 6.3. The twisted q-KZ equation is the following equation for the twisted intertwiner operator: Ψ (q −k−g z) = F−1 (q −k−g z)L+ (zq −g−k+uk )(L−i (zq −g ) ⊗ 1)F (z)Ψ (z)L− i . (6.11) It should be noted that the polarization used is the same as before deformation. To justify this we repeat that the definition of the intertwining operators is independent of normal ordering conventions, normal ordering is relevant only when the ordinary product does not exist, it is required to be well defined and covariant, nothing more. Of course, it is also true that, if Ψ is defined as in (6.9), then (6.11) is equivalent to (5.8). The top matrix element of Ψ is (v00 , Ψ v0 ) = (v00 , F−1 Ψv0 ) = (v00 , Ψv0 ) .
(6.12)
This shows that, in a complete description of, say, the eight-vertex model, both periodic and non-periodic functions appear. We had naively expected to encounter nothing but elliptic functions, that “the eight-vertex model lives on the torus”. Having thus discarded a prejudice, we are comfortable with the continued use, in the twisted case, of the original polarization based on the standard trigonometric R-matrix. The alternative of defining a normal-ordered product such that R Φ = (Ri ⊗ 1)Φ Ri is entirely redundant. Another idea is to replace matrix elements by traces, as suggested by Bernard [6] and in [12]. However, since we know that the elliptic quantum group, as an algebra, is the same as the standard quantum group (that is, a Kac–Moody algebra), there seems to be no reason to take less interest in the highest weight matrix elements in the elliptic case. Continuity of physics also suggests that we continue to work with the usual module structure, as was argued in [18, Sec. 4]. Trace functionals are interesting in themselves, but there seems to be no reason to neglect the matrix elements. The intertwiners of Kac–Moody modules, and the solutions of the KZ equation, know nothing about r-matrices. For all that we may derive different versions of the
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1047
equation, the solutions remain the same. To base the polarization on the R-matrix is not an imperative; more important is to adopt a workable definition that gives a meaning to the objects of interest; to wit, matrix elements of intertwiners. In the setting of conformal field theory twisting does not affect the quantization paradigm, but it does change the quantum fields (the intertwiners) and their operator product expansions. We shall need to know the twistor F . It is determined, uniquely, by the recursion relationsa [1 ⊗ fρ , Fm ] = m (Fm (fτ −m ρ ⊗ q −ϕ(ρ,.) ) − (fτ −m ρ ⊗ q ϕ(.,ρ) )Fm ) ,
(6.13)
with the initial conditions Fm = 1 − m
X
fτ −m ρ ⊗ f−ρ + · · ·
(6.14)
(ρ)
These equations were solved in a special case, and used to calculate the elliptic d in the fundamental representation [14]. Later, we shall exploit the R-matrix of sl(2) similarity between this relation and the recursion relation (2.4) for the universal R-matrix. 7. Quasi Hopf Twisting We are interested in the elliptic quantum groups, in the sense of Baxter [2] and Belavin [4]. This takes us out of the framework of Hopf algebras, but just barely so. The special nature of these quasi Hopf algebras is that they become Hopf algebras at level zero; that is, on the quotient by the ideal generated by the center. Quasi Hopf deformations are constructed in the same way as Hopf deformations, except that the element F need not satisfy the cocycle condition (6.1). The deformed R-matrix and coproduct are given by (6.5), but the former no longer satifies the Yang–Baxter relation and the latter is not coassociative, in general. m(ρ) If F(σ) are the coefficients of the elliptic Hopf twistor in (6.4), then the elliptic quasi Hopf twistor has the form [15] Y X m(ρ) F = Fm , Fm = nm F(σ) fσ1 . . . fσn ⊗ f−ρ1 . . . f−ρn Q(m, ρ) , m=1,2,...
(σ)
(7.1) where Q(m, ρ) ∈ A00 ⊗ A00 and A00 is the Cartan subalgebra of the quantized Kac– Moody algebra A0 . This factor is equal to unity in the Hopf case, and (7.1) then reduces to (6.4). The F -twisted algebra is a Hopf algebra when the parameters satisfy the condition ˆ1 , ϕ(σ, .) + ϕ(., τ σ) = 0 , σ ∈ Γ where now τ is the cyclic diagram automorphism that takes each simple root of sl(N ) to its neighbour. This condition can be satisfied on the loop algebra (when a The fact that the recursion relation (6.13) is necessary was shown in [14, 15]; sufficiency was proved quite recently, by Jimbo, Konno, Odake and Shiraishi [16].
1048
C. FRØNSDAL and A. GALINDO
c 7→ 0). We are interested in the full Kac–Moody algebra (c 6= 0); in that case the best that can be done is to choose parameters such that ϕ(σ, .) + ϕ(., τ σ) = [(1 − u)δσ0 + uδτ0σ ]c .
(7.2)
This algebra is what we mean by “elliptic quantum group in the sense of Baxter and Belavin”; it is a quasi Hopf algebra of a particularly benevolent type, where the deviation from coassociativity is confined to the center. Instead of the cocycle condition (6.1) we now have ((id ⊗ ∆21 )F )F12 = ((∆13 ⊗ id)F )F31,2 ,
(7.3)
where Fij,k is an extension of Fij , supported on the center, to the k’th space. In the case of interest, when we are dealing with modules with fixed level c 7→ k, this amounts to a modification of the coefficients in Fij . From (7.3) one gets the Cartan factors Q(m, ρ) [15] and the recursion relation [1 ⊗ fρ , Fm ] = m (Fm (fτ −m ρ ⊗ q −ϕ(ρ,.) ) − (fτ −m ρ ⊗ q ϕ(.,ρ) )Fm )Q(m, ρ) ,
(7.4)
with the initial conditions Fm = 1 − m
X
(fτ −m ρ ⊗ f−ρ )Q(m, ρ) + · · ·
(7.5)
ρ m m is known, F12,3 is obtained by means The solutions will be given later. Once F12 of the substitution 1 ⊗ c 7→ 1 ⊗ ∆(c) . (7.6)
8. Correlation Functions The main objects of interest, in conformal field theory as well as in the study of statistical models, are the correlation functions. In their simplest form they are matrix elements of products of intertwiners, fv0 v (z1 , . . . , zN ) = hv 0 , Φ(z1 ) . . . Φ(zN ) vi ∈ V1 (z1 ) ⊗ · · · ⊗ VN (zN ) , gv0 v (z1 , . . . , zN ) = hv 0 , Ψ(z1 ) . . . Ψ(zN ) vi ∈ V1 (z1 ) ⊗ · · · ⊗ VN (zN ) .
(8.1)
Here Φ(zp ) and Ψ(zp ) are intertwiners between highest weight modules, Φ(zp ), Ψ(zp ) : Vµp ,k → Vp (zp ) ⊗ Vµp−1 ,k ,
p = 1, . . . , N ,
with {Vp (zp )} a set of evaluation modules, and v ∈ VµN ,k , v 0 ∈ Vµ0 ,k . These “functions” are formal, V1 ⊗ · · · ⊗ VN -valued series in N distinct variables. Classical Correlation Functions We begin with the classical case and the polarization (4.18), X X z n L−n z −n Lna , Ja = Ja+ + Ja− = a + n>0
n≥0
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1049
and the normalization that leads to (4.22): (k + g)z
d Φ(z) + La ⊗ Ja+ Φ(z) + (La ⊗ 1)Φ(z)Ja− = 0 . dz
Then for any p ∈ {1, . . . , N }, (k + g)zp
d fv0 v (z1 , . . . , zN ) = −La(p) hv 0 , . . . Φ(zp−1 )Ja+ (zp )Φ(zp ) . . . vi dzp − La(p) hv 0 , . . . Φ(zp−1 )Φ(zp )Ja− (zp ) . . . vi .
(8.2)
Here La denotes the action of La in Vp . Suppose now that the vectors v0 and v00 are highest weight vectors of the respective highest weight modules. The intertwiners (p) satisfy [Lna , Φ(zp )] = −La zpn Φ(zp ); this allows us to permute J + through to the left, where it dies on the highest weight vector, and to permute J − towards the right, where only the zero modes survive, to contribute the last term in (p)
d fv0 v (z1 , . . . , zN ) dzp 0 0 X X zp n (p) 0 L(q) =− a La hv0 , . . . Φ(zp−1 )Φ(zp ) . . . v0 i z q n>0
(k + g)zp
1≤qp n≥0
zp
0 La(p) L(q) a hv0 , . . . Φ(zp−1 )Φ(zp ) . . . v0 i
+ La(p) hv 0 , . . . Φ(zp−1 )Φ(zp ) . . . La v0 i . Hence (k + g)
X d 1 fv00 v0 (z1 , . . . , zN ) = L(p) L(q) fv00 v0 (z1 , . . . , zN ) , p = 1, . . . , N , dzp zp − zq a a q6=p
(8.3) (N +1)
acts on v0 . The last where q takes the values 1, . . . , N + 1, zN +1 = 0, and La expression must be supplemented by the instruction X (1/z ) (zq /zp )n , q > p, p n≥0 1 (8.4) := X zp − zq n (−1/z ) (z /z ) , q < p . q p q n≥0
The domain of convergence is thus |z1 | > |z2 | > · · · > |zN +1 | = 0. In the simplest, nontrivial case N = 1. Projecting on a vector w ∈ V we get (k + g)
d w c fv0 v0 (z1 ) = fvw0 v0 (z1 ) , 0 dz1 z 0
c=
1 hw ⊗ v 0 , CΦ vi = (C(ν) − C(µ) − C(π)) , 0 hw ⊗ v , Φvi 2
1050
C. FRØNSDAL and A. GALINDO
which simply reflects the choice of grading of Φ. The case N = 2 is not much more complicated; the equations are (k + g)
df c12 f c13 f = + , dz1 z1 − z2 z1
(k + g)
df c12 f c23 f = + , dz2 z2 − z1 z2
(8.5)
d and funwhere cij = La La and “3” refers to the source space. In the case of sl(2) damental evaluation modules it is a simple matter to work out the hypergeometric solutions. The general structure of the solution was exploited by Khono [19] and Drinfel’d [8] to construct representations of the braid group and examples of quasi Hopf algebras. If instead we use the polarization of (4.27) we obtain from (4.30) and (4.31), on the vectors of highest weight, ! p−1 N X X d fv0 v (z1 , . . . , zN ) + Ap + rqp − rpq fv00 v0 (z1 , . . . , zN ) = 0 , (k + g)zp dzp 0 0 q=1 q=p+1 (i)
(j)
(8.6) Ap := (H +
ϕ(v00 , .)
+ ϕ(., v0 ))p ,
for p = 1, . . . , N . When N = 2, (k + g)z1
d fv0 v (z1 , z2 ) + A1 fv00 v0 (z1 , z2 ) − r12 fv00 v0 (z1 , z2 ) = 0 , dz1 0 0
(k + g)z2
d fv0 v (z1 , z2 ) + A2 fv00 v0 (z1 , z2 ) + r12 fv00 v0 (z1 , z2 ) = 0 . dz2 0 0
The solutions are, of course, the same, up to normalization. q-Deformed Correlation Functions We turn to the q-KZ equation (5.6), Φ(q −k−g z) = :L(z)Φ(z): . For functions of the type (8.1) the implication is Tp fv00 v0 (z1 , . . . , zN ) := fv00 v0 (. . . , zp−1 , q −k−g zp , zp+1 , . . .) = hv00 , . . . Φ(zp−1 )L−i (zp0 )L+ (zp )Φ(zp )L− i Φ(zp+1 ) . . . v0 i , with z 0 = q −g−uk z. More transparently, − Tp fv00 v0 (z1 , . . . , zN ) = [L−i (zp0 )L+j (zp )]hv00 , . . . Φ(zp−1 )L+ j Φ(zp )Li Φ(zp+1 ) . . . v0 i .
(8.7) For N = 2, 0
fv00 v0 (q −k−g z1 , z2 ) = [L−i (z10 )q −ϕ(v0 ,.) ]1 hv00 , Φ(z1 )L− i Φ(z2 )v0 i , fv00 v0 (z1 , q −k−g z2 ) = [q ϕ(.,v0 )+H L+i (z2 )]2 hv00 , Φ(z1 )L+ i Φ(z2 )v0 i .
(8.8)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1051
We reduce this using the quasi triangularity conditions in the Appendix. The final result is z2 g+k A1 −1 q fv00 v0 (z1 , z2 ) , q T1 fv00 v0 (z1 , z2 ) = R12 z1 z2 fv00 v0 (z1 , z2 ) , T2 fv00 v0 (z1 , z2 ) = q A2 R12 (8.9) z1 Ai := (ϕ(v00 , .) + ϕ(., v0 ) + H)i ,
i = 1, . . . , N .
These two equations can be combined in two ways. The result is the same in either case, in consequence of the fact that the operator A1 + A2 (the subscripts refer to the two evaluation modules) commutes with R12 . From the fact that the correlation function is invariant for the action of the Cartan subalgebra in the four spaces it follows in fact that we can replace 1 A1 + A2 → (C(µ) − C(ν)) . 2 The result is that T1 T2 fv00 v0 (z1 , z2 ) = q A1 +A2 fv00 v0 (z1 , z2 ) .
(8.10)
The two equations (8.9) are thus mutually consistent. For correlators with more than two intertwiners one obtains similar equations ([12] and below), and for them consistency depends on the fact that R satisfies the Yang–Baxter relation. For the other two-point function we have from (5.8), with z 00 = q −g−k+uk z, Tp gv00 ,v0 (z1 , . . . , zN ) − = L+i (zp00 )L−j (q −g zp )hv00 , . . . Ψ(zp−1 )L+ i Ψ(zp )Lj Ψ(zp+1 ) . . . v0 i ,
(8.11)
and, in particular, − T1 gv00 ,v0 (z1 , z2 ) = L+i (z100 )L−j (q −g z1 )hv00 , L+ i Ψ(z1 )Lj Ψ(z2 )v0 i ,
(8.12)
− T2 gv00 ,v0 (z1 , z2 ) = L+i (z200 )L−j (q −g z2 )hv00 , Ψ(z1 )L+ i Ψ(z2 )Lj v0 i ,
(8.13)
and with the help of the Appendix, −1 T1 gv00 ,v0 (z1 , z2 ) = q A1 R12
z2 z1
gv00 ,v0 (z1 , z2 ) ,
z2 −k−g A2 q gv00 ,v0 (z1 , z2 ) . q T2 gv00 ,v0 (z1 , z2 ) = R12 z1 The q-KZ equations for the 3-point functions are z2 k+g z3 k+g A1 −1 −1 T1 f (z1 , z2 , z3 ) = R12 R13 q f (z1 , z2 , z3 ) , q q z1 z1 z3 k+g A2 z2 −1 q R12 f (z1 , z2 , z3 ) , T2 f (z1 , z2 , z3 ) = R23 q z2 z1 z3 z3 A3 R23 f (z1 , z2 , z3 ) , T3 f (z1 , z2 , z3 ) = q R13 z1 z2
(8.14)
(8.15)
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C. FRØNSDAL and A. GALINDO
with A as before. Integrability is expressed as a cocycle condition that is precisely ˜ 12 is the Yang–Baxter relation for R, Eq. (A.6) with c2 = 0. (The tilde on R redundant.) Remarks 8. (1) It is interesting to note that 1
T1 T2 T3 f (z1 , z2 , z3 ) = q A1 +A2 +A3 f (z1 , z2 , z3 ) = q 2 (C(µ)−C(ν)) f (z1 , z2 , z3 ) . (8.16) This is what one expects, since the product of any number of intertwiners should have the universal property; see Remark 3.2, also (4.31) and (8.10). (2) The first and the last equations in (8.15) can be written as follows: −1 (k+g)d1 A1 q q f (z1 , z2 , z3 ) , T1 f (z1 , z2 , z3 ) = q −(k+g)d1 R1,32
T3 f (z1 , z2 , z3 ) = q A3 R21,3 f (z1 , z2 , z3 ) .
(8.17)
Here Ri,jk is the action of the universal R-matrix in the evaluation module via the opposite coproduct, R1,32 = (id ⊗ ∆0 )R, R21,3 = (∆0 ⊗ id)R. This too is an expression of universality; compare the first of (8.17) with the first of (8.9). Similarly one finds directly, using the formulas in the Appendix that, if g(z1 , z2 , z3) is the alternative 3-point function in (8.1), then z3 −k−g z3 −k−g A3 R13 q g(z1 , z2 , z3 ) q q T3 g(z1 , z2 , z3 ) = R23 z2 z1 = q −(k+g)d3 R12,3 q (k+g)d3 q A3 g(z1 , z2 , z3 ) .
(8.18)
The other two formulas cannot be obtained so directly, but the principle of universality encountered in Remarks 8 tells us that −1 g(z1 , z2 , z3 ) . T1 g(z1 , z2 , z3 ) = q A1 R1,23
(8.19)
Finally, from (8.16), T2 g(z1 , z2 , z3 ) = T1−1 q A1 +A2 +A3 T3−1 g(z1 , z2 , z3 ) −1 −(k+d)d1 q g(z1 , z2 , z3 ) . = q (k+g)d1 R12 q A2 R23
(8.20)
Summing up, we have −1 g(z1 , z2 , z3 ) , T1 g(z1 , z2 , z3 ) = q A1 R1,23 −1 T2 g(z1 , z2 , z3 ) = q −(k+g)d2 R12 q (k+g)d2 q A2 R23 g(z1 , z2 , z3 ) ,
(8.21)
T3 g(z1 , z2 , z3 ) = q −(k+g)d3 R12,3 q (k+g)d3 q A3 g(z1 , z2 , z3 ) . Twisting and Covariance Let us evaluate one of the two-point functions of the twisted model, g (z1 , z2 ) = hv 0 , Ψ (z1 )Ψ (z2 ) vi = hv 0 , F−1 Ψ(z1 )F−1 Ψ(z2 ) vi .
(8.22)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1053
On highest weight vectors, g (z1 , z2 ) = hv00 , Ψ(z1 )F−1 (z2 )Ψ(z2 ) v0 i = F−1,i (z2 )hv00 , ∆0 (Fi−1 )Ψ(z1 )Ψ(z2 ) v0 i . (8.23) Now ∆0 (f−ρ ) = f−ρ ⊗ 1 + q ϕ(.,ρ) ⊗ f−ρ and so, for Hopf deformations, when F is a series of the type (6.4), g (z1 , z2 ) = F−1 (z2 , z1 )g(z1 , z2 ) .
(8.24)
An alternative derivation of this result makes direct use of the cocycle condition. It can be written as follows −1 −1 ((id ⊗ ∆31 )F−1 ) = F21 ((∆12 ⊗ id)F−1 ) . F13
(8.25)
Applying v00 we get, because this vector is a highest weight vector, −1 ... , hv00 , (id ⊗ ∆31 )F−1 ) . . . = hv00 , F21
(8.26)
which is just what we need to reduce (8.23) to (8.24). The transformation formula (8.24) shows that the result (8.14) is not covariant with respect to twisting, in the following sense. The equation satisfied by g is z2 −1 F (z2 , z1 )g (z1 , z2 ) ; T1 g (z1 , z2 ) = F−1 (z2 , q −k−g z1 )q A1 R12 z1 the right-hand side is very different from the naive analogue of (8.14), z2 A1 −1 q R12 g (z1 , z2 ) . z1 Thus twisting does not preserve the form of the equations satisfied by matrix elements of intertwining operators; one cannot simply replace R in these equations by a twisted R-matrix. In fact, it is clear that our calculations made use of the specific form of the standard R-matrix. The factors q A , in particular, are characteristic of the standard R-matrix. Of course, we do not deny the existence of holonomic difference equations that involve R-matrices of a more general type. The claim is that the solutions to such equations are not, in general, matrix elements of intertwining operators for highest weight, quantized Kac–Moody modules. The elliptic correlation functions can be found by solving a “modified” q-KZ equation, but much more simply by the intermediary of the solutions of the standard q-KZ equations for the 6-vertex model, as in Eq. (8.24). For three-point functions the effect of twisting is g (z1 , z2 , z3 ) = hv00 , F−1 Ψ(z1 )F−1 Ψ(z2 )F−1 Ψ(z3 )v0 i −1 Ψ(z3 )v0 i = F−1,i (z2 )F−1,j (z3 )hv00 , Ψ(z1 )Fi−1 Ψ(z2 )Fj −1 )Ψ(z2 )Ψ(z3 )v0 i = F−1,i (z2 )F−1,j (z3 )hv00 , ∆41 (Fi−1 )Ψ(z1 )∆42 (Fj −1 −1 i )i (z2 )F−1,j (z3 )hv00 , Ψ(z1 )(Fj ) Ψ(z2 )Ψ(z3 )v0 i = F−1 (z2 , z1 )(Fj
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C. FRØNSDAL and A. GALINDO
−1 −1 i = F−1 (z2 , z1 )(Fj )i (z2 )F−1,j (z3 )hv00 , ∆41 ((Fj ) )Ψ(z1 )Ψ(z2 )Ψ(z3 )v0 i −1 −1 i )i (z2 )F−1,j (z3 )hv00 , ((Fj ) )(z1 )Ψ(z1 )Ψ(z2 )Ψ(z3 )v0 i = F−1 (z2 , z1 )(Fj −1 )(z1 , z2 )F−1,j (z3 )g(z1 , z2 , z3 ) = F−1 (z2 , z1 )∆12 (Fj
= F−1 (z2 , z1 )((id ⊗ ∆)F−1 )(z3 , z1 , z2 )g(z1 , z2 , z3 ) .
(8.27)
Thus we conclude that the twisted correlation functions can be obtained from the untwisted ones. The latter are found by solving equations that are known to be integrable by virtue of the fact that the standard R-matrix satisfies the Yang– Baxter relation. It is possible, but redundant and unrewarding, to write down the equations satisfied by the twisted correlation functions; they are complicated and uninstructive whether expressed in terms of R or R . 9. Correlation Function for the 8-Vertex Model Here we try to understand what, if any, are the qualitative new features that result from the fact that the elliptic quantum group is not a Hopf algebra. Technically, the difference is that the reduction of (8.23) to (8.24) is no longer valid, because the twistor is no longer of the type (6.4). Instead of the cocycle condition (6.1) that gave us (8.25) we now have the modified cocycle condition (7.3), which yields −1 −1 ((id ⊗ ∆31 )F−1 ) = F21,3 ((∆12 ⊗ id)F−1 ) (9.1) F13 and, instead of (8.24), −1 (z2 , z1 )g(z1 , z2 ) . g (z1 , z2 ) = F21,3
(9.2)
To calculate F21,3 see (7.6). This is the two-point function for the eight-vertex model. The quasi Hopf nature of the elliptic quantum group is parameterized by the level k of the highest weight module and the effect on the two-point function is in the numerical modification of the matrix F that is indicated by the third index. Equation (8.27) gets modified in the same manner. To get an idea of the importance of this effect it is enough to calculate the modified matrix in the case N = 2 with V the fundamental sl(2)-module. The result is as follows. The trigonometric R-matrix is given for comparison, with the two spaces interchanged: Rt = 2m F12,3 =
2m−1 = F12,3
A(q, x) ϕ q ((1 − q −2 x)H+ + (1 − x)H− + e1 ⊗ e−1 + e0 ⊗ e−0 ) , 1 − q −2 x
(9.3)
A2m (q, x, ) ((1 − q 2 αα0 x)H+ + (1 − αα0 x)H− − αf1 ⊗ f−1 − α0 f0 ⊗ f−0 ) , 1 − q 2 αα0 x (9.4) A2m−1 (q, x, ) ((1 − β 2 x)H+ + (1 − q 2 β 2 x)H− − βf1 ⊗ f−0 − βf0 ⊗ f−1 ) . 1 − q2 β 2 x (9.5)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1055
Here x = z1 /z2 , H± =
1 [(1 + H) ⊗ (1 ± H) + (1 − H) ⊗ (1 ∓ H)] 4
(9.6)
(in another notation, H+ = e11 ⊗ e11 + e22 ⊗ e22 , H− = e11 ⊗ e22 + e22 ⊗ e11 ), and α = q uk (2 q −k )m , α0 = q (1−u)k (2 q −k )m , β 2 = q k (2 q −k )2m−1 . d Remember that k denotes the level of the highest weight sl(2)-module. It enters here because it appears in the extension F21,3 of the twistor in Eq. (9.2). This operator acts in three spaces, but its action on the highest weight module is limited to the center. In the level zero case we recover the Hopf twistor. The calculation that leads to (9.3) is given in detail in the Appendix, with an explicit formula for the normalizing factor A(q, x). The matrix factors in (9.4) and (9.5) are obtained in the same way, and the scalar factors as follows: Proposition 9.1. (a) The normalizing factor in (9.4) is A2m (q, x, ) = A(1/q, αα0 x) .
(9.7)
(b) The normalizing factor in (9.5) is A2m−1 (q, x, ) = A(1/q, β 2 x) .
(9.8)
Proof of (a). Consider the universal R-matrix, and the algebra map generated by e1 → α−1 f1 ,
e−1 → −αf−1 ,
−1
e0 → α0 f0 ,
e−0 → −α0 f−0
(9.9)
in the second space, but ei → fi in the first space. This maps the original algebra to another algebra with the q replaced by q −1 . Now consider the factorization (2.1) of the universal R-matrix, R = q ϕ T . After (9.9), the first two terms in T t agree with the first two terms of F2m . The recursion relations (2.4) and (7.4) also agree, after replacing q by 1/q, and so do the solutions. Then we pass to the evaluation representation, setting f0 = zˆ1 f−1 . In the R-matrix (more precisely, in T t ) we have set e0 = z1 e−1 , which after the substitution (9.9) becomes α0 f0 = (z1 /α)f−1 , so that z1 = αα0 zˆ1 . Under these transformations, including transposition of the two spaces, the polynomial factor in (9.3) is transformed into that of (9.4), and the normalizing factor also agree. Proof of (b). In this case, in the second space let e1 → β −1 f0 , e−1 → −βf−0 , e0 → β −1 f1 , e−0 → −βf−1 , and in the first space ei → fi . Putting it all together, we have after a simple change of basis a dˆ b cˆ , x = z1 /z2 , F12,3 (z1 , z2 ) = A(F ) cˆ b ˆ d a
(9.10)
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C. FRØNSDAL and A. GALINDO
with A(F ) =
Y
A(q −1 , q k ¯4m+2 x) =
m≥0
=
Y (1 − xq k ¯4m+2 q 4n )(1 − xq k ¯4m+2 q 4n+4 ) (1 − xq k ¯4m+2 q 4n+2 )2
m,n≥0
(q k ¯2 x; q 4 , ¯4 )∞ (q k+4 ¯2 x; q 4 , ¯4 )∞ . (q k+2 ¯2 x; q 4 , ¯4 )2∞
and a ± dˆ =
Y (1 ± q −1+k/2 √x ¯2m−1 ) √ , (1 ± q 1+k/2 x ¯2m−1 ) m≥1
(9.11)
b ± cˆ =
Y (1 ± q −1+k/2 √x ¯2m ) √ , (1 ± q 1+k/2 x ¯2m ) m≥1
with ¯2 = 2 q −k . Finally, we give the result of projecting the universal elliptic d on the evaluation representation (k = 0), R-matrix of sl(2) α δ β γ R (z1 , z2 ) = ((Ft )−1 RF )(z1 , z2 ) = A (q, x−1 ) , γ β δ α where α+δ = q β+γ =
θ3 (u − ρ, τ ) θ2 (u − ρ, τ ) , α−δ =q , θ3 (u + ρ, τ ) θ2 (u + ρ, τ )
θ1 (u − ρ, τ ) , θ1 (u + ρ, τ )
β−γ =
θ(u − ρ, τ ) , θ(u + ρ, τ )
with x = z1 /z2 = e4πiu ,
q = e2πiρ ,
= eπiτ ,
A (q, x−1 ) = A(q, x−1 )A(F )/A(Ft ) .
In terms of the Jacobian elliptic functions one has α+δ :α−δ :β+γ :β−γ =
1 cn(2K(u − ρ), k) 1 sn(2K(u − ρ), k) dn(2K(u − ρ), k) :1: : , dn(2K(u + ρ), k) q cn(2K(u + ρ), k) q sn(2K(u + ρ), k)
where K, k are the real quarter-period and modulus, respectively, for the nome : 2 4 √ Y π Y 1 + 2n−1 1 − 2n 1 + 2n · , k = 4 . K= 2 1 − 2n−1 1 + 2n 1 + 2n−1 n≥1
n≥1
Acknowledgements We thank Olivier Babelon, Benjamin Enriquez, Moshe Flato, Tetsuji Miwa and Nikolai Reshetikhin for advice. We thank Moshe Flato for an incisive and constructive criticism of the original manuscript. A. G. thanks the Fundaci´on Del Amo for financial support and the Department of Physics of UCLA for hospitality.
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1057
Appendix Solving the Recursion Relations We shall solve the recursion relation (2.4) in the fundamental evaluation repred Here we set sentation of sl(2). 1 1 0 0 1 0 0 . e1 = κ , e−1 = κ , ϕ= H ⊗H, H = 0 −1 0 0 1 0 2 (A.1) The commutation relations hold with κ2 = q − q −1 . The factor T in R = q ϕ T has the form a b cx T = , x = z1 /z2 , c b a and (2.4) is equivalent to [T, 1 ⊗ e−γ ] = (e−γ ⊗ q ϕ(γ,.) )T − T (e−γ ⊗ q −ϕ(.,γ)) ,
γ = 1, 0 ,
with ϕ(1, .) = ϕ(., 1) = H, ϕ(0, .) = ϕ(., 0) = −H. Taking γ = 1 we get two relations, q(a − b) = c = (aq − b/q)/x , (A.2) and taking γ = 0 the same two relations. Hence R(q, x−1 ) =
A(q, x−1 ) ϕ q ((1 − q −2 x−1 )H+ + (1 − x−1 )H− + e−1 ⊗ e1 + e−0 ⊗ e0 ) . 1 − q −2 x−1
The matrices in (9.4) and (9.5) are found in the same way. In the special case of d the Cartan factors in (7.5) are, for m ≥ 1, sl(2) Q(2m, 1) = q (u−m)c ,
Q(2m, 0) = q (1−u−m)c ,
Q(2m − 1, 1) = Q(2m − 1, 0) = q (1−m)c . In the structure, R is determined uniquely by the recursion relations and the initial conditions, but in the evaluation representation the normalizing factor A(q, x−1 ) remains undetermined. Fortunately Levendorskii, Soibelman and Stukopin [21], starting from an equivalent expression for the standard, universal R-matrix for d obtain the following result, sl(2) X 1 q k − q −k xk . A(q, x) = exp (A.3) k q k + q −k k≥1
The sum converges for |q| 6= 1, |x| < 1 and the formula can be manipulated to yield (xq 2 ; q 4 )2∞ |q| < 1 , (x; q 4 )∞ (xq 4 ; q 4 )∞ , (A.4) A(q, x) = (x; q −4 )∞ (xq −4 ; q −4 )∞ , |q| > 1 . (xq −2 ; q −4 )2∞
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C. FRØNSDAL and A. GALINDO
Hence A(q, x)A(q −1 , x) = 1 ,
|q| 6= 1 .
(A.5)
This is also clear from (A.3). The inverse of R can also be represented as a series, similar to (2.1), X eˆ−α ⊗ eˆα + · · · , R−1 = q −ϕ Tˆ , Tˆ = 1 + α
with eˆα := q −ϕ(α,.) eα ,
eˆ−α = −e−α q ϕ(.,α) .
The commutation relations for the eˆ’s agree with those of the e’s, and the recursion relations for Tˆ agrees with that of T , all up to the sign of ϕ. (We get a recursion relation for Tˆ from the fact that R−1 also satisfies the Yang–Baxter relation.) Consequently, in the structure, R(ϕ, e)−1 = R(−ϕ, eˆ) , and in any evaluation representation, R(q, x)−1 = R(q −1 , x) . These results are quite general and imply, in particular, Eq. (A.5). Reduced Formulas We list here the formulas that are obtained from the Yang–Baxter relation R12 R13 R23 = R23 R13 R12 and the quasi triangular conditions (id ⊗ ∆)R = R12 R13 ,
(∆ ⊗ id)R = R23 R13
when the c, d factors are removed as in ˜, R = q uc ⊗ d+(1−u)d ⊗ c R namely ˜ 23 = R ˜ 23 (q −(1−u)d1 c2 R ˜ 12 ˜ 13 q uc2 d3 )R ˜ 13 q (1−u)d1 c2 )R ˜ 12 (q −uc2 d3 R R
(A.6)
and ˜ = (q −(1−u)d1 c3 R ˜ 13 , (∆ ⊗ id)R ˜ = (q −uc1 d3 R ˜ 13 . ˜ 12 q (1−u)d1 c3 )R ˜ 23 q uc1 d3 )R (id ⊗ ∆)R (A.7) These last two relations give us what we need to reduce (8.8), namely −1 z2 k−uk − ˜ L (z1 )13 Φ(z2 ) = R12 q (L−i (z1 ) ⊗ 1)Φ(z2 )L− i , z1 L
+i
(z2 )Φ(z1 )L+ i
˜ 12 = L (z2 )R +
z2 z1
(A.8)
Φ(z1 ) .
(A.9)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
For the other intertwiner, there is an analogue of (A.9), ˜ 12 z2 q −uk L+ (z2 )Ψ(z1 ) , = R L+i (z2 )Ψ(z1 )L+ i z1
1059
(A.10)
but we could not find an analogue of (A.8). To obtain (8.14) we used the method that was explained for the derivation of (8.21). References [1] O. Babelon and D. Bernard, “A Quasi Hopf interpretation of quantum 3-j and 6-j symbols and difference equations”, q-alg/9511019. [2] R. J. Baxter, “Partition Function of the Eight-Vertex Model”, Ann. Phys. 70 (1972) 193–228. [3] R. J. Baxter and S. B. Kelland, J. Phys. C: Solid State Phys. 7 (1974) L403–6. [4] A. A. Belavin, “Dynamical symmetry of integrable systems”, Nucl. Phys. 180 (1981) 198–200. [5] A. A. Belavin and V. G. Drinfeld, “Triangle equation and simple Lie algebras”, Sov. Sci. Rev. Math. 4 (1984) 93–165. [6] D. Bernard, “On the WZW model on the torus”, Nucl. Phys. B303 (1988) 77–174. [7] V. G. Drinfeld, “Quantum groups”, in Proc. Int. Congress Math. Berkeley, ed. A. M. Gleason, A. M. S., Providence, 1987. [8] V. G. Drinfeld, “Quasi Hopf Algebras”, Leningrad Math. J. 1 (1990) 1419–1457. [9] B. Enriquez and I. V. Rubtsov, “Quasi Hopf algebras associated with sl(2) and complex curves”, q-alg/9608005. [10] G. Felder, “Elliptic quantum groups”, hep-th/9412207. [11] O. Foda, K. Iohara, M. Jimbo, T. Miwa and H. Yan, “An elliptic algebra for sl(2)”, RIMS preprint 974. [12] I. B. Frenkel and N. Yu. Reshetikhin, “Quantum affine algebras and holonomic difference equations”, Commun. Math. Phys. 146 (1992) 1–60. [13] I. B. Frenkel, N. Yu. Reshetikhin and M. Semenov-Tian-Shansky, “Drinfeld–Sokolov reduction for difference operators and deformations of W -algebras I. The case of Virasoro algebra”, q-alg/9704011. [14] C. Frønsdal, “Generalization and deformations of quantum groups”, RIMS Publ. 33 (1997) 91–149 (q-alg/9606020). [15] C. Frønsdal, “Quasi Hopf deformation of quantum groups”, Lett. Math. Phys. 40 (1997) 117–134 (q-alg/9611028). [16] M. Jimbo, H. Konno, S. Odake and J. Shiraishi, “Quasi Hopf twistors for elliptic quantum groups”, q-alg/9712029. [17] M. Jimbo and T. Miwa, “Algebraic analysis of solvable lattice models”, Regional Conference Series in Math. (1995) Number 85. [18] M. Jimbo, T. Miwa and A. Nakayashiki, “Difference equations for the correlation functions of the eight-vertex model”, J. Phys. A: Math. Gen. 206 (1993) 2199–2209. [19] T. Kohno, “Monodromy representations of braid groups and Yang–Baxter equations”, Ann. Inst. Fourier (Grenoble) 37 (4) (1987) 139–160. [20] V. G. Knizhnik and A. B. Zamolodchikov, “Current algebra and Wess-Zumino model in two dimensions”, Nucl. Phys. B247 (1984) 83–103. [21] S. Levendorskii, Y. Soibelman and V. Stukopin, “The Quantum Weyl Group and (1) the Universal Quantum R-Matrix for Affine Lie Algebra A1 ”, Lett. Math. Phys. 27 (1993) 253–264. [22] N. Yu. Reshetikhin, “Multiparameter quantum groups and twisted quasitriangular Hopf algebras”, Lett. Math. Phys. 20 (1990) 331–336.
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS ` MICHELE IRAC-ASTAUD and GUY RIDEAU Laboratoire de Physique Th´ eorique de la mati` ere condens´ ee Universit´ e Paris VII 2 place Jussieu F-75251 Paris Cedex 05 France Received 15 July 1997 Generalizing the case of the usual harmonic oscillator, we look for Bargmann representations corresponding to deformed harmonic oscillators. Deformed harmonic oscillator algebras are generated by four operators a, a† , N and the unity 1 such as [a, N ] = a, [a† , N ] = −a† , a† a = ψ(N ) and aa† = ψ(N + 1). We discuss the conditions of existence of a scalar product expressed with a true integral on the space spanned by the eigenstates of a (or a† ). We give various examples, in particular we consider functions ψ that are linear combinations of q N , q −N and unity and that correspond to q-oscillators with Fock-representations or with non-Fock-representations.
1. Introduction The harmonic oscillator Lie-algebra is defined by four operators: the annihilation operator a, the creation operator a† , the energy operator N and the unity 1 satisfying the following commutation relations: [a, N ] = a ,
[a† , N ] = −a†
(1)
and [a, a† ] = 1 ,
(2)
†
where a is the adjoint of a and N is self-adjoint. This algebra has been deformed in many different ways (see in particular [1–6]) and the representations of the deformed algebras were widely studied. In this paper, the deformed harmonic oscillator is defined by the relations (1) and by the following relations between the three operators a, a† and N : a† a = ψ(N ) ,
aa† = ψ(N + 1) ,
(3)
where ψ is a real analytical function. In the other formulations encountered in the literature, (3) is replaced by [a, a† ] = f (N, q) ,
(4)
[a, a† ]q ≡ aa† − qa† a = fq (N, q) .
(5)
or
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M. IRAC-ASTAUD and G. RIDEAU
In these approach, the function ψ is not given but results of solving the equations: f (N, q) = ψ(N + 1) − ψ(N ) or fq (N, q) = ψ(N + 1) − qψ(N ) .
(6)
The resolution of these equations leads to some arbitrariness that is eliminated in our formulation, f and fq being uniquely determined in terms of ψ. Let us give some examples: — the usual harmonic oscillator defined by f (N ) = 1 corresponds to ψusual (N ) = N + σ. — the q-oscillator [1, 2] defined by fq (N, q) = q −N corresponds to ψqosc (N ) = −q −N /(q − q −1 ) + σq N /(q − q −1 ) ,
∀σ.
(7)
— the q-oscillator defined by fq (N, q) = 1 corresponds to 0 (N ) = (1 − q)−1 + σq N , ψqosc
— with the usual notation: [x] =
∀σ.
q x − q −x q − q −1
(8)
(9)
the function ψsuq (2) (N ) = σ − [N − 1/2]2 , ∀ σ, corresponds to f (N ) = −[2N ]; that is to the deformation suq (2) of the Lie-algebra su(2) after the identification a = L− , a† = L+ and Lz = N . — suq (1, 1) is obtained for the ψsuq (1,1) (N ) = −ψsuq (2) (N ). Generalizing the pioneer work of Bargmann [7] for the usual harmonic oscillator, the purpose of this paper is to study if the deformed harmonic oscillator defined by (1) and (3) admits representations on one space of complex variable functions. In [8], we restricted to the case where the function ψ does not vanish. The scalar product of the representations we are looking for, is written with a true integral as in [7–12] and contrarily to the works of [13–19] where a q-integration occurs. In Sec. 2, we recall how to build the irreducible representations on the basis of the eigenvectors of N . They are determined by the spectrum of N which is depending on the zeros of ψ. Then, in Sec. 3, we discuss the existence of the coherent states that are defined as the eigenstates of the operators a (or a† ). In Sec. 4, we study the possibility of Bargmann representations. The formulation of the problem is done in a general framework. We show on various examples how the construction works: Sec. 5 is devoted to strictly positive function ψ , other cases are considered in Sec. 6. 2. Representations Let |0i be the eigenvector of N with eigenvalue µ. We built the normalized vectors |ni ( λn a†n |0i , n ∈ N + (10) |ni = λn a−n |0i , n ∈ N −
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
with
λ−2 n = ψ(µ + n)! =
n Y ψ(µ + i) ,
1063
n ∈ N+
i=1
n+1 Y ψ(µ + i) , n ∈ N −
(11)
i=0
N + and N − are the set of integers ≥ 0 and < 0. The vectors |ni are the eigenvectors of N with eigenvalue µ + n and span the Hilbert space H. As hn|aa† |ni is necessarily positive or zero, the construction of the increasing states stops if it exists an integer ν+ such that ψ(µ + ν+ + 1) = 0
(12)
in which case the representation labelled by µ and ν+ admits a highest weight state |ν+ i. We have an analogous situation for the decreasing states built with a, when it exists an integer ν− such as ψ(µ + ν− ) = 0. The representation labelled by µ and ν− then admits a lowest weight state |ν− i. We get different types of representations [3, 4, 5, 6]: 1) ψ has no zero. The inequivalent representations are labelled by the decimal part of µ and are defined by: † 1/2 a |ni = (ψ(µ + n + 1)) |n + 1i a|ni = (ψ(µ + n))1/2 |n − 1i , N |ni = (µ + n) |ni
n∈Z
(13)
The spectrum of N , SpN , is µ + Z. The operator N has no lowest and no highest eigenstates. These representations, thus, are non equivalent to Fock-representations and are called non-Fock-representations [20, 21]. It is the case when ψ is equal to ψqosc with q ∈ [0, 1] and σ ≤ 0. An interpretation of this case [12] is obtained by identifying the states |ni to the functions on a circle. 2) ψ has zeros. We are interested in the intervals where ψ is positive: a) finite intervals We can associate a representation to the intervals that have a length equal to an T integer. The spectrum of N is [µ + ν− , µ + ν+ ] Z + µ. For example, in the case ψsuq (2) , when σ = [l + 1/2]2 , l being a positive integer, the dimension of the representation is 2l + 1 and verifies: 1 a† |l, mi= l + 1 2 − m + 1 − 1 2 2 |l, m + 1i 2 2 1 (14) 1 2 1 2 2 − m − |l, m − 1i a|l, mi = l + 2 2 N |l, mi= m|l, mi
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b) infinite intervals The representations are similar to the Fock-representation of the usual harmonic oscillator. The spectrum of N is µ + ν− + N + or µ + ν+ + N − . Let us give an example: when ψ is equal to ψqosc with σ = 1, we recover the usual q-oscillator case such as a† |ni=[n + 1]1/2 |n + 1i a|ni =[n]1/2 |n − 1i , n ∈ Z+ (15) N |ni= n |ni The first step to build a Bargmann representation requires to study the coherent vectors. 3. Coherent States We call coherent states [22], the eigenvectors of the operator a or a† . The state P |zi = p cp |pi is an eigenvector of a if the coefficients cp verify the recursive relation zcp = ψ(µ + p + 1)1/2 cp+1 .
(16)
— When the spectrum of N is upper bounded, (16) implies that all the cp vanish and then that a has no eigenvectors. If we look for the eigenvectors of a† , the situation is analogous: a† has no eigenvectors if the spectrum of N is lower bounded. Therefore, in the case (2.a) of the previous section as the spectrum of N is finite, a and a† have no eigenvectors, hence no Bargmann representation exists. — When the spectrum of N is no upper bounded, the eigenvectors |zi of a take the form: when SpN = Z + µ , −∞ ∞ X X zn n 1/2 |zi = z (ψ(µ + n)!) |ni + |ni , ψ(µ + n)!1/2 n=−1 n=0 when SpN = ν− + µ + N + , (17) ν− ∞ X X zn n 1/2 z (ψ(µ + n)!) |ni + |ni , ν− < 0 |zi = ψ(µ + n)!1/2 n=−1 n=0 ∞ X z n (ψ(µ + n)!)−1/2 |ni , ν− ≥ 0 |zi = n=ν−
with the convention ψ(µ)! = 1. The domain D of existence of the coherent states depends on the function ψ. Indeed, |zi belongs to the Hilbert space spanned by the basis |ni only if the series in the right-hand side of (17) are convergent in norm. — When SpN = Z + µ, this implies that: |z| < lim ψ(p)1/2 = r2 , p→∞
(18)
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1065
and |z| > lim ψ(p)1/2 = r1 . p→−∞
(19)
Thus when r2 = 0, the annihilation operator has no eigenvectors. When r1 is smaller than r2 , the eigenstates of a exist and their domain is r1 < |z| < r2 . When r1 is larger than r2 , the annihilation operator a has no eigenstates, but then we can establish by analogous reasoning that the creation operator a† has eigenstates if r1 6= 0. — When the spectrum of N is lower bounded, SpN = µ + ν− + N + , the second condition (19) does not exist and the eigenstates of a always exist provided r2 6= 0; their domain is defined by |z| < r2 . When the spectrum of N is upper bounded, SpN = µ + ν+ + N − , the eigenvectors of a† exist only if |z| < r1 . To summarize, the eigenvectors of a exist if: — SpN = µ + Z and r22 ≡ ψ(+∞) > r12 ≡ ψ(−∞), the domain of existence D of the coherent states is D = {z; r1 < |z| < r2 } or — SpN = µ + ν− + N + , then D = {z; |z| < r2 }. The eigenvectors of a† exist if: — SpN = µ + Z and r2 < r1 , then D = {z; r2 < |z| < r1 } or — SpN = µ + ν+ + N − , then D = {z; |z| < r1 }. The part played by a and a† is analogous, in the following we restrict to the case where the eigenstates of a exist, that is: — ψ is a strictly positive function with r1 < r2 — ∃x0 such that ψ(x0 ) = 0 and ψ(x) > 0 when x > x0 . We do not study here the case where r1 = r2 . Although µ is a significant quantity of labelling inequivalent representations, it does not play a part in the present problem. So we simplify the notation in assuming µ = 0 from now on. Indeed, this is equivalent to substituting N − µ to N and ψµ (N ) = ψ(µ + N ) to ψ(N ). 4. Bargmann Representation 4.1. Representation space Following the construction [7], in the Bargmann representation any state |f i of H: |f i =
X n∈SpN
fn |ni ,
X n∈SpN
|fn |2 < ∞
(20)
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M. IRAC-ASTAUD and G. RIDEAU
is represented as the function of a complex variable z, f (z) = hz|f i, with a Laurent expansion: X z n fn X + z n fn (ψ(n)!)1/2 , SpN = Z , 1/2 ψ(n)! n 0, that is mainly involved in the study of the q-oscillator (8) with non-Fock representations. The domain of existence of the coherent states is a ≤ |z|2 . The momentum Fˆ (n) reads Z +∞ ˆ F (n) = F (x)xn−1 dx . (57) a
We first prove that Eqs. (34) and (42) are not equivalent if F (x) is positive on the whole positive axis. Indeed, let us start with a solution of (42), Eq. (34) reads Z a F (x)xn−1 dx = 0 , (58) q−1 a
that is obviously impossible if F (x) is positive on [q −1 a, a]. Therefore in this case, the momentums deduced from the weight function solution of (42) are not the expected ones (solutions of (34)). Moreover, in [8], we proved that the solution of (42) is identically zero. Let us look for a solution of (35) that cannot have poles, due to (33): Fˆ (ρ + 1) = (q ρ + a)Fˆ (ρ) . We have as a convenient particular solution the following entire function: Y Fˆ (ρ) = aρ (1 + a−1 q ρ−p−1 ) , p≥0
(59)
(60)
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
1073
but it is not a Mellin transform of a true function F (x). Indeed if it has an inverse Mellin transform, it can be calculated on any parallel to the imaginary, for instance Q on Re ρ = ln a/ ln q. On this axis, |Fˆ (iy)| ≥ p≥0 (1 − q −p−1 ), so that (60) is not the Mellin transform of a true function. Nevertheless, we can write (60) in the form [24]: −n nρ X a q . Fˆ (ρ) = aρ 1 + (61) (q − 1) · · · (q n − 1) n≥1
The series is absolutely convergent as q > 1. It is easily verified that this expression can be seen as the Mellin transform of the following measure: F (x) =
X n≥0
a−n δ(ln a + ln q − ln x) . (q − 1) · · · (q n − 1)
(62)
Therefore, in this case we obtain a Bargmann representation if we accept the weight function to be a true measure. The same is true when we consider ψ(x) = 1/(q x + a), q < 1. Equation (35) reads 1 ˆ Fˆ (ρ + 1) = ρ F (ρ) . (63) q +a The domain of existence of the coherent states is the disc of radius 1/a. We then obtain F (x) =
X n≥0
(q −1
q −n a−n δ(− ln a + n ln q − ln x) . − 1) · · · (q −n − 1)
(64)
In this subsection, we gave examples where the Bargmann representations exist only if we admit that the scalar product be expressed by means of true measures. 6. ψ Vanishes In this section, we consider two cases where the spectrum of N is the set N + of the positive integers and where the coherent states are defined in the whole complex plane. 6.1. q-oscillators The first example corresponds to (7) with σ = 1 and the second one to (8) with σ = (q − 1)−1 . a) ψ(x) = [x] ≡ (q x − q −x )/(q − q −1 ) A resolution of the identity was shown to be obtained with a q-integration [13]. The Rx x∂x −q−x∂x q-integration 0 dq x is the inverse operator of the q-derivative Dq = x1 q q−q −1 that vanishes at the origin: Z x X q − q −1 −1 dq x = x∂x x = (q − q) q (2n+1)x∂x x , when q < 1 . (65) q − q −x∂x 0 n≥0
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The q-exponential is defined by Expq (qx) − Expq (q −1 x) x(q − q −1 )
Expq (x) = Dq Expq (x) =
(66)
with the condition that it is equal to one when x is zero. This function reads Expq (x) =
X xn , [n]!
(67)
n≥0
and vanishes on the negative axis [13]. Denoting by −ζ the first zero at the left of the origin the resolution of identity then reads Z
dθ 2
Z
ζ2
dq ρ2 Expq (−ρ2 )|ρe−iθ ihρeiθ |2ρdρ = 1 .
(68)
0
Here we look for a Bargmann representation where the scalar product involves a true integral. First, it is easy to verify that in this case as in the following, if F verifies (42), its Mellin transform is solution of (35) and the moments are the expected ones. In both cases, we choose to define the weight function, not through its Mellin transform but directly as solution of (42). Equation (42) for this particular case reads q −x∂x − q x∂x F (x) . q − q −1
xF (x) =
(69)
The obvious solution of this equation: F (x) = Expq (−x)
(70)
is not positive for all positive values of x and the Bargmann representation as defined in Sec. 4 does not exist. Following the trick used to get (68), we can try to limit the integration to the domain where Expq (x) is positive. Let us see if Z Fˆ (n) =
ζ
Expq (−x)xn−1 dx
(71)
0
could work. Equation (34) gives Z
ζ
Expq (−q −1 x)xn−1 dx −
qζ
Z
ζ
q−1 ζ
Expq (−qx)xn−1 dx = 0 .
(72)
The problem is symmetric under the change q into q −1 . Let us choose q > 1, (72) takes the form: Z ζ (Expq (−x)q n + Expq (−qx))xn−1 dx = 0 . (73) q−1 ζ
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
1075
The integrand of (73) reads (Expq (−x)(q n − x(q − q −1 )) + Expq (−q −1 x))xn−1 and is positive for n enough large; this leads to Expq (−x)(q n − x(q − q −1 ) + Expq (−q −1 x) = 0, that is impossible. Therefore in order to obtain a Bargmann representation, we must look for another solution of (42) that will be positive. As already noticed, the problem being symetric under the change q into q −1 , we assume q > 1. Let us start with (35) that reads qρ q ρ − q −ρ ˆ (1 − q −2ρ )Fˆ (ρ) . (74) F (ρ) = Fˆ (ρ + 1) = q − q −1 q − q −1 Let us write Fˆ on the form: ρ Fˆ (ρ) = φq 2 (ρ−1) (q − q −1 )−ρ fˆ(ρ) .
(75)
The function fˆ(ρ) must verify fˆ(ρ + 1) = (1 − q −2ρ )fˆ(ρ) , and is given by fˆ(ρ) =
X n≥0
(76)
q −2nρ . (1 − q −2 ) · · · (1 − q −2n )
(77)
The condition Fˆ (1) = 1, furnishes the normalization factor: φ = (q − q
−1
X
q −2n ) 1+ −2 (1 − q ) · · · (1 − q −2n ) n>0
!−1 .
(78)
Putting (77) and (78) in (75), we obtain Fˆ (ρ), and then we can calculate its inverse Mellin transform: 2 ! 1 1 ln x + ln(q − q −1 ) + ln q exp − 2 ln q 2 F (x) = X q −2n 1+ −2 (1 − q ) · · · (1 − q −2n ) n>0 ×
X q −n(2n+1) ((q − q −1 )x)−2n q − q −1 ×√ . −2 −2n (1 − q ) · · · (1 − q ) 2π ln q n≥0
(79)
This function being positive, we have obtained a Bargmann representation where the scalar product is written with a true integral. Let us stress that F (−x) is solution of (66) and is thus a possible candidate to write the resolution of identity with a q-integration and a positive function on the whole positive axis. The same is true in the next example where two resolutions of the identity coexist. b) ψ(x) = (x) ≡ (q x − 1)/(q − 1), with q > 1 First we show that the resolution of the identity can be obtained with a q-integral as in [13].
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The q-integration [23–25] is defined to be the inverse of the q-derivative Dq ≡ 1 qx∂x −1 x q−1 : Z x X q−1 x = (q − 1) dq x ≡ x∂x q −(n+1)x∂x x . (80) q − 1 0 n≥0
The q-exponential, solution of the equation: Expq (x) = Dq Expq (x) = is given by Expq (x) =
Y
Expq (qx) − Expq (x) x(q − 1)
(81)
X xn , (n)!
(82)
(1 + x(1 − q −1 )q −p ) =
p≥0
n≥0
and vanishes for x = −q p (1 − q −1 )−1 . The nearest zero on the left of the origin is of the identity takes the same form −ζ = −(1 − q −1 )−1 . Therefore the resolution Rx as in (68) with the new expressions for 0 dq x, Expq and ζ. Let us now look for a Bargmann representation as defined in Sec. 4. We see that the Eq. (42) can be written: F (q −1 x) = (x(q − 1) + 1)F (x) .
(83)
We easily prove that the weight function is given by F (x) =
1 . Expq (qx)
(84)
It is a positive function when x > 0 and its Mellin transform fulfills qρ − 1 ˆ F (ρ) . Fˆ (ρ + 1) = q−1
(85)
This ensures that the momentum Fˆ (n) are the expected one (32). Thus, in this case, coexist two resolutions of the identity ,one involving a true integral and a weight function F (x) = (Expq (qx))−1 and one with a q-integral, the weight function being Expq (−x). 6.2. ψ(x) = xn , n > 0 The Mellin transform of the weight function is solution of the equation deduced from (35): (86) Fˆ (ρ + 1) = ρn Fˆ (ρ) , R ∞ −t z−1 and can be expressed with the gamma-function Γ(z) = 0 e t dt: Fˆ (ρ) = (Γ(ρ))n . When n is an integer, the inverse Mellin transform gives F (x): Z ∞ Z ∞ F (x) = ··· e−(t1 +···+tn ) dt1 · · · dtn δ(x − t1 × · · · × tn ) . 0
0
(87)
(88)
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
1077
On this expression, we see that F (x) is a positive function so that the Bargmann representation exists. In the case n = 1, we recover the usual harmonic oscillator where F (x) = e−x . 7. Conclusion We have studied the possibility of Bargmann representations for any deformed oscillator algebra characterized by a function ψ. We gave the conditions to be verified by this function for admitting representations with coherent states. We get the unique functional equation to be satisfied by the Mellin transform of the weight function defining the scalar product. We were able to get definite and positive answer in many cases including in particular some types of q-oscillators. Although we did not succeed in obtaining a general characterization of the function ψ leading to Bargmann representations, we underline two points: — We exhibited cases where the Bargmann representations do not exist even when coherent states do (Subsec. 5.2); — The analysis of Subsec. 5.3 showed that the scope of our study have to be extended up to include true measures for writing the scalar product. Finally let us remark that we have obtained scalar products for the Bargmann representations of the usual q-oscillators, involving true integrals instead of q-integrations as previously proposed in literature. References [1] L. C. Biedenharn, J. Phys. A: Math. Gene. 22 (1989) L873. [2] A. J. Mac Farlane, J. Phys. A: Math. Gene. 22 (1989) 4581. [3] M. Irac-Astaud and G. Rideau, “Deformed quantum harmonic oscillator”, Proc. Third Int. Wigner Symposium, Oxford, 1993, to appear. [4] M. Irac-Astaud and G. Rideau, “On the existence of quantum bihamiltonian systems: the harmonic oscillator case”, preprint PAR-LPTM 92, Lett. Math. Phys. 29 (1993) 197, Theor. Math. Phys. 99 (1994) 658. [5] C. Quesne and N. Vansteenkiste, “Representation theory of deformed oscillator”, Helv. Phys. Acta 69 (1996) 141, and many references therein. [6] P. Kosinski, M. Mazewski and P. Maslanka, “Representations of generalized oscillator algebra”, Czech. J. Phys. 47 (1997) 41. Fifth Colloquium on quantum groups and integrable systems, Prague, 20–22 June, 1996. [7] V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform”, Commun. Pure and Appl. Math. 14 (1961) 187. [8] M. Irac-Astaud and G. Rideau, “Bargmann representation for some deformed harmonic oscillators with non-Fock representation”, Proc. Symposium in honor of Jiri Patera and Pavel Winternitz for their 60th birthday, Algebraic Methods and Theoretical Physics, January 9–11, 1997, Centre de recherches math´ e matiques, Universit´e de Montr´eal. [9] A. D. Janussis, P. Filippakis and J. C. Papaloucas, “Commutation relations and coherent states”, Lettere al Nuovo Cimento 29(15) (1980) 481. [10] J. A. de Azcarraga and D. Ellinas, “Complex analytic realizations for quantum algebras”, J. Math. Phys. 35(3) (1994) 1322. [11] V. Spiridonov, “Coherent states of the q-Weyl algebra”, Lett. Math. Phys. 35 (1995) 179.
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[12] K. Kowalski, J. Rembielinski and L. C. Papaloucas, “Coherent states for a particle on a circle”, J. Phys. A: Math. Gene. 29 (1996) 4149. [13] R. W. Gray and C. A. Nelson, “A completeness relation for the q-analogue coherent states by q-integration”, J. Phys. A: Math. Gen. 23 (1990) L945. [14] M. Chaichian, D. Ellinas and P. Kulish, “Quantum algebra as the dynamical symmetry of deformed Jaynes–Cummings model”, Phys. Rev. Lett. 65 (1990) 980. [15] A. J. Bracken, D. S. McAnally, R. B. Zhang and M. D. Gould, “A q-analogue of Bargmann space and its scalar product”, J. Phys. A: Math. Gen. 24 (1991) 1379. [16] B. Jurco, “On coherent states for the simplest quantum groups”, Lett. Math. Phys. 21 (1991) 51. [17] C. Quesne, “Coherent states, K-matrix theory and q-boson realizations of the quantum algebra suq (2)”, Phys. Lett. A153 (1991) 303. [18] A. Odzijewicz, “Quantum algebras and q-special functions related to coherent states maps of the disc”, preprint IFT 18/95. [19] A. M. Perelomov, Helv. Phys. Acta 68 (1996) 554; A. M. Perelomov,“On the completeness of some subsystems of q-deformed coherent states”, preprint FTUV 96-38, IFIC 96-46. [20] P. P. Kulish, “Contraction of quantum algebra and q-oscillators”, Theor. Math. Phys. 86 (1991) 108. [21] G. Rideau, “On the representations of quantum oscillator algebra” Lett. Math. Phys. 24 (1992) 147. [22] J. R. Klauder and B. S. Skagerstam, Coherent States, World Scientific, 1985. [23] E. H. Jackson, “On q-definite integrals”, Q. J. Pure Appl. Math. 41 (1910) 193. [24] H. Exton, q-Hypergeometric Functions and Applications, Ellis Horwood Series, New York, 1983. [25] D. S. McAnally, “q-exponential and q-gamma functions”, J. Math. Phys. 36(1) (1995) 546. [26] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products, Academic Press, 1965.
SOME PROPERTIES OF MASSLESS PARTICLES IN ARBITRARY DIMENSIONS MOURAD LAOUES Laboratoire Gevrey de Math´ ematique Physique Universit´ e de Bourgogne 9, avenue Alain Savary B.P. 400, F-21011 Dijon Cedex, France E-mail : [email protected] [email protected] Received 26 January 1998 Various properties of two kinds of massless representations of the n-conformal (or (n+1)˜n = g SO0 (2, n) are investigated for n ≥ 2. It is found that, for spaceDe Sitter) group G time dimensions n ≥ 3, the situation is quite similar to the one of the n = 4 case for SO0 (2, n − 1). These representations Sn -massless representations of the n-De Sitter group g ˜ n . The main difference is that they are not are the restrictions of the singletons of G contained in the tensor product of two UIRs with the same sign of energy when n > 4, whereas it is the case for another kind of massless representations. Finally some examples of Gupta–Bleuler triplets are given for arbitrary spin and n ≥ 3.
1. Introduction The (ladder) representations D(s + 1, s, s), 2s ∈ N and || = 1 of the universal g0 (2, 4) of the conformal group remain irreducible when restricted covering C˜4 = SO g0 (1, 3) n T4 of the Poincar´e group and each nonto the universal covering P˜4 = SO trivial positive energy representation of the conformal group with that property is equivalent to one of them. However the restriction to the universal covering S˜4 of the De Sitter group is irreducible only if s > 0; indeed one has D(s + 1, s) if s > 0; D(s + 1, s, s)| ˜ = S4 D(1, 0) ⊕ D(2, 0) if s = 0. These representations are called massless (relatively to the De Sitter group) for a variety of reasons [2]. In the present paper we call them S4 -massless representations g0 (2, 3) because, as indicated in [2, 11, 13] they satisfy of the De Sitter group S˜4 = SO the following masslessness conditions: (a) They contract smoothly to a massles discrete helicity representation of the g0 (1, 3) n R4 ; Poincar´e group P˜4 = SO (b) Any massless discrete helicity representation U P of the Poincar´e group has ˆ (called C4 -massless representation in this a unique extension to a UIR U ˆ to the SO0 (2, 4). The restriction of U paper) of the conformal group C˜4 = g De Sitter group is precisely one of the massless representations of S˜4 recalled above;
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M. LAOUES
(c) For spin s ≥ 1 one may construct a gauge theory on the Anti-de Sitter space for massles particles, quantizable only by the use of an indefinite metric and a Gupta Bleuler triplet; (d) The massless representations in question distinguish themselves by the fact that the physical signals propagate on the Anti-de Sitter light cone. Other interesting representations of S˜4 are the Dirac singletons Di = D(1, 12 ) and Rac = D( 12 , 0) (which are also C3 -massless representations in the sense defined below). Some of their properties are: 1. Dirac singletons are, up to equivalence, the only unitary irreducible positive energy representations of S˜4 which remain irreducible when restricted to the ˜ 4 of the Lorentz group; universal covering L ˜ 4 ) they contract 2. In the limit of zero curvature (of the De Sitter space S˜4 /L ˜ to unitary irreducible representations (UIR) of P4 that are trivial on the translation part T4 ; 3. Let χ(µ1 ) ⊗ π(µ2 ) denote the IR (up to equivalence), with highest weight ˜ 4 of the maximal compact subgroup of (µ1 , µ2 ) of the universal covering K ˜ 4 of the Dirac singletons UIRs the De Sitter group. Then the restriction to K is given by M 1 1 1 + s, s | ˜ = +s+l ⊗ π(s + l), s = 0 or . D χ − K 2 2 2 4 l∈N
4. Finally the Dirac singletons satisfy the following [10]: Rac ⊗ Rac =
M
D(s + 1, s);
s∈N
Rac ⊗ Di =
M
D(s + 1, s);
s− 12 ∈N
Di ⊗ Di =
M
D(s + 1, s) ⊕ D(2, 0).
s−1∈N
Note [2] that the Dirac singletons are not massless representations of the De Sitter group. But if one considers S4 as the conformal group of the 3-dimensional Minkowski space then the Dirac singletons are massless, i.e. their restriction to the corresponding Poincar´e group P3 is irreducible [2, 3, 13]. In this case it is clear from the context what kind of masslessness is considered. However, for general n, some confusion may arise. To avoid it we shall introduce a prefix to the word “massless” (see Definition (1)), to distinguish between “conformal masslessness” and “De Sitter masslessness” in any dimension, to precise which group we are representing. A common property to both types of massless representations is the existence of Gupta–Bleuler (GB) quantization; see for example [2, 6, 14, 15]. The purpose of this work is to continue the study performed in [3] and more specifically to look for properties of maslessness (both types) which persist when
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
1081
the space-time dimension becomes an arbitrary integer n ≥ 2. In Sec. 2 we fix the notations and recall some results. In Sec. 3 we discuss the irreducibility of a massless representation of the n-conformal group when restricted to the (n + 1)Lorentz group and its contractibility to UIRs of the n-Poincar´e group. Reduction to the maximal compact subgroup of the conformal group is studied in Sec. 4. Finally Dirac singletons and Gupta–Bleuler triplets are treated in (respectively) Secs. 5 and 6. It is found that almost all the properties of massless representations in dimension n = 4 are conserved when n ≥ 3; however the property that massless representations are, when n = 4, contained in the tensor product of two positive energy UIRs (of the De Sitter group) fails for general n. After a first version of this paper was written appeared a preprint [9] with somewhat different conclusions, based on a less-demanding notion of masslessness in higher dimensions. Since we need the definitions and results of this paper to compare both notions, we shall discuss this point at the end of the paper. 2. Generalities We suppose n ≥ 2. Let R1,n−1 be the n-dimensional Minkowski space-time, Tn its group of translations, Ln = SO0 (1, n − 1) the n-Lorentz group, Pn = Ln n Tn the n-Poincar´e group and Sn = SO0 (2, n − 1) the n-De Sitter group. We write Tn , Ln , Pn and Sn the corresponding Lie algebras. Let Gn = SO0 (2,n). The preceding groups may be considered as subgroups of Gn . Indeed let Mab −1≤a 0. Thus one must consider the products D0 (α, ±α) ⊗ D0 (β, ±β) D0 (α, ±α) ⊗ D0 (β, ∓β),
α, β > 0
or, equivalently, the products h i h i D(0) D(α) ⊗ D(0) D(β) h i h i D(0) D(α) ⊗ D(β) D(0) h i h i D(α) D(0) ⊗ D(β) D(0) h i h i D(α) D(0) ⊗ D(0) D(β) . Now using D(α) ⊗ D(β) ∼
∞ M
D(α + β + l),
(32)
(33)
(34)
l=0
one finds D0 (α, ±α) ⊗ D0 (β, ±β) =
∞ M
D0 (α + β + l, ±[α + β + l]),
(35)
l=0
D0 (α, ±α) ⊗ D0 (β, ∓β) = D0 (α + β, ±[α − β]).
(36)
Finally it is easily seen that D0 (1/4, 1/4) ⊗ D0 (1/4, −1/4) = D0 (1/2, 0),
(37)
D0 (3/4, 3/4) ⊗ D0 (3/4, −3/4) = D0 (3/2, 0),
(38)
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
1089
thus h i h i D0 (1/4, 1/4) ⊕ D0 (3/4, 3/4) ⊗ D0 (1/4, −1/4) ⊕ D0 (3/4, −3/4) i h i h = D0 (1/2, 0) ⊕ D0 (3/2, 0) ⊕ D0 (1, 1/2) ⊕ D0 (1, −1/2) .
(39)
The right-hand side is a sum of S3 -massless representations. Using (35) and (36), one can see that this is a unique solution (up to equivalence) of the problem Singleton⊗Singleton = ⊕S3 -massless for unitary Dirac singletons, which are, here, D0 (1/4, ±1/4) ⊕ D0 (3/4, ±3/4). They are not irreducible, but each component is irreducible on both S3 and L3 . Let n = 2. Then C2 ' so(2, 2), S2 ' so(2, 1) and L2 ' so(1, 1). A (HW) C2 massless representation of C2 has the form D0 (α, ±α), α > 0. Thus the S2 -massless ∼ D(α). Now C1 -masslessness representations of S2 have the form D0 (α, ±α)| S2 on D(β) (or irreducibility on the 2-Lorentz group L2 ) implies β = 0, so that, instead of the n = 4 and n = 3 cases, Dirac singletons are not compatible with C1 -masslessness. But one has D(α/2) ⊗ D(α/2) ∼ D(α) ⊕
∞ M
D(α + 1 + l).
(40)
l=0
Thus S2 -massless representations occur in the tensor product of two S2 -massless ones. 6. Indecomposability. Gupta Bleuler Triplets Gupta–Bleuler triplets are used to quantize gauge theories, in a way similar to the quantization of (4-dimensional flat) QED. This kind of quantization is done on an indefinite metric space which carries indecomposable representations, as in the Gupta–Bleuler quantization of the electromagnetic field. Let us see how it works in the case of our massless representations. If U2 is a massless representation of Gn then it can be obtained as a component of an indecomposable representation. Indeed one can find UIRs Uε , ε > 0, and U3 such that limε→0 Uε is a non trivial extension U2 → U3 (i.e. we have an exact sequence 0 → H3 → H → H2 → 0 where Hi is the carrying space of Ui , i = 2 or 3). The elements of H3 , the gauge states, are obtained from those of H by applying a constraint similar to the Lorentz condition in QED. The elements of H2 , the physical states, are realized on the quotient H/H3 . Now the indecomposable representation (U2 + U3 , H) has no invariant nondegenerate metric, thus covariant quantization is not possible. But if one extends the representation U3 by U2 + U3 in a non trivial way (U3 → U2 → U3 ) to a bigger space endowed with an invariant nondegenerate (but indefinite) Hermitian form then quantization of the gauge theory under construction becomes possible. In the following we construct some examples of Gupta–Bleuler triplets for the massless representations when n ≥ 3.
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6.1. Massless representations and indecomposability Let us recall that the massless representations for Gn = so(2, n) are the Cn -massless and the Sn+1 -massless ones. Below we write them again, according to the parity of n. In analogy with 4-dimensional physics we call the parameter s the spin of the representation. Case 1: n is even Cn -massless representations are n−2 , s, . . . , ±s , D s+ 2
2s ∈ N.
(41)
Sn+1 -massless representations are 1 1 n 1 ⊕D , ,..., ,− D 2 2 2 2 n−1 n+1 D , 0, . . . , 0 ⊕ D , 0, . . . , 0 2 2
1 n 1 , ,..., 2 2 2
1 2
(42)
for spin 0.
(43)
for spin
Case 2: n is odd Cn -massless representations are n−2 , s, . . . , s , D s+ 2
s∈
1 0, . 2
(44)
Sn -massless representations are n+1 n−1 , 0, . . . , 0 ⊕ D , 0, . . . , 0 for spin 0 2 2 n−1 D s+ , s, . . . , s , 2s ∈ N and s ≥ 12 . 2
D
(45) (46)
Some of the above irreducible representations correspond to the limit of unitarity [1, 7]. It is the case of the Cn -massless ones and, when n is odd, of the Sn+1 massless representations for which s ≥ 1. Then one can look for indecomposability and Gupta–Bleuler (GB) triplets. That is what we do in the next subsections. In the next two subsections the cases of the representations D( n−2 2 , 0, . . . , 0) and 1 n−2 1 1 D( 2 + 2 , 2 , . . . , 2 ) are treated without separating the n even and n odd cases, since those representations are Cn -massless for both n even and n odd. Finally, when s ≥ 1 the Cn -massless D(s + n−2 2 , s, . . . , s) for n even and the Sn+1 -massless , s, . . . , s) for n odd are investigated successively. D(s + n−1 2
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MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
6.2. Cn -masslessness, spin 0 6.2.1. Reduction of D(E0 , 0, . . . , 0) to kn and its indecomposability Recall that kn ' so(2) ⊕ so(n) is the maximal compact subalgebra of Gn . Let Gn = kn + pn the Cartan decomposition of Gn . Let Xjk −r≤j,k≤r a basisd of GnC such that (47) Xjk = −Xkj , and [Xjk , Xj 0 k0 ] = δj,−j0 Xkk0 + δk,−k0 Xjj0 − δj,−k0 Xkj 0 − δk,−j0 Xjk0 ,
(48)
n± = hX±j,±k , (1 − ν) ≤ j, k ≤ ri + hX±j,∓k , 1 ≤ j < k ≤ ri,
(49)
and let
h = hX−j,j = Hj , 1 ≤ j ≤ ri.
(50)
Then GnC = n+ + h + n− is a triangular decomposition of GnC . ± C + C − Let p± = pC n ∩ n . Then Gn = p + kn + p . The basis (Xjk )jk is chosen such that p± = hX±1,j , −r ≤ j ≤ r
and |j| 6= 1i,
(51)
kC n = hXjk , −r ≤ j, k ≤ r
and |j|, |k| 6= 1i.
(52)
The root system ∆n+2 is defined by the set of positive roots ∆+ n+2 which is given ]]) instead of n + 1 (resp. r0 = [[ n+1 by (13), but with n + 2 (resp. r = [[ n+2 2 2 ]]). P (±α) ± the The new basis is also chosen such that in the decomposition n = α>0 Gn (e ±e )
(e )
subspace Gn j k is, for 1 ≤ j < k ≤ r, generated by Xj,±k and, if n is odd, Gn j is, for 1 ≤ j ≤ r, generated by X0j . The roots which correspond to kC n are the compact roots and the others the noncompact ones. The set of positive compact +n (resp. noncompact) roots is denoted by ∆+c n+2 (resp. ∆n+2 ). +c Let λ = (λ1 , . . . , λr ) a ∆n+2 -dominant integer weight and let K(λ) denote the irreducible (finite dimensional) HW kn -module. We write N (λ) for the induced HW Gn -module, with HW λ, and L(λ) for the irreducible quotient. The HW vectors for both N (λ) and L(λ) are, for simplicity, identified and denoted by vλ . Proposition 5. Let E0 > 0, λ = (−E0 , 0, . . . , 0), Oλ = D(E0 , 0, . . . , 0), P Z = |h|6=1 X−1,h X−1,−h ∈ U(GnC ) and, for l, k ∈ N, vlk = (X−1,2 )l Zk vλ ∈ N (λ). Then ∞ M U(kC (53) N (λ) = n )vlk l,k=0
and / N (λ) is irreducible ⇐⇒ E0 ∈
n n − 1, . . . , − 2 2
n−1 2
d This basis is more appropriate to the triangular decomposition of G C than the n
. Mab
(54) a,b
basis.
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M. LAOUES
− j for some j ∈ {1, . . . , [[ n−1 2 ]]}, then n N (λ) L(λ) = L − + j, 0, . . . , 0 ' ∞ M 2 U(kC n )vl,j+k
Moreover if E0 =
n 2
l,k=0
'
j−1 ∞ M M
U(kC n )vlk .
(55)
l=0 k=0
Corollary 2. Let us write χ(µ1 ) ⊗ π(µ2 , . . . , µr ) for the irreducible representation, with HW µ, of kn on K(µ). Then D(E0 , . . . , 0)| = kn
∞ M ∞ M
if E0 ∈ / D(E0 , . . . , 0)| = kn
χ(−[E0 + l + 2k]) ⊗ π(l, 0, . . . , 0)
l=0 k=0
j−1 ∞ M M
n−1 − 1, . . . , 2
(56)
,
χ(−[E0 + l + 2k]) ⊗ π(l, 0, . . . , 0)
l=0 k=0
if E0 =
n 2
n 2
n−1 . − j for some j ∈ 1, . . . , 2
(57)
Remark 2. 1. The value j = 1 corresponds to the Cn -massless case: ∞ M n−2 , 0, . . . , 0 | = χ(−[E0 + l]) ⊗ π(l, 0, . . . , 0) D kn 2 l=0
which is a particular case of Proposition 3. 2. Thanks to the preceding results one can see that indecomposability arises when E0 reaches the value n2 − j (we use the same notations): n n − j, 0, . . . , 0 + D + j, 0, . . . , 0 . (58) D D(E0 , 0, . . . , 0) −→ n E0 → 2 −j 2 2 Proof of the Proposition. The vlk ’s are maximal vectors for D(E0 , 0, . . . , 0)| ; kn C + C indeed one has n+ ∩ kC n , X−1,2 = 0 and kn , Z = 0, thus n ∩ kn vlk = 0. N (λ) is r Q qj vλ , where (qj )|j|6=1 is a family of natural generated by the monomials j=−r X−1,j |j|6=1
integers and, if |j| = 6 1: 1 X−2,j vl+1,k l+1 X−1,j vlk = vl+1,k X l 1 v − X−2,−h X−2,h vl+1,k l−1,k+1 l+1 (l + 1)(l + n) |h|6=1,2
if |j| 6= 2, if j = 2, if j = −2,
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
1093
− C C where v−1,k = 0; thus one has p− vlk ⊂ U(kC n )vl−1,k+1 +U(kn )vl+1,k . Since p , kn ⊂ p− one can conclude that N (λ) = U(p− )vλ ⊂
∞ M
U(kC n )vlk .
l,k=0
Now p+ = h X1j , −r ≤ j ≤ r and |j| 6= 1 i and |j| 6= 1 implies n X1j vlk = δj,−2 l(E0 + 2k + l − 1)vl−1,k + 2k E0 − + k X−1,j vl,k−1 , 2 with v−1,k = vl,−1 = 0; thus for a maximal vector for which the weight is strictly less than λ, necessarily proportional to some vlk , one must have k(E0 − n2 + k) = 0 and l = 0, i.e. l = 0, k 6= 0 and E0 − n2 + k = 0. E0 being strictly positive one has 1 ≤ k ≤ [ n−1 2 ]. L∞ L∞ n C Finally let j ∈ {1, . . . , [ n−1 l=0 k=j U(kn )vlk . Then 2 ]}, E0 = 2 − j and Kj = the relation C pC n vlk ⊂ U(kn )h vl−1,k ; vl+1,k ; vl−1,k+1 ; vl+1,k+1 i implies U(GnC )Kj ⊂ Kj , so that
h i N − n − j , 0, . . . , 0 i hn 2 − j , 0, . . . , 0 = L − . 2 Kj
6.2.2. A Gupta–Bleuler triplet for the Cn -massless D( n−2 2 , 0, . . . , 0) Using the preceding notations and results one can see that D( n−2 2 + ε, 0, . . . , 0) sends the operator Z to zero if ε = 0 but it does not if ε 6= 0. It is precisely this fact which gives us the desired indecomposable representations. Indeed, let ε > 0 and E0 = n−2 2 + ε. Then D(E0 , 0, . . . , 0) is irreducible, but when ε → 0 one obtains, from Remark 2 and for j = 1, an indecomposable representation: n+2 n−2 n−2 + ε, 0, . . . , 0 −→ D , 0, . . . , 0 + D , 0, . . . , 0 . (59) D ε→0 2 2 2 In order to construct explicitly a Gupta–Bleuler (GB) triplet [4], let ρ > 0 and let ( ) n X 2,n a 2,n 2 Hρ = y, y = y ea ∈ R such that y = 1/ρ , a=−1 2 − y02 − y2 . The De Sitter space-time is the universal covering where y 2 = y a ya = y−1 of Hρ2,n . The action of Gn on C ∞ -functions defined on Hρ2,n is well known:
Uλ (Mab ) = Lab = ya ∂b − yb ∂a , where ∂c =
∂ ∂y c .
(60)
Let ∂ 2 = ∂ a ∂a and δ = y a ∂a . Then one has 1 Uλ (C2 ) = − Lab Lab = −y 2 ∂ 2 + δ(δ + n). 2
(61)
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M. LAOUES
Now the resolution of the Laplace–Beltrami equation on Hρ2,n is standard [16]. One finds that the following solutions form a Hilbertian basis for L2µ (Hρ2,n ), with dµ(y) = ρ−11+y2 dtdn y:
1/2
n−2 E0 ψklm (t, y) = ρ−(2k+E0 − 2 )
Γ(k + E0 + l)Γ(k + 1) n − 2 Γ(k + n2 + l)Γ k + E0 − 2 E0 +l
×e−i(E0 +l+2k)t (ρ−1 + y2 )− 2 (y2 ) 2 ! −1 2 n − y y ρ (l+ n−2 ,E − ) 0 l 2 Ym p ×Pk 2 , ρ−1 + y2 (y2 ) l
(62)
(α,β)
are the Jacobi polynomials, l = (l2 , . . . , l[ n+1 ] ) and m = (m1 , . . . , m[ n2 ] ) where Pk 2 are vectors, in Nr−1+ν and Nr−1 respectively, subject to certain conditions, l = l2 , −1 0 1/2 l Ym are the spherical harmonics on S n−1 and eit = yy−1 +iy . The scalar +iy 0 product we use to normalize these functions is given by Z ←→ dn y 0 (ψ, ψ ) = ψ(y) : i∂t : ψ 0 (y) −1 , (63) ρ + y2 Rn ↔
E0 where ψ(y)Aψ 0 (y) = Aψ(y)ψ 0 (y) + ψ(y)Aψ 0 (y). We extend the functions ψklm to 2,n 2,n H+ = ∪ρ>0 Hρ by fixing the degree of homogeneity: δψ = −E0 ψ. Then, ψ being in the kernel of ∂ 2 , one has
Uλ (C2 )ψ = E0 (E0 − n)ψ. Let
x±j
i √ (y −1 ± iy 0 ) 2 1 = √ (y 2j−1 ± iy 2j ) 2 n y ∂j =
∂ ∂x−j
(64)
if j = 1, if 2 ≤ j ≤ r, if n is odd and j = 0, and
∂−j = ∂j .
P P P Then one has y 2 = − rj=−r x−j xj , ∂ 2 = − rj=−r ∂−j ∂j , δ = rj=−r x−j ∂j and one can choosee Xjk such that Uλ (Xjk ) = xk ∂j − xj ∂k .
(65)
0 . Then ϕ2 is, up to a multiplicative constant, the maximal vector Let ϕ2 (y) = x−E 1 E0 of Uλ thus ψklm ∈ U(Gn )ϕ2 . Moreover one finds that 0 −2 0 −1 − 2εE0 x−1 x−E , (Zϕ2 )(y) = −E0 (E0 + 1)y 2 x−E 1 1
e We use the notations of the preceding subsubsection.
(66)
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MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
thus lim (Zϕ2 )(y) = −
ε→0
n − 2 n 2 − n+2 y x1 2 . 2 2 −n 2
Now assume ε = 0 and let ϕ1 (y) = x−1 x1
− n+2 2
and ϕ3 (y) = y 2 x1
2 −n
1 nZ
(67) . Then
2 n−2 Z
−−→ ϕ2 p−−−−−−−− −−→ ϕ3 ϕ1 p−−−−−−−− where Z =
P |j|6=1
(68)
X1j X1,−j and one has ∂ 2 ϕ2 = ∂ 2 ϕ3 = 0,
whereas ∂ 2 ϕ1 =
1 ϕ 6= 0, y2 3
but (∂ 2 )2 ϕ1 = 0.
H
Let cl(V ) denotes the closure of any topological space V and let i = cl(U(GnC )ϕi ), i taking the value 1, 2 or 3. Then it is not difficult to prove the following
H
(0)
Proposition 6. 1. 1 (0) invariant subspace of i−1 ;
H and H 2. H /H (0) 1
(0) 2
⊃
H
(0) 2
⊃
H
(0) 3
and
H
(0) i
(0)
, i = 2 or 3, is a closed
H /H
(0) 3
carry the IR D( n+2 2 , 0, . . . , 0), while , 0, . . . , 0). ries the Cn -massless representation D( n−2 2 3. (n − 2)(n + 2) ϕi = 0 if i = 2 or 3, Uλ (C2 ) + 4 (n − 2)(n + 2) ϕ1 = nϕ3 6= 0, Uλ (C2 ) + 4 2 (n − 2)(n + 2) Uλ (C2 ) + ϕ1 = 0. 4
H
(0) 2
(0) 3
car-
(69)
(0)
4. limy2 →0 ϕ(y) = 0 ∀ϕ ∈ 3 . Thus the Cn -massless D( n−2 2 , 0, . . . , 0) may be realized irreducibly on the cone Q2,n = {y, y ∈ R2,n such that y 2 = 0}. Definition 4. In analogy with QED on 4-dimensional Minkowski space we (0) (0) (0) (0) (0) (0) (0) call the elements of S = 1 / 2 (resp. P = 2 / 3 , resp. G = (0) 3 ) scalar (resp. physical, resp. gauge) states.
H
H
K
H H
H
H H
H
− n+2
Remark 3. Let (0) the closure of the GnC -module generated by y 7−→ x1 2 ; 4 2 2 2 it carries the IR D( n+2 2 , 0, . . . , 0). Let ∂ = (∂ ) and let us identify y to the corresponding operator. Then the GB triplet n−2 n+2 n+2 , 0, . . . , 0 −→ D , 0, . . . , 0 −→ D , 0, . . . , 0 D 2 2 2
1096
M. LAOUES
defined by ϕ1 , ϕ2 and ϕ3 may be defined by (0) = positive energy solutions f of ∂ 4 f = 0 1
H H H
(0) 2 (0) 3
= f∈ = f∈
H H
(0) 1
such that ∂ 2 f = 0 ,
(0) 2
such that
f ∈ y2
K
(0)
n−2 f and δf = − 2
,
.
H
←→ R n (0) y Now, for ϕ and ϕ0 in 1 , define (ϕ, ϕ0 )1 = S 1 ×Rn ϕ(y)y 2 ∂ 2 ϕ0 (y) ρdtd −1 +y 2 and R n−2 (ϕ, ϕ0 )2 = S 1 ×S n−1 (y2 ) 2 ϕ(y)ϕ0 (y)dtdΩ, where y belongs to some Hρ2,n (resp. Q2,n ) in the first (resp. second) integral. Then it is not difficult to choose the constant c such that the form defined by hϕ, ϕ0 i = (ϕ, ϕ0 )1 + c(ϕ, ϕ0 )2 is an invariant non degenerate indefinite metric such that hϕi , ϕj i 6= 0 if and only if (i, j) ∈ {(1, 3), (3, 1), (2, 2)}.
Definition 5. Again in analogy with 4-dimensional Minkowskian QED, the (0) (0) condition ∂ 2 f = 0, on f ∈ 1 , which fixes the space 2 will be called Lorentz condition; the equation ∂ 4 f = 0 will be called the dipole equation.
H
H
6.3. Cn -masslessness, spin 1/2 6.3.1. Reduction on kn and indecomposability of D(E0 , 12 , . . . , 12 ) The following result is known; see for example [1, 7]. Proposition 7. D(E0 , 12 , . . . , 12 ) is unitarizable if and only if E0 ≥ Here we consider only the unitary case, i.e. E0 ≥
n−1 2 .
n−1 2 .
Proposition 8. Let λ = (−E0 , 12 , . . . , 12 ) and recall that ν = 0 (resp. 1) if n is even (resp. odd). 1. If E0 >
n−1 2
then D(E0 , 12 , . . . , 12 ) is irreducible and one has
∞ M 1 1 1 = χ (−[E + l + 2k]) ⊗ π D E0 , , . . . , 0 2 2 |kn 2 l,k=0 ∞ M 1 1 1 + l, , . . . , , ν − χ (−[E0 + l + 2k + 1]) ⊗ π ⊕ 2 2 2 l,k=0
1 1 + l, , . . . , 2 2 1 . 2
(70) then N (λ) is not simple; it contains a maximal submodule 2. If E0 = n−1 2 1 1 1 n+1 1 isomorphic to L(− n+1 2 , 2 , . . . , 2 , ν − 2 ) which carries the UIR D( 2 , 2 , . . . , 1 1 n−1 1 1 2 , ν − 2 ). The irreducible one D( 2 , 2 , . . . , 2 ) is carried by the quotient.
1097
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
1 1 1 1 Proof. 1. If E0 > n−1 2 , then D(E0 + 2 , 0, . . . , 0) ⊗ D(− 2 , 2 , . . . , 2 ) = 1 1 1 1 1 D(E0 , 2 , . . . , 2 ) ⊕ D(E0 + 1, 2 , . . . , 2 , ν − 2 ). If we denote by vσ the maximal vector of D(− 12 , 12 , . . . , 12 ) one finds that, for l, k ∈ N, the vectors vlk ⊗ vσ and vlk ⊗ (X−1,[ν−1]r vσ ) generate a submodule (of the tensor product) isomorphic to L(λ). Pr 1 ν 2. Now assume E0 = n−1 j=2 X−1,j X−j,[ν−1]r . 2 and let Y = ν+1 X−1,[ν−1]r − Then one can see that Yν (v00 ⊗vσ ) generates an irreducible submodule of U(GnC )(v00 ⊗ 1 1 n−1 1 1 vσ ) isomorphic to L(− n+1 2 , 2 , . . . , ν − 2 ) while D( 2 , 2 , . . . , 2 ) is carried by the quotient U(GnC )(v00 ⊗ vσ )/U(GnC )Yν (v00 ⊗ vσ ). 1 1 6.3.2. A Gupta–Bleuler triplet for D( n−1 2 , 2, . . . , 2) ν Let ε ≥ 0 such that E0 = n−1 2 + ε. Proposition 8 says that if ε = 0 then Y is 1 1 sent to 0 by Uλ = D(E0 , 2 , . . . , 2 ). Now assume ε > 0, then Uλ is irreducible; but when ε → 0 one obtains an indecomposable representation: 1 1 1 1 1 n+1 1 n−1 1 n−1 + ε, , . . . , −→ D , ,..., +D , ,..., ,ν − . D 2 2 2 ε→0 2 2 2 2 2 2 2 (71) To construct a Gupta–Bleuler triplet we need explicit realizations of the representations concerned. Let σ = ( 12 , . . . , 12 ) and let, if n is even, σ − = ( 12 , . . . , 12 , − 21 ). We denote by Sσ the irreducible spinor representation D(− 12 , 12 , . . . , 12 ) and, when n is even, by Sσ− the irreducible one D(− 12 , 12 , . . . , 12 , − 21 ). Let + be the carrier space of Sσ and − the carrier one of Sσ− when n is even (resp. {0} when n is = + ⊕ − be the spinor module of Gn . odd). Finally let Let γ−1 , . . . , γ2r−2 be 2r matrices in gl( ) such that [γa , γb ]+ = 2ηab ,f where 2 = −1. Then [A, B]+ = AB + BA, and let γ2r−1 ∈ Cγ−1 · · · γ2r−2 such that γ2r−1
S
S S S S
S
[γa , γb ]+ = 2ηab The following realization of Sσ on
∀a, b ∈ {−1, . . . , n}.
S is well known:
−−→ Sab = Mab p−−−−−−−−
1 1 [γa , γb ] = (γa γb − ηab ). 4 2
Later we shall also need the generators ωj defined by i √ (γ−1 ± iγ0 ) if j = 1, 2 1 ω±j = √ (γ2j−1 ± iγ2j ) if 2 ≤ j ≤ r, 2 if n is odd and j = 0. γn Thus one has [ωj , ωk ]+ = −2δj,−k f We identify the identity of
gl(S ) with 1.
∀j, k ∈ {−r, . . . , r},
1098
M. LAOUES
and the preceding realization of Sσ may be written: Xjk p−−−−−−−− −−→
1 1 [ωj , ωk ] = (ωj ωk + δj,−k ). 4 2
S
2,n −→ + such that We realize D(E0 , 12 , . . . , 12 ) on spinor fields Ψ : H+ 1 Ψ. ∂2Ψ = 0 and δΨ = − E0 + 2
The action of Gn on spinor fields is given by Uλ (Mab ) = Lab + Sab . Let y= /
n X a=−1
Theng
r X
y a γa =
x−j ωj
and ∂/ =
n X
∂ a γa = −
a=−1
j=−r
r X
∂−j ωj .
j=−r
(n + 1)(n + 2) y − /∂/ Ψ Uλ (C2 )Ψ = −y 2 ∂ 2 + δ(δ + n + 1) + 8 (n + 1)(n + 2) y 1 1 = E0 + E0 − − n + − /∂/ Ψ. 2 2 8
(72)
It is easy to prove the following Lemma. y and ∂/ commute with the action of Gn ; Lemma 1. 1./ y 2. [/ , ∂/ ]+ = 2δ + n + 2; 3. −y 2 ∂ 2 = y/ ∂/ (y/ ∂/ − 2δ − n); 4. if ε > 0, then (−2ε)−1 (∂//y − 2) and(−2ε)−1 /y∂/ are projectors on the irreducible subspaces of the tensor product L(−[E0 + 12 ], 0, . . . , 0) ⊗ L( 12 , . . . , 12 ), namely the spaces L(−E0 , 12 , . . . , 12 ) and L(−[E0 + 1], 12 , . . . , 12 , ν − 12 ) respectively. Let us consider the spinor fields Ψ2 and Ψ3 defined by −E0 − 12
Ψ2 (y) = x1
vσ
and
−E0 − 32
Ψ3 (y) = /y x1
ω[ν−1]r vσ .
Then one has 1 1 1 , 0, . . . , 0 ⊗ L ,..., ' U(GnC )Ψ2 ⊕ U(GnC )Ψ3 . L − E0 + 2 2 2 −E0 − 12
Moreover, let Ψ1 (y) = x1
ω−1 ω[ν−1]r . 1 ν E0 + Y Ψ2 = 2
thus
Then 1 1 Ψ3 − ε Ψ1 , 2 2
1 1 E0 + Ψ3 . lim Y Ψ2 = ε→0 2 2 ν
g We identify y 2 with the function y 7−→ y 2 , / y with y 7−→ y /, and so on.
(73)
(74)
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MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
From now on we assume ε = 0, i.e. E0 =
n−1 2 .
1 − 2−ν X1,−[ν−1]r
Then
ν 4 nY
−−→ Ψ2 p−−−−−−−− −−→ Ψ3 . Ψ1 p−−−−−−−−
H
(1/2)
Let i to prove.
= cl(U(GnC )Ψi ), i being 1, 2 or 3. The next proposition is not difficult
H
(1/2)
Proposition 9. 1. 1 closed invariant subspace of
H
(1/2) 1
2.
H
(1/2) 2
3.
(75)
/H
/H
(1/2) 3
(1/2) 2
H
⊃
H
(1/2) 2
⊃
(1/2) i−1 ;
and
H
(1/2) 3
H
(1/2) 3
and
H
(1/2)
i
, i = 2 or 3, is a
1 1 carry the IR D( n+1 2 , 2 , . . . , ν − 2 ), while
1 1 carries the Cn -massless representation D( n−1 2 , 2 , . . . , 2 );
∂/ Ψi = 0
if
y/∂/ Ψ1 = nΨ3 6= 0
i = 2 or i = 3, but
(y/∂/ )2 Ψ1 = 0.
(76)
H
(1/2) y Ψ)(y) = 0 ∀Ψ ∈ 4. limy2 →0 (/ and limy2 →0 (/y Ψ2 )(y) 6= 0. Thus the Cn 3 n−1 1 1 massless representation D( 2 , 2 , . . . , 2 ) may be realized irreducibly on the cone Q2,n .
H
H
(1/2)
(1/2)
H
(1/2)
= / 2 (resp. Definition 6. The elements of the space S 1 (1/2) (1/2) (1/2) (1/2) (1/2) = 2 / 3 , resp. G = 3 ) are called scalar (resp. physical, P resp. gauge) states.
H
H
H
Remark 4. Let − n+1 2
H
K
(1/2)
H
be the closure of the GnC -module generated by the field
1 1 1 ω[ν−1]r vσ ; it carries the IR D( n+1 y 7−→ Φ(y) = x1 2 , 2 , . . . , 2 , − 2 ). Then the Gupta–Bleuler triplet
D
n+1 1 1 1 , ,..., ,ν − 2 2 2 2
−→ D
n−1 1 1 , ,..., 2 2 2
−→ D
n+1 1 1 1 , ,..., ,ν − 2 2 2 2
defined by Ψ1 , Ψ2 and Ψ3 may be redefined by
H H H
(1/2) 1 (1/2) 2 (1/2) 3
n = positive energy solutions of ∂ 2 Ψ = 0, δΨ = − Ψ and (/y∂/ )2 Ψ = 0 , 2 (1/2) = Ψ∈ 1 such that ∂/Ψ = 0 , (77) (1/2) = Ψ∈ 2 such that Ψ ∈ /y (1/2) .
H H
K
H
←→ R n (1/2) y , define (Ψ, Ψ0 )1 = ρ−1 S 1 ×Rn Ψ∗ (y) /y∂/ Ψ0 (y) ρdtd Now, for Ψ and Ψ0 in 1 −1 +y 2 R n and (Ψ, Ψ0 )2 = S 1 ×S n−1 (y2 ) 2 Ψ∗ (y)Ψ0 (y)dtdΩ, y being in some Hρ2,n (resp. Q2,n )
1100
M. LAOUES
in the first (resp. second) integral. Again it is not difficult to choose the constant c such that the form defined by hΨ, Ψ0 i = (Ψ, Ψ0 )1 + c(Ψ, Ψ0 )2 is an invariant non degenerate indefinite metric such that hΨi , Ψj i 6= 0 if and only if (i, j) ∈ {(1, 3), (3, 1), (2, 2)}. Definition 7. The equation ∂/Ψ = 0, which fixes the space called the Lorentz condition.
H
(1/2) , 2
will be
6.4. Indecomposability and GB triplets for spin s ≥ 1 We assume in this subsection that s ≥ 1 and 2s ∈ N. 6.4.1. Indecomposability of D(E0 , s, . . . , s, sν ) Let λ = (−E0 , s, . . . , s, sν ), where |sν | = s and, if n is odd, sν ≥ 0. Proposition 10. 1. D(E0 , s, . . . , s, sν ) is unitarizable ⇐⇒ E0 ≥ n−2+ν + s; 2 n−2+ν 2. if E0 > 2 + s, then N (λ) is simple; + s, then N (λ) contains, up to a multiplicative constant, a 3. if E0 = n−2+ν 2 ν unique maximal vector of weight (−E0 −1, s, . . . , s, sν − ssν ); it is given by Y−1,− sν vλ , s r where 0 Y−1,±r = 2sX−1,±r −
r X
X−1,j X−j,±r ,
j=2 1 = 2sX−1,−r + 2X−1,0 X−r,0 Y−1,−r
2(s − 1) X 2 X − X−1,j X−j,−r − X−1,j X−j,0 X−r,0 . 2s − 1 j=2 2s − 1 j=2 r
r
Since, for n even, the treatment of Uλ is similar for both signs of sν we shall consider from now on that sν = s. Proof of the Proposition. For the first two items see [1, 7]. For the last one, a maximal vector of weight (−E0 − 1, s, . . . , s, s − 1) for n even has the general form: v 0 = aX−1,−r +
r X
bj X−1,j X−j,−r vλ ,
j=2 a and n+ v 0 = 0 implies bj = − 2s for each j. The same technique works for odd n.
Remark 5. The situation for s ≥ 1, for both n even and n odd, is more complicated than for the spin 0 and spin 1/2 cases. Indeed more than one submodule for N (λ) exists when E0 = n−2+ν + s, thus it is a priori possible to construct very 2 different examples of Gupta–Bleuler triplets U 0 −→ Uλ −→ U 0 with U 0 unitary.
1101
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
6.4.2. A GB triplet for D( n−2+ν + s + i, s, . . . , s, s − i), i = 1 or 2 2 + s + ε, ε ≥ 0. To realize our Gupta–Bleuler triplet we need Let E0 = n−2+ν 2 explicitly the representations D(E0 , s, . . . , s) and D(E0 + 1, s, . . . , s, s − 1), especially for ε = 0. Both of them are contained in the reduction of the tensor product D(E0 + s, 0, . . . , 0) ⊗ D(−s, s, . . . , s). The representation S[2sσ] = D(−s, s, . . . , s) itself is contained in the tensor power Sσ⊗2s of the irreducible spinorial representation. We define the action of Mab ∈ Gn on a tensor v1 ⊗ · · · ⊗ v2s ∈ +⊗2s by
S
Sab (v1 ⊗ · · · ⊗ v2s ) =
=
2s X
1 v1 ⊗ · · · ⊗ [γa , γb ]vt ⊗ · · · ⊗ v2s 4 t=1
2s X
(78)
(t)
Sab (v1 ⊗ · · · ⊗ vt ⊗ · · · ⊗ v2s ).
t=1
S
Let γa (t) be defined on the tensorsh of
⊗2s
S ⊕S
=(
⊗2s −)
+
by
γa (t) (v1 ⊗ · · · ⊗ v2s ) = v1 ⊗ · · · ⊗ γa vt ⊗ · · · ⊗ v2s . Then the action defined in (78) may be written more simply: −−→ Sab = Mab p−−−−−−−−
2s X t=1
S
Let Sym( 0
γ a (t) γa (t ) = −
⊗2s + ) r X
(t)
Sab =
2s X 1 (t) (t) γa , γb . 4 t=1
be the space of symmetric tensors in (t)
S
⊗2s +
(79) 0
and let γ (t) ·γ (t ) =
(t0 )
ω−j ωj .
j=−r
Proposition 11. S[2sσ] = D(−s, s, . . . , s) is realized irreducibly on the space:
S
V S = Sym(
⊗2s + )
∩
h\
0
ker γ (t) · γ (t ) − ν
i .
t,t0
t6=t0
2,n We realize the unitary representations of interest on tensor-spinors Ψ : H+ −→ V S such that ∂2Ψ = 0 and δΨ = −(E0 + s)Ψ.
To this effect, we define the action of Gn on them by Mab 7−→ Lab + Sab . Let y(t) = y a γa (t) = /
r X j=−r
h Recall that
(t)
x−j ωj
S− = {0} for n odd.
and ∂/(t) = ∂ a γa (t) = −
r X j=−r
(t)
∂−j ωj ,
1102
M. LAOUES
then one has 2s i h X y/(t)∂/(t) Ψ Uλ (C2 )Ψ = −y 2 ∂ 2 + δ(δ + n + 2s) + rs(s + r − 1 + ν) − t=1 2s i h X y/(t)∂/(t) Ψ. (80) = (E0 + s)(E0 − s − n) + rs(s + r − 1 + ν) − t=1
Lemma 2. 1. For fixed t, /y (t) and ∂/(t) satisfy the three first items of Lemma 1; y (t) , / y (t0 ) ] = 0 and [∂/(t) , ∂/(t0 ) ] = 0; 2. if t 6= t0 , then [/ 0 0 3. if t 6= t0 , then [∂/(t) , y/(t ) ] = γ (t) · γ (t ) (= ν on V S ). Let us define, for non-negative integers k, l and spinors v1 , . . . , vk , symmetric tensors in ⊗2s by
S
v1 · · · vk =
1 X τ (v1 ) ⊗ · · · ⊗ τ (vk ), k! τ ∈Sk
v1l = v1 · · · v1 | {z }
(81)
l terms
and let Ψ1 , Ψ2 and Ψ3 be defined by h i 0 −s Ψ1 (y) = x−E ω−1 ω−r vσ vσ − ν ω−1 vσ ω−r vσ vσ2s−2 , 1 0 −s 2s Ψ2 (y) = x−E vσ , 1 h i Ψ3 (y) = x1−E0 −s−1 /y ω−r vσ vσ − ν /yvσ ω−r vσ vσ2s−2 .
Then one has U(GnC )Ψ2 ⊕ U(GnC )Ψ3 ⊂ L(−[E0 + s], 0, . . . , 0) ⊗ L(s, . . . , s) and one finds that ν Ψ2 = s(E0 + s)Ψ3 − εsΨ1 , Y−1,−r
(82)
ν lim Y−1,−r Ψ2 = s(E0 + s)Ψ3 .
(83)
thus ε→0
From now on we assume E0 = − 12 X1,r
n−2+ν 2
+ s. Then ν 2 Y−1,−r s(n−2+ν+4s)
Ψ1 p−−−−−−−− −−→ Ψ2 p−−−−−−−− −−→ Ψ3 .
H
(84)
= cl(U(GnC )Ψi ), i being equal to 1, 2 or 3. The next proposition is Let i straightforward: (s)
1103
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
H
(s)
Proposition 12. 1. 1 (s) invariant subspace of i−1 .
H
⊃
H
(s) 2
⊃
H
(s) 3
H /H and H carry the H /H carries the representation D( (s) 1 (s) 3
2.
(s) 2
(s) 2
(s) 3
and
H
(s)
i
, i = 2 or 3, is a closed
IR D( n+ν 2 + s, s, . . . , s, s − 1), while + s, s, . . . , s);
n−2+ν 2
3. ∂/(t) Ψi = 0 ∀t ∈ {1, . . . , 2s} 2s X
if
i = 2 or i = 3;
y (t)∂/ (t) Ψ1 = (n − 2 + ν + 4s)Ψ3 6= 0 /
but
t=1
2s X
(85) !2
/y (t)∂/(t)
Ψ1 = 0;
t=1
H
(s) y (1) · · · / y (2s) Ψ)(y) = 0 ∀Ψ ∈ and limy2 →0 (/y (1) · · · /y(2s) Ψ2 )(y) 6= 4. limy2 →0 (/ 3 n−2+ν 0. Thus the representation D( 2 + s, s, . . . , s) may be realized irreducibly on the cone Q2,n .
H
(s)
H H (s)
(s)
H
(s)
Definition 8. The elements of the space S = 1 / 2 (resp. P = (s) (s) (s) = 3 ) are called scalar (resp. physical, resp. gauge) 3 , resp. G states.
H H
H
(s) 2 /
H
Let, for t ∈ N, v t ⊗ (v ∧ v 0 ) = v t+1 ⊗ v 0 − v t ⊗ v 0 ⊗ v, and let τ(t,t0 ) t≤t0 be the system of generators (permutations t ↔ t0 if t 6= t0 and identity if t = t0 ) of the group-algebra of S2s . Let
Y
i 1 hP τ(t,2s) /y(2s) 1≤t≤2s 2s hP 1 = 1≤t≤2s−1 τ(t,2s−1) 2s(2s − 1) i P + y/(2s−1) − y/(2s) 0 ,2s−1) τ τ 0 (t,2s) (t 1≤t j0 1. J (i, j) ∩ [a, b] = ∅, 2. m− (Q(j, λ∇2 H(si ))) = 2N . Remark 3.2. Applying the Hartman–Grobman theorem we can easily prove a necessary condition for the existence of bifurcation point of non-stationary 2π1 × R is a periodic solutions of Hamiltonian system (3.1). Namely, if (si , λ0 ) ∈ H2π bifurcation point then there is j ∈ N such that λ0 ∈ J (i, j). Only the elements of S S the set ki=1 {si } × ∞ j=1 J (i, j) are suspected to be bifurcation points. Let us fix si ∈ {s1 , . . . , sk }, j ∈ N and λ0 ∈ J (i, j). By Remark 3.1 one can choose ε > 0 such that [λ0 − ε, λ0 + ε] ∩ J (i, j) = {λ0 } . L The bifurcation index η(si , λ0 ) ∈ Z ⊕ ( ∞ i=1 Z) is defined as follows: η(si , λ0 )S1 = 0 ,
(3.3)
and η(si , λ0 )Zj = (−1)m
−
(−λ0 ∇2 H(si ))
m− (Q(j, (λ0 + ε)∇2 H(si ))) − m− (Q(j, (λ0 − ε)∇2 H(si ))) . (3.4) 2 From Remark 3.1 it follows that our bifurcation index is well defined, i.e. only a finite number of coordinates of our bifurcation index is different from 0. We can now formulate the main theorem of this subsection. ·
Theorem 3.1 (Global bifurcation theorem for Hamiltonian systems, I). Let us fix j ∈ N, i ∈ {1, . . . , k}, λ0 ∈ J (i, j) and choose ε > 0, such that [λ0 − ε, λ0 + ε] ∩ J (i, j) = {λ0 }. If m− (Q(j, (λ0 + ε)∇2 H(si ))) 6= m− (Q(j, (λ0 − ε)∇2 H(si ))), then continuum C(si , λ0 ) is
1132
A. MACIEJEWSKI and S. RYBICKI
1 1. either unbounded in H2π × R, 1 2. or bounded in H2π × R, and Sk Sk m (a) C(si , λ0 ) ∩ m=1 {sm } × R = m=1 ({sm } × {λm 1 , . . . , λim }) , im k X X
(b)
η(sm , λm i ) = Θ.
(3.5)
m=1 i=1 1 × R is bounded. Let |x|∞ = Proof. Suppose that continuum C(si , λ0 ) ⊂ H2π maxt∈[0,2π] |x(t)|. It is known that there exists a positive constant c, such that for 1 , any x ∈ H2π 1 . |x|∞ ≤ c · |x|H2π
As an immediate consequence of the above inequality, we obtain that the corresponding set of solution curves is bounded in R2N × R. We suppose that this set is included in an open disk Dα = (x, λ) ∈ R2N × R : |x|2 + λ2 < α2 , of a sufficiently large radius α 1. Let us define a smooth function ψ : R2N × R → R, such that ψ(x, λ) = 1 in D2α , and ψ(x, α) = 0 in (R2N × R) − D4α . We define H1 (x, λ) = λψ(x, λ)H(x) and modify Hamiltonian system (3.1) as follows: x˙ = J∇H1 (x, λ) .
(3.6)
Since the vector fields in systems (3.1) and (3.6) coincide in D2α , the set C(si , λ0 ) is also a bounded continuum of solutions of system (3.6). Moreover, the vector field in system (3.6) vanishes outside of D4α , and that is why sup (x,λ)∈R2N ×R
|∇2 H1 (x, λ)| < ∞ .
(3.7)
The rest of this proof falls naturally into two steps. In the first step, we will apply the S1 -equivariant Amann–Zehnder reduction given in [5]. Applying this reduction, we obtain a parametrized family of S1 -equivariant functions, defined on a finitedimensional representation of the group S1 . Step 1. Following the Amann–Zehnder reduction, see [5], we define functional 1 × R → R as follows: f : H2π Z Z 2π 1 2π (−J x, ˙ x) dt − H1 (x, λ) dt . (3.8) f (x, λ) = 2 0 0 It is well known that f is a C 2 -functional. Since ∇f (x, λ) = −J x˙ − ∇H1 (x, λ), looking for the 2π-periodic solutions of system (3.6) is equivalent to looking for the critical points of f with respect to x. In order to find the critical points of the functional f , it is enough to find critical points of a finite-dimensional function a ∈ C 2 (Z ⊕ R). Namely, similarly to the proof of Theorem 5 in [5], and thanks to Remark 2.2 of [4], in order to prove this theorem, it is enough to establish critical
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
...
1133
points of a function a ∈ C 2 (Z ⊕ R), where linear space Z is defined as follows. Let A(x) = J x. ˙ It is known, see Lemma 3 of [5], that operator A is self-adjoint, and that σ(A) = Z, where σ(A) denotes the spectrum of A. By (3.7) one can choose ( ) jinf = inf
j∈N:
sup (x,λ)∈R2N ×R
|∇2 H1 (x, λ)| < j
and define a positive number β in the following way: 1 β= 2
|∇ H1 (x, λ)| + jinf 2
sup (x,λ)∈R2N ×R
,
! .
It is evident that β 6∈ σ(A), and that, for any (x, λ) ∈ R2N ⊕ R, we have σ(∇2 H1 (x, λ)) ⊂ (−β, β) , where σ(B) denotes the spectrum of matrix B. Let us denote by E(µ) a linear 1 , which is an eigenspace of A corresponding to the eigenvalue µ, subspace of H2π spanned by the following vectors: cos(µt)ek + sin(µt)Jek
for k = 1, 2, . . . , 2N .
We put n = max{j ∈ N : j < β}, and Z = E(0) ⊕
n M
(E(−j) ⊕ E(j)) .
j=1
Using notation of the Classification Theorem it can be easily seen that Z≈
n M
R[2N, j].
j=0
It was shown that the map a is constant on the orbits of the action of the group S1 (see p. 177 in [5]). From now on, we will be interested in the orbits of zeros of an S1 -equivariant, gradient map ∇a : Z ⊕ R → Z , where ∇a denotes the gradient of a with respect to the first coordinate. Let us list the main properties of the map ∇a (see Lemma 4 of [5]). There exists an S1 1 defined equivariant, C 1 -map y : Z × R → Z ⊥ , such that a map x : Z × R → H2π by the formula x(z, λ) = z + y(z, λ), satisfying x(si , λ) = si for any λ ∈ R and i = 1, . . . , k, has the following properties: 1. orbits of zeros of ∇a (critical orbits of a) are in an one-to-one correspondence with the critical points of f given by (3.8), i.e. ∇a(z, λ) = 0 iff x(z, λ) is a critical point of functional f . Moreover, a is of the form: Z Z 2π 1 2π (−J w, ˙ w) dt − H1 (w(t), λ) dt , a(z, λ) = 2 0 0 where w(t) = x(z, λ)(t),
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A. MACIEJEWSKI and S. RYBICKI
2. ∇a(si , λ) = 0 for all λ ∈ R and i = 1, . . . , k, 3. ∇2 a(si , λ)|R[2N,0] = −∇2 H1 (si , λ) = −λ∇2 H(si ), for all λ ∈ R and i = 1, . . . , k, 4. ∇2 a(si , λ)|R[2N,j] = Q(j, ∇2 H1 (si , λ)) = Q(j, λ∇2 H(si )), for all (si , λ) ∈ Dα and i = 1, . . . , k. Hence, for any i = 1, . . . , k 1
ker(∇2 a(si , λ)) ∩ Z S = ker(∇2 a(si , λ)) ∩ R[2N, 0] ( {0} for any λ 6= 0 , = R[2N, 0] for λ = 0 . Step 2. Now it is not difficult to verify that the map ∇a : Z ⊕ R → Z satisfies all the assumptions of Theorem 2.2. Thus, applying Theorem 2.2, we complete our proof. A similar theorem has been announced by Dancer in [13]. However, Dancer used as a tool another version of degree for S1 -equivariant gradient maps. Remark 3.3. The period of periodic solution whose existence is guaranteed by the above theorem need not be minimal period. This is the usual feature of results obtained by means of variational methods. 3.2. Case ∇H(x, λ)6= 6 λ∇H(x) In this subsection, we are interested in finding sufficient conditions for the existence of bifurcation points of non-stationary periodic solutions of a fixed period 2π for a parametrized family of Hamiltonian systems of the following form: x˙ = J∇H(x, λ) ,
(3.9)
where H : R2N × R → R is a C 2 -map, such that ∇H −1 (0) = {ϕ1 (R1 ), . . . , ϕk (Rk )}, where for any i = 1, . . . , k 1. Ri is either an open half-line or R, 2. ϕi : Ri → R2N × R are continuous maps, and ϕi (λ) = (xi (λ), λ), where xi : Ri → R2N , is a continuous map. Definition 3.2. A set ϕ1 (R1 ) ∪ · · · ∪ ϕk (Rk ) is said to be the set of trivial 1 × R is said solutions of system (3.9). A point ϕi (λ0 ) ∈ ϕ1 (R1 ) ∪ · · · ∪ ϕk (Rk ) ⊂ H2π to be a bifurcation point of non-stationary 2π-periodic solutions of Hamiltonian 1 × R there exists a nonsystem (3.9) if for its any open neighborhood O ⊂ H2π stationary 2π-periodic solution of system (3.9) in O. Let us denote by C(ϕi (λ0 )) a connected component of the set 1 × R : x(t) is a non-stationary solution of (3.9) on level λ} , closure{(x(t), λ) ∈ H2π
containing ϕi (λ0 ).
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
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As in the previous subsection, for (i, j) ∈ {1, . . . , k} × N, we define matrices Q(j, ∇2 H(ϕi (λ))) as follows: # " jJ T −∇2 H(ϕi (λ)) 2 , Q(j, ∇ H(ϕi (λ))) = jJ −∇2 H(ϕi (λ)) and sets J (i, j) = {λ ∈ Ri : det Q(j, ∇2 H(ϕi (λ))) = 0}. Assume that for any (i, j) ∈ {2, . . . , k} × N 1. #(J (i, j)) < ∞, 2. if λ0 ∈ J (i, j) then ∇2 H(ϕi (λ0 )) is non-degenerate. S∞ Sk Remark 3.4. Notice that only elements of the set i=1 q j=1 ϕi (J (i, j)) can be the bifurcation points of non-stationary 2π-periodic solutions of system (3.9). For fixed (i, j) ∈ {2, . . . , k} × N, λ0 ∈ J (i, j) we choose ε > 0, such that [λ0 − ε, λ0 + ε] ∩ J (i, j) = {λ0 } , L∞ and define the bifurcation index η(ϕi (λ0 )) ∈ Z ⊕ ( i=1 Z) as follows η(ϕi (λ0 ))S1 = 0, and η(ϕi (λ0 ))Zj = (−1)m ·
−
(−∇2 H(ϕi (λ0 )))
m− (Q(j, ∇2 H(ϕi (λ0 + ε))) − m− (Q(j, ∇2 H(ϕi (λ0 − ε))) . (3.10) 2
The bifurcation index is well defined. Theorem 3.2 (Global bifurcation theorem for Hamiltonian systems, II). Let us assume that m− (Q(j, ∇2 H(ϕi (λ0 + ε)))) 6= m− (Q(j, ∇2 H(ϕi (λ0 − ε)))), for fixed i ∈ {2, . . . , k}, j ∈ N, λ0 ∈ J (i, j), and ε > 0 chosen is such a way that [λ0 − ε, λ0 + ε] ∩ J (i, j) = {λ0 }. Then either C(ϕi (λ0 )) ∩ ϕ1 (R1 ) 6= ∅ or continuum C(ϕi (λ0 )) is 1 1. either unbounded in H2π × R, 1 2. or bounded in H2π × R and Sk Sk m (a) C(ϕi (λ0 )) ∩ m=2 ϕm (Rm ) = m=2 {ϕm (λm 1 ), . . . , ϕm (λim )} ,
(b)
im k X X
η(ϕm (λm i )) = Θ .
(3.11)
m=2 i=1
Theorem 3.2 is a slight generalization of Theorem 3.1. The proof of this theorem is in fact the same as the proof of Theorem 3.1 (see Remark 2.2). Remark 3.5. Let us fix j ∈ N. Then it is easy to check that C(ϕi (λ0 ))Zj = 1 Zj 1 ) × R ⊂ H2π × R is a continuum of 2π C(ϕi (λ0 )) ∩ (H2π j -periodic solutions of system (3.9).
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4. Global Bifurcations of Periodic Solutions of H´ enon Heiles System. Solutions of an Arbitrary Period In this section, we consider H´enon–Heiles system and, using the theory developed in the previous sections, study the global behavior of branches of periodic solutions, which emanate from the stationary solutions. The H´enon–Heiles Hamiltonian system has the form: x˙ = J∇H(x) ,
(4.12)
where the Hamiltonian H : R4 → R is given by the formula: H(x1 , x2 , x3 , x4 ) =
1 2 1 (x + x22 + x23 + x24 ) + x33 − x3 x24 . 2 1 3
(4.13)
System (4.12), written explicitly has the following form: x˙1 = −x3 − x23 + x24 , x˙2 = −x4 + 2x3 x4 , x˙3 = x1 , x˙4 = x2 . Lemma 4.1. The H´enon–Heiles system has the following properties: 1. if
2πi cos 3 2πi sin 3 g(i) = 0 0
− sin cos
2πi 3
0
2πi 3
0
0
cos
2πi 3
0
sin
2πi 3
0
0 , 2πi − sin 3 2πi cos 3
then J∇H(g(i)x) = g(i)J∇H(x) , for i = 0, 1, 2, 2. there are four stationary solutions
s3 =
s1 = (0, 0, 0, 0) , √ ! 1 3 = g(1)s2 , 0, 0, , 2 2
s2 = (0, 0, −1, 0) , √ ! 1 3 = g(2)s2 , s4 = 0, 0, , − 2 2
3. σ(∇2 H(s1 )) = {1, 1, 1, 1}, σ(∇2 H(s2 )) = σ(∇2 H(s3 )) = σ(∇2 H(s4 )) = {−1, 1, 1, 3},
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
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4. (a) σ(Q(j, λ∇2 H(s1 ))) = {−λ + j, −λ − j}, multiplicity of any eigenvalue equals 4, p p p (b) p σ(Q(j, λ∇2 H(si ))) = { λ2 + j 2 , − λ2 + j 2 , −2λ + λ2 + j 2 , −2λ − λ2 + j 2 , multiplicity of any eigenvalue equals 2, for i = 2, 3, 4. The following lemma is a direct consequence of Lemma 4.1. Lemma 4.2. It is evident that 1. J (1, j) = {±j}, 2. for sufficiently small ε > 0 m− (Q(j, (j + ε)∇2 H(s1 )) = 8 , m− (Q(j, (j − ε)∇2 H(s1 )) = 4 , m− (Q(j, (−j + ε)∇2 H(s1 )) = 4 , m− (Q(j, (−j − ε)∇2 H(s1 )) = 0 , −
2
H(s1 )) = 1, 3. (−1)m (±j∇ √ 3 4. J (i, j) = {± 3 j} for i = 2, 3, 4, 5. for sufficiently small ε > 0
! !! √ 3 2 j + ε ∇ H(si ) Q j, = m 3 ! !! √ 3 j − ε ∇2 H(si ) = m− Q j, 3 ! !! √ 3 − 2 j + ε ∇ H(si ) Q j, − = m 3 ! ! √ 3 − 2 j − ε ∇ H(si ) = Q(j, − m 3 −
6,
4,
4,
2,
for i = 2, 3, 4, √ − 2 3 6. (−1)m (± 3 j∇ H(si )) = −1, for i = 2, 3, 4. Let us change the study of periodic solutions of an arbitrary period of system (4.12) into the bifurcation problem of periodic solutions with a fixed period 2π. Namely, applying the change of variables s(t) = λt , we obtain ( x˙ = λJ∇H(x), (4.14) x(0) = x(2π) . It is clear that 2π-periodic solutions of system (4.14) on level λ correspond to 2πλperiodic solutions of system (4.12). We are now in a position to formulate the main theorem of this section. This theorem ensures the existence of unbounded (in
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A. MACIEJEWSKI and S. RYBICKI
period or amplitude) components of non-stationary periodic solutions of H´enon– Heiles system which emanate from stationary solutions of this system. Theorem 4.1. Let us fix any i ∈ {1, 2, 3, 4}, j ∈ N and λ0 ∈ J (i, j). Then continuum C(si , λ0 ) of non-stationary 2π-periodic solutions of system (4.14) is 1 × R. unbounded in H2π Proof. Our proof starts with the observation that the H´enon–Heiles system has Z3 × S1 symmetry, where Z3 -symmetries come from the symmetry of Hamiltonian (4.13). In fact, the H´enon–Heiles Hamiltonian is fixed on the orbits of the action 1 × R, but we do not need such rich symmetries in our of the group D3 × S1 on H2π proof. Functional (3.2) constructed for system (4.14) is fixed on the orbits of the action of the group Z3 × S1 given by (g(i), ϕ) ? (x(t), λ) := (g(i)x(t + ϕ), λ) , 1 × R is a solution where g(i) is defined in Lemma 4.1. Hence, if (x(t), λ) ∈ H2π 1 of system (4.14), and (g(i), ϕ) ∈ Z3 × S , then (g(i), ϕ) ? (x(t), λ) is a solution of system (4.14). By (3.3), (3.4) and Lemma 4.2, we compute the bifurcation indices ( 2 Q = Zj , η(s1 , j)Q = 0 Q 6= Zj ,
and
√
3 j η si , 3
!
( = Q
−1
Q = Zj ,
0
Q 6= Zj ,
for i = 2, 3, 4 and j ∈ N. Let us suppose, contrary to our claim, that the continuum C(si , λ0 ) is bounded in 1 × R. From Lemma 4.1 it follows that system (4.14) satisfies all the assumptions H2π of Theorem 3.1. Notice that the bifurcation indices η(s1 , j) are nontrivial and their √ nonzero coordinates are positive. On the other hand, the bifurcation indices η(si , 33 j) are nontrivial but their nonzero coordinates are negative. By the above and formula (3.5) there exists j ∈ N such that (s1 , j) ∈ C(si , λ0 ). In other words C(s1 , j) = C(si , λ0 ). The proof will be completed if we show that C(s1 , j) is unbounded for any j ∈ N. Let us fix j ∈ N. Since (g(i), 0) ? (s1 , j) = (s1 , j), (g(i), 0) ? C(s1 , j) = C(s1 , j) for i = 0, 1, 2. √Therefore, by Lemma 4.1, √ we have: if s1 6= sm and √ there exists j0 such that (sm , 33 j0 ) ∈ C(s1 , j) then (s2 , 33 j0 ) ∈ C(s1 , j), (s3 , 33 j0 ) ∈ C(s1 , j) and √ (s4 , 33 j0 ) ∈ C(s1 , j). Thus, by Theorem 3.1, we obtain (√ √ ) 3 3 j1 , . . . , jp ∪ {s1 } C(s1 , j) ∩ ({s1 , s2 , s3 , s4 } × R) = {sm } × 3 3 m=2 4 [
×{j, l1 , . . . , lq } ,
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
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1139
and, consequently, by (3.5) √ ! q X 3 ji + η sm , η(s1 , li ) + η(s1 , j) = Θ ∈ Z ⊕ 3 i=1 i=1
p 4 X X
∞ M
m=2
i=1
It is clear that p X
√
3 ji η sm , 3· 3 i=1
! +
q X
η(s1 , li ) + η(s1 , j) = Θ ∈ Z ⊕
∞ M
i=1
! Z
.
! Z
.
(4.15)
i=1
Let us look at the coordinate of (4.15) which corresponds to the isotropy group Zj . We check at once that this coordinate is equal to 2 or −1, contrary to (4.15). Corollary 4.1. As an immediate consequence of Theorem 4.1, we obtain the following. For any stationary solution si ∈ {s1 , s2 , s3 , s4 } of H´enon–Heiles system there exists a connected set C(si ) of periodic solutions of this system, emanating from si and satisfying at least one of the following conditions: 1. for any arbitrary large period T there exists a T -periodic function in C(si ), 2. there exists a function in C(si ) of arbitrary large amplitude. The results of this section were obtained earlier in [11, 17], see the remark below. The aim of this section was to show how our general machinery works in the case of the well-known H´enon–Heiles Hamiltonian system. However, we would like to point out that our methods also work for more general class of Hamiltonian systems for which approach used in [11, 17] cannot be applied, see Sec. 5. In Sec. 5 we prove new results on periodic solutions of H´enon–Heiles family of Hamiltonian systems. Remark 4.1. The sets C(si ) have the following properties: 1. C(s1 ) has property 1. stated in Corollary 4.1 (see [17]), 2. C(s1 ) has property 2. stated in Corollary 4.1, because C(s1 ) ∩ H −1 (h) 6= ∅ for any h > 0 (see [11]), 3. C(si ) has property 2. stated in Corollary 4.1, because C(si ) ∩ H −1 (h) 6= ∅ for any h > 0 (see [11]), for i = 2, 3, 4. 5. Global Bifurcations of Periodic Solutions of H´ enon Heiles System. Solutions of Fixed Period T = 2π In this section, we consider an 1-parameter H´enon–Heiles family of Hamiltonian systems and, using the theory developed in previous sections, study the global behavior of branches of 2π-periodic solutions, which emanate from the curves of stationary solutions. Let us consider a family of Hamiltonian systems of the form: x˙ = J∇H(x, λ) ,
(5.16)
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A. MACIEJEWSKI and S. RYBICKI
where the Hamiltonian H : R4 × R → R is given by the formula: H(x1 , x2 , x3 , x4 , λ) =
1 2 λ (x1 + x22 + x23 + x24 ) + x33 − x3 x24 . 2 3
Family (5.16), written explicitly, has the form: x˙1 = −x3 − λx23 + x24 , x˙2 = −x4 + 2x3 x4 , x˙3 = x1 , x˙4 = x2 . It was considered in [9, 19]. Lemma 5.1. The H´enon–Heiles family (5.16) has the following properties: 1. there are five curves of stationary solutions of family (5.16), namely ϕ1 (λ) = ((0, 0, 0, 0), λ) 1 0, 0, − , 0 , λ ϕ2 (λ) = λ 1 0, 0, − , 0 , λ ϕ3 (λ) = λ 1 √λ+2 ,λ 0, 0, , 2 ϕ4 (λ) = 2 √ 1 λ+2 ,λ ϕ5 (λ) = 0, 0, , − 2 2
R1 = R , R2 = {λ ∈ R : λ > 0} , R3 = {λ ∈ R : λ < 0} , R4 = {λ ∈ R : λ ≥ −2} , R5 = {λ ∈ R : λ ≥ −2} ,
2. moreover, σ(Q(j, ∇2 H(ϕ1 (λ)))) = {±j − 1} ,
for λ ∈ R1 ,
σ(Q(j, ∇2 H(ϕ2 (λ)))) ) ( p 2 λ2 p 2λ + 2 ± 1 + j , = ± 1 + j2, − 2λ σ(Q(j, ∇2 H(ϕ3 (λ)))) ) ( p 2 λ2 p 2λ + 2 ± 1 + j , = ± 1 + j2, − 2λ
for λ ∈ R2 ,
for λ ∈ R3 ,
σ(Q(j, ∇2 H(ϕ4 (λ)))) ) ( p p 3 + λ ± (λ + 1)2 + 4j 2 2 , = ± 1 + j ,− 2
for λ ∈ R4 ,
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
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σ(Q(j, ∇2 Hϕ5 (λ)))) ) ( p 2 + 4j 2 p 3 + λ ± (λ + 1) , = ± 1 + j2, − 2
1141
for λ ∈ R5 ,
and the multiplicity of any eigenvalue is equal to 4, 2, 2, 2, 2, respectively. For any (i, j) ∈ {1, 2, 3, 4, 5} × N, we define J (i, j) = {λ : det(Q(j, ∇2 H(ϕi (λ)))) = 0} . The following lemma is a direct consequence of Lemma 5.1. Lemma 5.2. It is easy to check that ( J (1, j) =
1.
for j = 1 ,
∅
for j > 1 ,
∅ ∅ J (2, j) = ( ) p 4 + 3j 2 + 4 λ2,j = j2 − 4
2.
for j = 1 , for j = 2 , for j > 2 ,
( ) √ √ 4+ 7 4− 7 λ3,0 = − , λ3,1 = − 3 3 3 λ3,2 = − J (3, j) = 8 ) ( p 4 − 4 + 3j 2 λ3,j = j2 − 4
3.
4. 5. 6. 7.
R
J (4, j) = {λ4,j = −2 + j 2 } for j ≥ 1, J (5, j) = {λ5,j = −2 + j 2 } for j ≥ 1, − 2 (−1)m (−∇ H(ϕ1 (λ))) = 1, − 2 (−1)m (−∇ H(ϕ2 (λ))) = −1, ( (−1)m
8. −
2
−
(−∇2 H(ϕ3 (λ)))
=
1 −1
9. (−1)m (−∇ H(ϕ4 (λ))) = −1 for λ > −2, − 2 10. (−1)m (−∇ H(ϕ5 (λ))) = −1 for λ > −2,
for j = 1 ,
for j = 2 ,
for j > 2 ,
for λ ∈ (−2, 0) , for λ ∈ (−∞, −2) ,
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A. MACIEJEWSKI and S. RYBICKI
11. for j > 2: (−1)m
−
(−∇2 H(ϕ2 (λ2,j )))
· (m− (Q(j, ∇2 H(ϕ2 (λ2,j + ε))))
−m− (Q(j, ∇2 H(ϕ2 (λ2,j − ε)))) = −1 · (4 − 6) = 2 , 12.
(−1)m
−
(−∇2 H(ϕ3 (λ3,0 )))
(5.17)
· (m− (Q(2, ∇2 H(ϕ3 (λ3,0 + ε))))
−m− (Q(2, ∇2 H(ϕ3 (λ3,0 − ε)))) = −1 · (4 − 6) = 2 , (−1)m
−
(−∇2 H(ϕ3 (λ3,1 )))
(5.18)
· (m− (Q(2, ∇2 H(ϕ3 (λ3,1 + ε))))
−m− (Q(2, ∇2 H(ϕ3 (λ3,1 − ε)))) = 1 · (2 − 4) = −2 , 13.
(−1)m
−
(−∇2 H(ϕ3 (λ3,2 )))
(5.19)
· (m− (Q(2, ∇2 H(ϕ3 (λ3,2 + ε))))
−m− (Q(2, ∇2 H(ϕ3 (λ3,2 − ε)))) = 1 · (2 − 4) = −2 ,
(5.20)
14. for j > 2: (−1)m
−
(−∇2 H(ϕ3 (λ3,j )))
· (m− (Q(j, ∇2 H(ϕ3 (λ3,j + ε))))
−m− (Q(j, ∇2 H(ϕ3 (λ3,j − ε)))) = 1 · (2 − 4) = −2 ,
(5.21)
15. for j ≥ 1: (−1)m
−
(−∇2 H(ϕ4 (λ4,j )))
· (m− (Q(j, ∇2 H(ϕ4 (λ4,j + ε))))
−m− (Q(j, ∇2 H(ϕ4 (λ4,j − ε)))) = −1 · (6 − 4) = −2 ,
(5.22)
16. for j ≥ 1: (−1)m
−
(−∇2 H(ϕ5 (λ5,j )))
· (m− (Q(j, ∇2 H(ϕ5 (λ5,j + ε))))
−m− (Q(j, ∇2 H(ϕ5 (λ5,j − ε)))) = −1 · (6 − 4) = −2 .
(5.23)
Now we are in a position to formulate the main theorem of this section. Namely, we describe topological properties of continua of non-stationary 2π-periodic solutions of H´enon–Heiles family (5.16) which bifurcate from the set of stationary solutions. Theorem 5.1. Continua of 2π-periodic solutions of family (5.16) have the following properties.
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
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1143
1. If C(ϕ3 (λ3,0 )) ∩ ϕ1 (R1 ) = ∅, then 1 ×R (a) either C(ϕ3 (λ3,0 )) is unbounded is H2π (b) or C(ϕ3 (λ3,0 )) is bounded and there is i ∈ {3, 4, 5}, such that ϕi (λi,1 ) ∈ C(ϕ3 (λ3,0 )); moreover, either C(ϕk (λk,1 )) ∩ ϕ1 (R1 ) 6= ∅ or C(ϕk (λk,1 )) 1 × R, for k ∈ {3, 4, 5} − {i}, is unbounded in H2π 3. for any i ∈ {3, 4, 5}, if C(ϕi (λi,1 )) ∩ ϕ1 (R1 ) = ∅, then 1 × R, (a) either C(ϕi (λi,1 )) in unbounded is H2π (b) or C(ϕi (λi,1 )) is bounded, and ϕ3 (λ3,0 ) ∈ C(ϕi (λi,1 )); moreover, either 1 × R, for C(ϕk (λk,1 )) ∩ ϕ1 (R1 ) 6= ∅ or C(ϕk (λk,1 )) is unbounded in H2π k ∈ {3, 4, 5} − {i}, 3. for any i ∈ {3, 4, 5}; if C(ϕi (λi,2 )) ∩ ϕ1 (R1 ) = ∅, then C(ϕi (λi,2 )) is un1 × R, bounded in H2π 4. for any j ≥ 3, if C(ϕ2 (λ2,j )) ∩ ϕ1 (R1 ) = ∅, then 1 × R, (a) either C(ϕ2 (λ2,j )) is unbounded in H2π (b) or C(ϕ2 (λ2,j )) is bounded, and there exists i ∈ {3, 4, 5}, such that ϕi (λi,j ) ∈ C(ϕ2 (λ2,j )); moreover, either C(ϕk (λk,j )) ∩ ϕ1 (R1 ) 6= ∅ or 1 × R, for k ∈ {3, 4, 5} − {i}, C(ϕk (λk,j )) is unbounded in H2π 5. for any i ∈ {3, 4, 5} and j ≥ 3, if C(ϕi (λi,j )) ∩ ϕ1 (R1 ) = ∅, then 1 ×R (a) either C(ϕi (λi,j )) is unbounded in H2π (b) or C(ϕi (λi,j )) is bounded, and ϕ2 (λ2,j ) ∈ C(ϕi (λi,j )); moreover, either 1 × R, for C(ϕk (λk,j )) ∩ ϕ1 (R1 ) 6= ∅ or C(ϕk (λk,j )) is unbounded in H2π k ∈ {3, 4, 5} − {i}. Proof. 1. By (5.18), (5.19), (5.22), (5.23) and (3.10), we obtain ( ( 2 Q = Z1 , −2 Q = Z1 , η(ϕ3 (λ3,1 ))Q = η(ϕ3 (λ3,0 ))Q = 0 Q 6= Z1 , 0 Q 6= Z1 , ( η(ϕ4 (λ4,1 ))Q =
−2
Q = Z1 ,
0
Q 6= Z1 ,
( η(ϕ5 (λ5,1 ))Q =
−2
Q = Z1 ,
0
Q 6= Z1 .
(5.24)
(5.25)
Moreover, for any i = 2, 3, 4, 5, and j > 1 η(ϕi (λi,j ))Z1 = 0 .
(5.26)
Let us suppose that C(ϕ3 (λ3,0 )) ∩ ϕ1 (R1 ) = ∅ and that C(ϕ3 (λ3,0 )) is bounded 1 × R. in H2π By (3.11) C(ϕ3 (λ3,0 ) ∩ {ϕ3 (λ3,1 ), ϕ4 (λ4,1 ), ϕ5 (λ5,1 )} 6= ∅, because the sum of L∞ bifurcation indices equals the trivial element in Z ⊕ ( i=1 Z). We fix i ∈ {3, 4, 5} such that ϕi (λi,1 ) ∈ C(ϕ3 (λ3,0 ). By (5.24)–(5.26) and (3.11), we have / C(ϕ3 (λ3,0 ) ϕk (λk,1 ) ∈
for
k ∈ {3, 4, 5} − {i} .
We fix k ∈ {3, 4, 5} − {i} and suppose that C(ϕk (λk,1 )) ∩ ϕ1 (R1 ) = ∅. By (5.24)– (5.26) and (3.11), continuum C(ϕk (λk,1 )) is unbounded.
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A. MACIEJEWSKI and S. RYBICKI
2. The proof for 2. is similar to the proof of 1. 3. By (5.20), (5.22), (5.23) and (3.10), we obtain ( ( −2 Q = Z2 , −2 η(ϕ4 (λ4,2 ))Q = η(ϕ3 (λ3,2 ))Q = 0 Q 6= Z2 , 0 ( η(ϕ5 (λ5,2 ))Q =
−2
Q = Z2 ,
0
Q 6= Z2 .
Q = Z2 , Q 6= Z2 ,
(5.27)
(5.28)
Moreover, for any i = 2, 3, 4, 5, and j 6= 2 η(ϕi (λi,j ))Z2 = 0 .
(5.29)
We fix i ∈ {3, 4, 5} and suppose that C(ϕi (λi,2 )) ∩ ϕ1 (A1 ) = ∅. By (5.27)–(5.29) and (3.11) continuum C(ϕi (λi,2 )) is unbounded. 4. The proof of 4. is similar to the proof of 1. 5. The proof of 5. is similar to the proof of 1. Combining Theorem 3.2 and Remark 3.5 with the fact that ∇2 H(ϕ1 (λ)) = Id for any λ ∈ R1 , one can prove the following two theorems. These theorems yield information about 2π/j-periodic solutions of family (5.16). The proofs of these theorems are in fact similar to the proof of Theorem 5.1. The only difference is that we must restrict functional (3.2) to the set of fixed points of the action of the group Z2 in Theorem 5.2 and Zj , j > 2 in Theorem 5.3. Theorem 5.2. Continua of π-periodic solutions of family (5.16) have the following properties: 1 × 1. continua C(ϕ3 (λ3,2 ))Z2 , C(ϕ4 (λ4,2 ))Z2 , C(ϕ5 (λ5,2 ))Z2 are unbounded in H2π R, 2. for any even j > 2 (a) either C(ϕ2 (λ2,j ))Z2 , C(ϕ3 (λ3,j ))Z2 , C(ϕ4 (λ4,j ))Z2 , C(ϕ5 (λ5,j ))Z2 are 1 × R, unbounded in H2π Z2 (b) or C(ϕ2 (λ2,j )) is bounded and there is i ∈ {3, 4, 5} such that ϕi (λi,j ) ∈ 1 × R, for C(ϕ2 (λ2,j ))Z2 ; moreover, C(ϕk (λk,j ))Z2 is unbounded in H2π k ∈ {3, 4, 5} − {i},
3. for any i ∈ {3, 4, 5} and even j > 2 1 × R, (a) either C(ϕi (λi,j ))Z2 is unbounded in H2π Z2 (b) or C(ϕi (λi,j )) is bounded and ϕ2 (λ2,j ) ∈ C(ϕi (λi,j ))Z2 ; moreover, 1 × R, for k ∈ {3, 4, 5} − {i}. C(ϕk (λk,j ))Z2 is unbounded in H2π Theorem 5.3. Let j ≥ 3, then continua of have the following properties. For any k ∈ N:
2π j -periodic
solutions of family (5.16)
1. either continua C(ϕ2 (λ2,kj))Zj , C(ϕ3 (λ3,kj))Zj , C(ϕ4 (λ4,kj))Zj , C(ϕ5 (λ5,kj))Zj 1 are unbounded in H2π × R,
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
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2. or C(ϕ2 (λ2,kj ))Zj is bounded and there is i ∈ {3, 4, 5}, such that ϕi (λi,kj ) ∈ 1 × R for r ∈ C(ϕ2 (λ2,kj ))Zj ; moreover, C(ϕr (λr,kj ))Zj are unbounded in H2π {3, 4, 5} − {i}.
References [1] J. F. Adams, Lectures on Lie Groups, Benjamin, New York, 1969. [2] H. Amann, “Multiple positive fixed points of asymptotically linear maps”, J. Funct. Anal. 17 (1974) 174–213. [3] H. Amann, “Saddle points and multiple solutions of differential equations”, Math. Z. 169 (1979) 127–166. [4] H. Amann and E. Zehnder, “Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations”, Ann. Sc. Norm. Super. Pisa, CC. Sci. IV, Ser 7, (1980) 539–603. [5] H. Amann and E. Zehnder, “Periodic solutions of asymptotically linear Hamiltonian systems”, Manuscr. Math. 32 (1980) 149–189. [6] M. Antonowicz and S. Rauch-Wojciechowski, “Bi-Hamiltonian formulation of the Henon–Heiles system and its multidimensional extension”, Phys. Lett. A163 (1992) 167. [7] T. Bartsch, “Topological methods for variational problems with symmetries”, Lect. Notes in Math. 1560, Springer-Verlag, Berlin-Heidelberg-New York, 1993. [8] M. Blaszak and S. Rauch-Wojciechowski, “A generalized Henon–Heiles system and related integrable Newton equations”, J. Math. Phys. 35 (1994) 1693. [9] M. Braun, “On the applicability of third integral of motion”, J. Diff. Eq. 13 (1973) 300–318. [10] R. C. Churchill, G. Pecelli and D. L. Rod, “Hyperbolic periodic orbits”, J. Diff. Eq. 24 (1977) 329–348. [11] R. C. Churchill, G. Pecelli and D. L. Rod, “A survey of the H´ enon–Heiles Hamiltonian with applications to related examples”, Lectures Notes in Phys. 93 (1979) 76–136. [12] R. C. Churchill and D. L. Rod, “Pathology in dynamical systems II: Applications”, J. Diff. Eq. 21 (1976) 66–112. [13] E. N. Dancer, “A new degree for S 1 -invariant mappings and applications”, Ann. Inst. H. Poincar´e, Analyse Nonlin´eaire 2 (1985) 473–486. [14] A. P. F. Fordy, “The Henon–Heiles system revisited”, Physica D52 (1991) 204. [15] M. H´enon and Heiles, “The applicability of the third integral of motion: Some numerical experiments”, Astronomical J. 69(1) (1964) 73–79. [16] H. Ito, “Non-integrability of H´ enon–Heiles system and a theorem of Ziglin”, Kodai Math. J. 8(1) (1985) 120–138. [17] R. H. G. Helleman, “Periodic solutions of arbitrary period, variational methods”, Lectures Notes in Phys. 93 (1979) 353–375. [18] S. Kasperczuk, “Homoclinic chaos in generalized H´enon–Heiles system”, Acta Phys. Polonica A88 (1995) 1073–1079. [19] M. Kummer, “On resonant nonlinearly coupled oscillators with two equal frequencies”, Commun. Math. Phys. 48 (1976) 53–79. [20] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Berlin Heidelberg New York, Springer, 1989. [21] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Inst. of Mathematical Sciences, New York, 1974. [22] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Reg. Conf. Ser. Math. 35, Providence, R.I., Am. Math. Soc., 1986.
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[23] D. L. Rod and R. C. Churchill, “A guide to the H´ enon–Heiles Hamiltonian”, in Singularities and Dynamical Systems, ed. S. N. Pnevmatikos, Elsevier Sci. Publ., 1985, 385–395. [24] S. Rybicki, “S1 -degree for orthogonal maps and its applications to bifurcation theory”, Nonlinear Anal. TMA 23(1) (1994) 83–102. [25] S. Rybicki, “On periodic solutions of autonomous Hamiltonian systems via degree for S1 -equivariant gradient maps”, to appear in Nonlinear Anal. TMA, (1998). [26] S. Rybicki, “Applications of degree for S1 -equivariant gradient maps to variational nonlinear problems with S1 -symmetries”, Top. Meth. in Nonlin. Anal. 9(2) (1997) 383–417. [27] M. Struwe, “Variational methods; Applications to nonlinear partial differential equations and Hamiltonian systems”, A series of modern surveys in mathematics 34, Springer, (1996).
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES IN 1+1 DIMENSIONS ∗ ¨ MICHAEL MUGER
Dipartimento di Matematica Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica I-00133 Roma Italy E-mail : [email protected] Received 6 May 1997 Revised 2 December 1997 We show that a large class of massive quantum field theories in 1 + 1 dimensions, characterized by Haag duality and the split property for wedges, does not admit locally generated superselection sectors in the sense of Doplicher, Haag and Roberts. Thereby the extension of DHR theory to 1 + 1 dimensions due to Fredenhagen, Rehren and Schroer is vacuous for such theories. Even charged representations which are localizable only in wedge regions are ruled out. Furthermore, Haag duality holds in all locally normal representations. These results are applied to the theory of soliton sectors. Furthermore, the extension of localized representations of a non-Haag dual net to the dual net is reconsidered. It must be emphasized that these statements do not apply to massless theories since they do not satisfy the above split property. In particular, it is known that positive energy representations of conformally invariant theories are DHR representations.
1. Introduction It is well known that the superselection structure, i.e. the structure of physically relevant representations or “charges”, of quantum field theories in low dimensional spacetimes gives rise to particle statistics governed by the braid group and is described by “quantum symmetries” which are still insufficiently understood. The meaning of “low dimensional” in this context depends on the localization properties of the charges under consideration. In the framework of algebraic quantum field theory [?, ?] several selection criteria for physical representations of the observable algebra have been investigated. During their study of physical observables obtained from a field theory by retaining only the operators invariant under the action of a gauge group (of the first kind), Doplicher, Haag and Roberts were led to singling out the class of locally generated superselection sectors. A representation is of this type if it becomes unitarily equivalent to the vacuum representation when restricted to the observables localized in the spacelike complement of an arbitrary double cone (intersection of future and past directed light cones): ∗ Supported
by the Studienstiftung des deutschen Volkes and the CEE. 1147
Reviews in Mathematical Physics, Vol. 10, No. 8 (1998) 1147–1170 c World Scientific Publishing Company
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π A(O0 ) ∼ = π0 A(O0 ) ∀ O ∈ K .
(1.1)
Denoting the set of all double cones by K we consider a quantum field theory to be defined by its net of observables K 3 O 7→ A(O). This is a map which assigns to each double cone a C ∗ -algebra A(O) satisfying isotony: O1 ⊂ O2 ⇒ A(O1 ) ⊂ A(O2 ) .
(1.2)
This net property allows the quasilocal algebra to be defined by A=
[
A(O)
k·k
.
(1.3)
O∈K
The net is local in the sense that [A(O1 ), A(O2 )] = {0}
(1.4)
if O1 , O2 are spacelike to each other. The algebra A(G) associated with an arbitrary subset of Minkowski space is understood to be the subalgebra of A generated (as a C ∗ -algebra) by all A(O) where G ⊃ O ∈ K. Furthermore, the Poincar´e group acts on A by automorphisms αΛ,x such that αΛ,x (A(O)) = A(ΛO + x)
∀ O.
(1.5)
This abstract approach is particularly useful if there is more than one vacuum. One requires of a physically reasonable representation that at least the translations (Lorentz invariance might be broken) are unitarily implemented: π ◦ αx (A) = Uπ (x)π(A)Uπ (x)∗ ,
(1.6)
the generators of the representation x 7→ U (x), i.e. the energy-momentum operators, satisfying the spectrum condition (positivity of the energy). Vacuum representations are characterized by the existence of a unique (up to a phase) Poincar´e invariant vector. Furthermore we assume them to be irreducible and to satisfy the Reeh–Schlieder property, the latter following from the other assumptions if weak additivity is assumed. In the analysis of superselection sectors satisfying (??) relative to a fixed vacuum representation one usually assumes the latter to satisfy Haag dualitya π0 (A(O))0 = π0 (A(O0 ))00
∀ O ∈ K,
(1.7)
which may be interpreted as a condition of maximality for the local algebras. In [?, ?], based on (??), a thorough analysis of the structure of representations satisfying (??) was given, showing that the category of these representations a M0 = {X ∈ B(H)|XY = Y X ∀ Y ∈ M} denotes the algebra of all bounded operators commuting with all operators in M. If M is a unital ∗-algebra then M00 is known to be the weak closure of a M.
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
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together with their intertwiners is monoidal (i.e. there is a product or, according to current fashion, fusion structure), rigid (i.e. there are conjugates) and permutation symmetric. In particular, the Bose–Fermi alternative, possibly with parastatistics, came out automatically although the analysis started from observable, i.e. strictly local, quantities. A lot more is known in this situation (cf. [?]) but we will not need that. A substantial part of this analysis, in particular concerning permutation statistics and the Bose–Fermi alternative, is true only in at least 2 + 1 spacetime dimensions. The generalization to 1 + 1 dimensions, where in general only braid group statistics obtains, was given in [?] and applied to conformally invariant theories in [?]. Whereas for the latter theories all positive energy representations are of the DHR type [?], it has been clear from the beginning that the criterion (??) cannot hold for charged sectors in gauge theories due to Gauss’ law. Implementing a programme initiated by Borchers, Buchholz and Fredenhagen proved [?] for every massive one-particle representation (where there is a mass gap in the spectrum followed by an isolated one-particle hyperboloid) the existence of a vacuum representation π0 such that π A(C 0 ) ∼ = π0 A(C 0 ) ∀ C .
(1.8)
Here the C’s are spacelike cones which we do not need to define precisely. In ≥ 3 + 1 dimensional spacetime the subsequent analysis leads to essentially the same structural results as the original DHR theory. Due to the weaker localization properties, however, the transition to braid group statistics and the loss of group symmetry occur already in 2 + 1 dimensions, see [?]. In the 1 + 1 dimensional situation with which we are concerned here, spacelike cones reduce to wedges (i.e. translates of WR = {x ∈ R2 | x1 ≥ |x0 |} and the spacelike complement WL = WR0 ). Furthermore, the arguments in [?] allow us only to conclude the existence of two a priori different vacuum representations π0L , π0R such that the restriction of π to left handed wedges (translates of WL ) is equivalent to π0L and similarly for the right handed ones. As for such representations, of course long well-known as soliton sectors, an operation of composition can only be defined if the “vacua fit together” [?], there is in general no such thing as permutation or braid group statistics. For lack of a better name soliton representations with coinciding left and right vacuum, i.e. representations which are localizable in wedges, will be called “wedge representations (or sectors)”. There have long been indications that the DHR criterion might not be applicable to massive 2d-theories as it stands. The first of these was the fact, known for some time, that the fixpoint nets of Haag-dual field nets with respect to the action of a global gauge group do not satisfy duality even in simple sectors, whereas this is true in ≥ 2 + 1 dimensions. This phenomenon has been analyzed thoroughly in [?] under the additional assumption that the fields satisfy the split property for wedges. This property, which is expected to be satisfied in all massive quantum field theories, plays an important role also in the present work which we summarize briefly.
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In the next section we will prove some elementary consequences of Haag duality and the split property for wedges (SPW), in particular strong additivity and the time-slice property. The significance of our assumptions for superselection theory derives mainly from the fact that they preclude the existence of locally generated superselection sectors. More precisely, if the vacuum representation satisfies Haag duality and the SPW then every irreducible DHR representation is unitarily equivalent to the vacuum representation. This important and perhaps surprising result, to be proved in Sec. 3, indicates that the innocent-looking assumptions of the DHR framework are quite restrictive when they are combined with the split property for wedges. Although this may appear reasonable in view of the non-connectedness of O0 , our result also applies to the wedge representations which are only localizable in wedges provided left and right handed wedges are admitted. In Sec. 4 we will prove the minimality of the relative commutant for an inclusion of double cone algebras which, via a result of Driessler, implies Haag duality in all locally normal irreducible representations. In Sec. 5 the facts gathered in the preceding sections will be applied to the theory of quantum solitons thereby concluding our discussion of the representation theory of Haag-dual nets. Summing up the results obtained so far, the representation theory of such nets is essentially trivial. On the other hand, dispensing completely with a general theory of superselection sectors including composition of charges, braid statistics and quantum symmetry for massive theories is certainly not warranted in view of the host of more or less explicitly analyzed models exhibiting these phenomena. The only way to accommodate these models seems to be to relax the duality requirement by postulating only wedge duality. In Sec. 6 Roberts’ extension of localized representations to the dual net will be reconsidered and applied to the theories considered already in [?], namely fixpoint nets under an unbroken inner symmetry group. In this work we will not attempt to say anything concerning the quantum symmetry question. 2. Strong Additivity and the Time-Slice Axiom Until further notice we fix a vacuum representation π0 (which is always faithful) on a separable Hilbert space H0 and omit the symbol π0 (·), identifying A(O) ≡ π0 (A(O)). Whereas we may assume the algebras A(O), O ∈ K to be weakly closed, for more complicated regions X, in particular infinite ones like O0 , we carefully distinguish between the C ∗ -subalgebra A(X) ≡
[
A(O)
k·k
(2.1)
O∈K,O⊂X
of A ≡ π0 (A) and its ultraweak closure R(X) = A(X)00 . Definition 2.1. An inclusion A ⊂ B of von Neumann algebras is standard [?] if there is a vector Ω which is cyclic and separating for A, B, A0 ∧ B. Due to the Reeh–Schlieder property, the inclusion A(O1 ) ⊂ A(O2 ) (R(W1 ) ⊂ R(W2 )) is standard whenever O1 ⊂⊂ O2 (W1 ⊂⊂ W2 ), i.e. the closure of O1 is
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
...
1151
contained in the interior of O2 . (W1 ⊂⊂ W2 is equivalent to the existence of a double cone O such that W1 ∪ W20 = O0 .) Definition 2.2. An inclusion A ⊂ B of von Neumann algebras is split [?], if there exists a type-I factor N such that A ⊂ N ⊂ B. A net of algebras satisfies the split property (for double cones) [?] if the inclusion A(O1 ) ⊂ A(O2 ) is split whenever O1 ⊂⊂ O2 . The importance of these definitions derives from the following result [?, ?]: Lemma 2.3. Let A ⊂ B be a standard inclusion. Then the following are equivalent: (i) The inclusion A ⊂ B is split. (ii) The is a unitary Y such that Y ab0 Y ∗ = a ⊗ b0 , a ∈ A, b0 ∈ B 0 . Remarks. 1. The implication (ii)⇒(i) is trivial, an interpolating type-I factor being given by N = Y ∗ (B(H0 ) ⊗ 1)Y . 2. The natural spatial isomorphism A(O1 ) ∨ A(O2 )0 ∼ = A(O1 ) ⊗ A(O2 )0 implied by the split property whenever O1 ⊂⊂ O2 clearly restricts to A(O1 ) ∨ R(O20 ) ∼ = A(O1 ) ⊗ R(O20 ) .
(2.2)
As an important consequence, every pair of normal states φ1 ∈ A(O1 )∗ , φ2 ∈ R(O20 )∗ extends to a normal state φ ∈ (A(O1 ) ∨ R(O20 ))∗ . Physically this amounts to a form of statistical independence between the regions O1 and O20 . 3. We emphasize that in the case where Haag duality fails (A(O) ( A(O0 )0 ), requiring (??) whenever O1 ⊂⊂ O2 defines a weaker notion of split property since one can conclude only the existence of a type-I factor N such that A(O1 ) ⊂ N ⊂ A(O20 )0 = Ad (O2 ). In 1 + 1 dimensions (and only there, cf. [?, p. 292]) the split property may be strengthened by extending it to wedge regions. In this paper we will examine the implications of the split property for wedges (SPW). The power of this assumption in combination with Haag duality derives from the fact that one obtains strong results on the relation between the algebras of double cones and of wedges. Some of these have already been explored in [?], where, e.g., it has been shown that the local algebras associated with double cones are factors. We recall some terminology introduced in [?]: the left and right spacelike complements of O are denoted by O O O 0 O 0 and WRR , respectively. Furthermore, defining WLO = WRR and WRO = WLL WLL we have O = WLO ∩ WRO . Before we turn to the main subject of this section, we remark on the relation between the two notions of Haag duality which are of relevance for this paper. In [?], as apparently in a large part of the literature, it was implicitly assumed that Haag duality for double cones implies duality for wedges, i.e. R(W )0 = R(W 0 ) ∀ W ∈ W ,
(2.3)
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where W is the set of all wedge regions. Whereas there seems to be no general proof of this claim, for theories in 1 + 1 dimensions satisfying the SPW we can give a straightforward argument, thereby also closing the gap in [?]. In view of Remark 3 after Lemma ?? the following definition of the split property for wedges is slightly weaker than the obvious modification of Definition ??, but seems more natural from a physical point of view (cf. Remark 2): Definition 2.4. A net of algebras satisfies the split property for wedges if the map x ⊗ y 7→ xy, x ∈ R(W1 ), y ∈ R(W2 ) extends to an isomorphism between R(W1 ) ⊗ R(W2 ) and R(W1 ) ∨ R(W2 ) whenever W1 ⊂⊂ W20 . By standardness this isomorphism is automatically spatial in the sense of Lemma ?? (ii). In the case O O , W2 = WRR the canonical implementer [?] will be denoted Y O . where W1 = WLL Proposition 2.5. Let A(O) be a net of local algebras in 1 + 1 dimensions, satisfying Haag duality (for double cones) and the SPW. Then A satisfies wedge duality and the inclusion R(W1 ) ⊂ R(W2 ) is split whenever W1 ⊂⊂ W2 . Proof. Appealing to the definition (??), duality for double cones is clearly equivalent to O O ) ∨ R(WRR ) ∀ O ∈ K. A(O)0 = R(WLL
(2.4)
Given a right wedge W , let Oi , i ∈ N be an increasing sequence of double cones all of which have the same left corner as W and satisfying ∪i Oi = W . Then we clearly W have R(W ) = i A(Oi ) and ^ ^ Oi R(W )0 = R(W 0 ) ∨ R(WRR A(Oi )0 = ) . (2.5) i
i
O1 O1 Using the unitary equivalence Y O1 R(W 0 ) ∨ R(WRR ) Y O1 ∗ = R(W 0 ) ⊗ R(WRR ), the right-hand side of (??) is equivalent to O^ ^ R(W 0 ) ⊗ R(WROi ) = R(W 0 ) R(WROi ) = R(W 0 ) ⊗ C1 , (2.6) i
i
∧i R(WROi )
where we have used the consequence = C1 of irreducibility. This clearly proves R(W )0 = R(W 0 ). The final claim follows from Lemma ??. Now we are prepared for the discussion of additivity properties, starting with the easy Lemma 2.6. O ) ∨ A(O) = R(WLO ) , R(WLL
(2.7)
O ) ∨ A(O) = R(WRO ) . R(WRR
(2.8)
O O Remark. Equivalently, the inclusions R(WLL ) ⊂ R(WLO ), R(WRR ) ⊂ R(WRO ) are normal.
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
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O O O O Proof. Under the unitary equivalence R(WLL )∨R(WRR )∼ )⊗R(WRR ) = R(WLL O O O O ∼ O ∼ we have R(WLL ) = R(WLL ) ⊗ 1 and A(O) = R(WL ) ∩ R(WR ) = R(WR ) ⊗ O O ) ∨ A(O) ∼ ) ∨ R(WRO )) ⊗ R(WLO ). Due to wedge R(WLO ). Thus R(WLL = (R(WLL duality and factoriality of the wedge algebras this equals B(H0 )⊗R(WLO ) ∼ = R(WLO ). We emphasize that all above equivalences are established by the same unitary transformation. The second equation is proved in the same way.
Remark. The proof of factoriality of wedge algebras in [?] relies, besides the usual net properties, on the spectrum condition and on the Reeh–Schlieder theorem. This is the only place where positivity of the energy and weak additivity enter into our analysis. ˜ are spacelike Consider now the situation depicted in Fig. 1. In particular, O, O separated double cones the closures of which share one point. Such double cones will be called adjacent.
@@ @@ @@ @@W @@ @@ W @ O @ O˜ @ @@ @@ @@ @ @ @ O L
O LL
Fig. 1. Double cones sharing one point.
ˆ = sup(O, O) ˜ be the smallest double cone containing O, O. ˜ Lemma 2.7. Let O Then ˜ = A(O) ˆ . A(O) ∨ A(O)
(2.9) ˜
ˆ = W O ∩ W O . Under the unitary Proof. In the situation of Fig. 1 we have O L R ∼ ˜ ˜ since O ˜ ⊂ W O . Thus equivalence considered above we have A(O) = 1 ⊗ A(O) RR ˜ O O O O ∼ ˜ ˜ A(O)∨A(O) = R(WR )⊗(R(WL )∨A(O)). But now WL = WLL leads to R(WLO )∨ ˜ ˜ ∼ ˜ = R(W O˜ ) via the preceding lemma. Thus A(O) ∨ A(O) A(O) = R(WRO ) ⊗ R(WLO ) L ˜ O O ˆ which in turn is unitarily equivalent to R(W ) ∧ R(W ) = A(O). R
L
Remark. In analogy to chiral conformal field theory we denote this property strong additivity. With these lemmas it is clear that the quantum field theories under consideration are n-regular in the sense of the following definition for all n ≥ 2. Definition 2.8. A quantum field theory is n-regular if R(W1 ) ∨ A(O1 ) ∨ · · · ∨ A(On−2 ) ∨ R(W2 ) = B(H0 ) ,
(2.10)
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whenever Oi , i = 1, . . . , n − 2 are mutually spacelike double cones such that the sets Oi ∩ Oi+1 , i = 1, . . . , n − 3 each contain one point and where the wedges W1 , W2 are such that !0 n−2 [ Oi . (2.11) W1 ∪ W2 = i=1
Corollary 2.9. A quantum field theory in 1 + 1 dimensions satisfying Haag duality and the SPW fulfills the (von Neumann version of the) time-slice axiom, i.e. R(S) = B(H0 ) ,
(2.12)
whenever S = {x ∈ R2 | x · η ∈ (a, b)} where η ∈ R2 is timelike and a < b. Proof. The time-slice S contains an infinite string Oi , i ∈ Z of mutually spacelike double cones as above. Thus the von Neumann algebra generated by all these double cones contains each A(O), O ∈ K from which the claim follows by irreducibility. Remarks. 1. We wish to emphasize that this statement on von Neumann algebras is weaker than the C ∗ -version of the time-slice axiom, which postulates that the C ∗ -algebra A(S) generated by the algebras A(O), O ⊂ S equals the quasilocal algebra A. We follow the arguments in [?, Sec. III.3] to the effect that this stronger assumption should be avoided. 2. It is interesting to confront the above result with the investigations concerning the time-slice property [?] and the split property [?, Theorem 10.2] in the context of generalized free fields (in 3 + 1 dimensions). In the cited works it was proved that generalized free fields possess the time-slice property iff (roughly) the spectral measure vanishes sufficiently fast at infinity. On the other hand, the split property imposes strong restrictions on the spectral measure, in particular it must be atomic without an accumulation point at a finite mass. The split property (for double cones) is, however, neither necessary nor sufficient for the time-slice property. 3. Absence of Localized Charges Whereas the results obtained so far are intuitively plausible, we will now prove a no-go theorem which shows that the combination of Haag duality and the SPW is extremely strong. Theorem 3.1. Let O 7→ A(O) be a net of observables satisfying Haag duality and the split property for wedges. Let π be a representation of the quasilocal algebra A which satisfies π A(W ) ∼ = π0 A(W )
∀ W ∈W,
(3.1)
where W is the set of all wedges (left and right handed). Then π is equivalent to an at most countable direct sum of representations which are unitarily equivalent
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
to π0 : π=
M
πi ∼ = π0 .
πi ,
...
1155
(3.2)
i∈I
In particular, if π is irreducible it is unitarily equivalent to π0 . Remark. A fortiori, this applies to DHR representations (??).
@
@@
@ @@O @@W @ @W @ @@ W @@ @@ @ @ @ O2
@@
1
1
2
Fig. 2. A split inclusion of wedges.
Proof. Consider the geometry depicted in Fig. 2. If π is a representation satisfying (??) then there is a unitary V : Hπ → H0 such that, setting ρ = V π(·)V ∗ , we have ρ(A) = A if A ∈ A(W 0 ). Due to normality on wedges and wedge duality, ρ continues to normal endomorphisms of R(W ), R(W1 ). By the split property there are type-I factors M1 , M2 such that R(W ) ⊂ M1 ⊂ R(W1 ) ⊂ M2 ⊂ R(W2 ) .
(3.3)
Let x ∈ M1 ⊂ R(W1 ). Then ρ(x) ∈ R(W1 ) ⊂ M2 . Furthermore, ρ acts trivially on M10 ∩ R(W2 ) ⊂ R(W )0 ∩ R(W2 ) = A(O2 ), where we have used Haag duality. Thus ρ maps M1 into M2 ∩ (M10 ∩ R(W2 ))0 ⊂ M2 ∩ (M10 ∩ M2 )0 = M1 , the last identity following from M1 , M2 being type-I factors. By [?, Corollary 3.8] every endomorphism of a type-I factor is inner, i.e. there is a (possibly infinite) family of P isometries Vi ∈ M1 , i ∈ I with Vi∗ Vj = δi,j , i∈I Vi Vi∗ = 1 such that ρ(A) = η(A) where η(A) ≡
X
∀ A ∈ M1 ,
Vi A Vi∗ , A ∈ B(H0 ) .
(3.4)
(3.5)
i∈I
(The sum over I is understood in the strong sense.) Now, ρ and thus η act trivially on M1 ∩ R(W )0 ⊂ R(W1 ) ∩ R(W )0 = A(O1 ), which implies Vi ∈ M1 ∩ (M1 ∩ R(W )0 )0 = R(W ) .
(3.6)
ˆ ⊃⊃ W Thanks to Lemma ?? we know that for every wedge W ˆ ) = R(W ) ∨ A(O) , R(W
(3.7)
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ˆ ∩ W 0 . From the fact that ρ acts trivially on A(W 0 ) it follows that where O = W ˆ ) which (??) is true also for A ∈ A(O). By assumption, ρ is normal also on A(W ˆ ). As this holds for every W ˆ ⊃⊃ W , we conclude that leads to (??) on A(W X V ∗ Vi A Vi∗ V ∀ A ∈ A . (3.8) π(A) = i∈I
Remarks. 1. The main idea of the proof is taken from [?, Proposition 2.3]. 2. The above result may seem inconvenient as it trivializes the DHR/FRS superselection theory [?, ?, ?] for a large class of massive quantum field theories in 1 + 1 dimensions. It is not so clear what this means with respect to field theoretical models since little is known about Haag duality in nontrivial models. 3. Conformal quantum field theories possessing no representations besides the vacuum representation, or “holomorphic” theories, have been the starting point for an analysis of “orbifold” theories in [?]. In [?], which was motivated by the desire to obtain a rigorous understanding of orbifold theories in the framework of massive two-dimensional theories, the present author postulated the split property for wedges and claimed it to be weaker than the requirement of absence of nontrivial representations. Whereas this claim is disproved by Theorem ??, as far as localized (DHR or wedge) representations of Haag dual theories are concerned, none of the results of [?] is invalidated or rendered obsolete. 4. Haag Duality in Locally Normal Representations A further crucial consequence of the split property for wedges is observed in the following: Proposition 4.1. Let O 7→ A(O) be a net satisfying Haag duality (for double ˆ we have cones) and the split property for wedges. Then for every pair O ⊂⊂ O ˆ ∧ A(O)0 = A(OL ) ∨ A(OR ) , A(O)
(4.1)
where OL , OR are as in Fig. 3.
@@
@@ @@ @@ @W@ @@ @@ @W@ W @@O @@O @@O @@W @@ @@ @@ @@ ˆ O
O LL
ˆ O LL
O RR
L
R
Fig. 3. Relative commutant of double cones.
ˆ O RR
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Proof. By the split property for wedges there is a unitary operator Y O : H0 → O O O O ) ∨ R(WRR ) = Y O∗ (R(WLL ) ⊗ R(WRR ))Y O . More H0 ⊗ H0 such that R(WLL specifically, Y O xy Y O∗ = x ⊗ y
O O ∀ x ∈ R(WLL ), y ∈ R(WRR ).
(4.2)
O O O O ˆ 0= By Haag duality A(O)0 = R(WLL ) ∨ R(WRR )∼ ) ⊗ R(WRR ) and A(O) = R(WLL ˆ ˆ ˆ ˆ O O O O O 0 ∼ ˆ R(WLL ) ∨ R(WRR ). Now R(WLL/RR ) ⊂ R(WLL/RR ) implies A(O) = R(WLL ) ⊗ ˆ R(W O ) under the same equivalence ∼ = provided by Y O , and thus RR
ˆ ˆ ˆ ˆ O O ˆ ∼ ) ⊗ R(WRR ))0 = R(WRO ) ⊗ R(WLO ) , A(O) = (R(WLL
(4.3)
where we have used wedge duality and the commutation theorem for tensor products. Now we can compute the relative commutant as follows: ˆ ˆ O O ˆ ∧ A(O)0 ∼ A(O) ) ⊗ R(WRR )) = (R(WRO ) ⊗ R(WLO )) ∧ (R(WLL ˆ
ˆ
O O )) ⊗ (R(WLO ) ∧ R(WRR )) = (R(WRO ) ∧ R(WLL
= A(OL ) ⊗ A(OR ) ∼ = A(OL ) ∨ A(OR ) .
(4.4) ˆ
O We have used Haag duality in the form R(WRO ) ∧ R(WLL ) = A(OL ) and similarly for A(OR ).
Remarks. 1. Readers having qualms about the above computation of the intersection of tensor products are referred to [?, Corollary 5.10], which also provides the justification for the arguments in Sec. 2. 2. Recalling that R(O) = A(O) and that the algebras of regions other than double cones are defined by additivity, (??) can be restated as follows: ˆ ∩ O0 ) . ˆ ∩ R(O)0 = R(O R(O)
(4.5)
In conjunction with the assumed properties of isotony, locality and Haag duality for double cones (??) entails that the map O 7→ R(O) is a homomorphism of orthocomplemented lattices as proposed in [?, Sec. III.4.2]. While the discussion in [?, Sec. III.4.2] can be criticized, the class of models considered in this paper provides examples where the above lattice homomorphism is in fact realized. The proposition should contribute to the understanding of Theorem ?? as far as DHR representations are concerned. In fact, it already implies the absence of DHR sectors as can be shown by an application of the triviality criterion for local 1-cohomologies [?] given in [?], see also [?]. Sketch of proof. Let z ∈ Z 1 (A) be the local 1-cocycle associated according to [?, ?] with a representation π satisfying the DHR criterion. Due to Proposition ?? it satisfies z(b) ∈ A(|∂0 b|) ∨ A(|∂1 b|) for every b ∈ Σ1 such that |∂0 b| ⊂⊂ |∂1 b|0 . Thus
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the arguments in the proof of [?, Theorem 3.5] are applicable despite the fact that we are working in 1+1 dimensions. We thereby see that there are unique Hilbert spaces H(O) ⊂ A(O), O ∈ Σ0 ≡ K of support 1 such that z(b)H(∂1 b) = H(∂0 b) ∀ b ∈ Σ1 . Each of these Hilbert spaces implements an endomorphism ρO of A such that ρO ∼ = π. This implies that ρ is either reducible or an inner automorphism. Remark. This argument needs the split property for double cones. It is not completely trivial that the latter follows from the split property for wedges. It is clear that the latter implies unitary equivalence of A(O1 ) ∨ A(O2 ) and A(O1 ) ⊗ A(O2 ) if O1 , O2 are double cones separated by a finite spacelike distance. The split ˆ 0 property for double cones requires more, namely unitary equivalence of A(O)∨A(O) 0 ˆ ˆ and A(O)⊗A(O) whenever O ⊂⊂ O, which is equivalent to the existence of a type-I ˆ factor N such that A(O) ⊂ N ⊂ A(O). Lemma 4.2. Let A be a local net satisfying Haag duality and the split property for wedges. Then the split property for double cones holds. Proof. Using the notation of the preceding proof we have A(O) ∼ = R(WRO ) ⊗ R(WLO ) ,
(4.6)
ˆ ˆ ˆ ∼ A(O) = R(WRO ) ⊗ R(WLO ) .
(4.7) ˆ
By the SPW there are type-I factors NL , NR such that R(WLO ) ⊂ NL ⊂ R(WLO ) ˆ and R(WRO ) ⊂ NR ⊂ R(WRO ). Thus Y O∗ (NR ⊗ NL )Y O is a type-I factor sitting ˆ between A(O) and A(O). Having disproved the existence of nontrivial representations localized in double cones or wedges, we will now prove a result which concerns a considerably larger class of representations. Theorem 4.3. Let O 7→ A(O) be a net of observables satisfying Haag duality and the SPW. Then every irreducible, locally normal representation of the quasilocal algebra A fulfills Haag duality. Proof. We will show that our assumptions imply those of [?, Theorem 1]. A satisfies the split property for double cones (called “funnel property” in [?, ?]) by Lemma ??, whereas we also assume condition (1) of [?, Theorem 1] (Haag duality and irreducibility). Condition (3), which concerns relative commutants A(O2 ) ∩ A(O1 )0 , O2 ⊃⊃ O1 in the vacuum representation, is an immediate consequence of Proposition ?? (we may even take O = O1 , O2 = O3 ). Finally, Lemma ?? implies ˆ 0 ∨ A(OL ) ∨ A(OR ) , A(O)0 = A(O)
(4.8)
where we again use the notation of Fig. 3. This is more than required by Driessler’s condition (2). Now [?, Theorem 1] applies and we are done.
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Remarks. 1. In [?] a slightly simplified version of [?, Theorem 1] is given which dispenses with condition (2) at the price of a stronger form of condition (3). This condition is still (more than) fulfilled by our class of theories. 2. Observing that soliton representations are locally normal with respect to both asymptotic vacua [?, ?], we conclude at once that Haag duality holds for every irreducible soliton sector where at least one of the vacua satisfies Haag duality and the SPW. Consequences of this fact will be explored in the next section. We remark without going into details that our results are also of relevance for the construction of soliton sectors with prescribed asymptotic vacua in [?]. 5. Applications to the Theory of Quantum Solitons In [?] it has been shown that every factorial massive one-particle representation (massive one-particle representation) in ≥ 2 + 1 dimensions is a multiple of an irreducible representation which is localizable in every spacelike cone. (Here, massive one-particle representation means that the lower bound of the energy-momentum spectrum consists of a hyperboloid of mass m > 0 which is separated from the rest of the spectrum by a mass gap.) In 1 + 1 dimensions one is led to irreducible soliton sectors [?] which we will now reconsider in the light of Theorems ?? and ??. In this section, where we are concerned with inequivalent vacuum representations, we will consider a QFT to be defined by a net of abstract C ∗ -algebras instead of the algebras in a concrete representation. Given two vacuum representations π0L , π0R , a representation π is said to be a soliton representation of type (π0L , π0R ) if it is translation covariant and L/R A(WL/R ) , π A(WL/R ) ∼ = π0
(5.1)
where WL , WR are arbitrary left and right handed wedges, respectively. An obvious consequence of (??) is local normality of π0L , π0R with respect to each other. In order to formulate a useful theory of soliton representations [?] one must assume L/R to satisfy wedge duality. After giving a short review of the formalism in [?], π0 we will show in this section that considerably more can be said under the stronger assumption that one of the vacuum representations satisfies duality for double cones and the SPW. (Then the other vacuum is automatically Haag dual, too.) Let π0 be a vacuum representation and W ∈ W a wedge. Then by A(W )π0 we denote the W ∗ -completion of the C ∗ -algebra A(W ) with respect to the family of seminorms given by kAkT = |tr T π0 (A)| ,
(5.2)
where T runs through the set of all trace class operators in B(Hπ0 ). Furthermore, R we define extensions AL π0 , Aπ0 of the quasilocal algebra A by AL/R = π0
[ W ∈WL/R
A(W )π0
k·k
,
(5.3)
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where WL , WR are the sets of left and right wedges, respectively. Now, it has been demonstrated in [?] that, given a (π0L , π0R )-soliton representation π, there are to AR such that homomorphisms ρ from AR πR πL 0
0
π∼ = π0L ◦ ρ .
(5.4)
(Strictly speaking, π0L must be extended to AR , which is trivial since A(W )π0 is π0L 00 isomorphic to π0 (A(W )) .) The morphism ρ is localized in some right wedge W in the sense that ρ A(W 0 ) = id A(W 0 ) .
(5.5)
Provided that the vacua of two soliton representations π, π 0 “fit together” π0R ∼ = π00L 0 L 0R one can define a soliton representation π × π of type π0 , π0 via composition of the corresponding morphisms: π × π0 ∼ = π0L ◦ ρρ0 A .
(5.6)
Alternatively, the entire analysis may be done in terms of left localized morphisms to AL . As proved in [?], the unitary equivalence class of the composed η from AL π0L π0R representation depends neither on the use of left or right localization nor on the concrete choice of the morphisms. Whereas for soliton representations there is no analog to the theory of statistics [?, ?, ?], there is still a “dimension” ind(ρ) defined by ind(ρ) ≡ [A(W )π0L : ρ(A(W )π0R )] ,
(5.7)
where ρ is localized in the right wedge W and [M : N ] is the Jones index of the inclusion N ⊂ M . Proposition 5.1. Let π be an irreducible soliton representation such that at least one of the asymptotic vacua π0L , π0R satisfies Haag duality and the SPW. Then π and both vacua satisfy the SPW and duality for double cones and wedges. The associated soliton-morphism satisfies ind(ρ) = 1. Proof. By symmetry it suffices to consider the case where π0L satisfies HD + SPW. By Theorem ?? also the representations π and π0R satisfy Haag duality since they are locally normal w.r.t. to π0L . Let now W1 ⊂⊂ W2 be left wedges. By Proposition ??, wedge-duality holds for π0L and π0L (A(W1 ))00 ⊂ π0L (A(W2 ))00 is split. Since π0L (A(W2 ))00 is unitarily equivalent to π(A(W2 ))00 , also π(A(W1 ))00 ⊂ π(A(W2 ))00 splits. A fortiori, π satisfies the SPW in the sense of Definition ?? and thus wedge duality by Proposition ??. By a similar argument the SPW is carried over to π0R . Now, for a right wedge W we have π0L ◦ ρ(A(W ))− = π0L ◦ ρ(A(W 0 ))0 = π0L (A(W 0 ))0 = π0L (A(W ))− .
(5.8)
By ultraweak continuity on A(W ) of π0L and of ρ this implies ρ(A(W )π0R ) = A(W )π0L , whence the claim.
(5.9)
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This result rules out soliton sectors with infinite index so that [?, Theorem 3.2] applies and yields equivalence of the various possibilities of constructing antisoliton sectors considered in [?]. In particular the antisoliton sector is uniquely defined up to unitary equivalence. Now we can formulate our main result concerning soliton representations. Theorem 5.2. Let π0L , π0R be vacuum representations, at least one of which satisfies Haag duality and the SPW. Then all soliton representations of type (π0L , π0R ) are unitarily equivalent. Remark. Equivalently, up to unitary equivalence, a soliton representation is completely characterized by the pair of asymptotic vacua. Proof. Let π, π 0 be irreducible soliton representations of types (π0 , π00 ) and respectively. They may be composed, giving rise to a soliton representation of type (π0 , π0 ) (or (π00 , π00 )). This representation is irreducible since the morphisms ρ, ρ0 must be isomorphisms by the proposition. Now, π × π 0 is unitarily equivalent to π0 on left and right handed wedges, which by Theorem ?? and irreducibility implies π × π 0 ∼ = π0 . We conclude that every (π00 , π0 )-soliton is an antisoliton of 0 every (π0 , π0 )-soliton. This implies the statement of the theorem since for every soliton representation with finite index there is a corresponding antisoliton which is unique up to unitary equivalence.
(π00 , π0 ),
Remark. The above proof relies on the absence of nontrivial representations which are localizable in wedges. Knowing just that DHR sectors do not exist, as follows already from Proposition ??, is not enough. 6. Solitons and DHR Representations of Non-Haag Dual Nets 6.1. Introduction and an instructive example We have observed that the theory of localized representations of Haag-dual nets of observables which satisfy the SPW is trivial. There are, however, quantum field theories in 1 + 1 dimensions where the net of algebras which is most naturally considered as the net of observables does not fulfill Haag duality in the strong form (??). As mentioned in the introduction, this is the case if the observables are defined as the fixpoints under a global symmetry group of a field net which satisfies (twisted) duality and the SPW. The weaker property of wedge duality (??) remains, however. This property is also known to hold automatically whenever the local algebras arise from a Wightman field theory [?]. However, for the analysis in [?, ?, ?] as well as Sec. 4 above one needs full Haag duality. Therefore it is of relevance that, starting from a net of observables satisfying only (??), one can define a larger but still local net Ad (O) ≡ R(WLO ) ∧ R(WRO )
(6.1)
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¨ M. MUGER
which satisfies Haag duality, whence the name dual net. Here WLO , WRO are wedges such that WLO ∩ WRO = O and duality is seen to follow from the fact that the wedge algebras R(W ), W ∈ W are the same for the nets A, Ad . (For observables arising as group fixpoints the dual net has been computed explicitly in [?].) It is known [?, ?] that in ≥ 2+1 dimensions representations π satisfying the DHR criterion (??) extend uniquely to DHR representations π ˆ of the (appropriately defined) dual net. Furthermore, the categories of DHR representations of A and Ad , respectively, and their intertwiners are isomorphic. Thus, instead of A one may as well study Ad to which the usual methods are applicable. (The original net is needed only to satisfy essential duality, which is implied by wedge duality.) In 1 + 1 dimensions things are more complicated. As shown in [?] there are in general two different extensions ˆ R . They coincide iff one (thus both) of them is a DHR representation. Even π ˆL, π before defining precisely these extensions we can state the following consequence of Theorem ??. Proposition 6.1. Let A be a net of observables satisfying wedge duality and the SPW. Let π be an irreducible DHR or wedge representation of A which is not unitarily equivalent to the defining (vacuum) representation. Then there is no extension π ˆ to the dual net Ad which is still localized in the DHR or wedge sense. Proof. Assume π to be the restriction to A of a wedge-localized representation π ˆ of Ad . As the latter is known to be either reducible or unitarily equivalent to π0 , the same holds for π. This is a contradiction. The fact that the extension of a localized representation of A to the dual net Ad cannot be localized, too, partially undermines the original motivation for considering these extensions. Nevertheless, one may entertain the hope that there is something to be learnt which is useful for a model-independent analysis of the phenomena observed in models. Before we turn to the general examination of the extensions ˆ R we consider the most instructive example. π ˆL, π It is provided by the fixpoint net under an unbroken global symmetry group of a field net as studied in [?]. We briefly recall the framework. Let O 7→ F (O) be a (for simplicity) bosonic, i.e. local, net of von Neumann algebras acting on the Hilbert space H and satisfying Haag duality and the SPW. On H there are commuting strongly continuous representations of the Poincar´e group and of a group G of inner symmetries. Both groups leave the vacuum Ω invariant. Defining the fixpoint net A(O) = F (O)G = F (O) ∩ U (G)0
(6.2)
A(O) = A(O) H0
(6.3)
and its restriction
to the vacuum sector (= subspace of G-invariant vectors) we consider A(O) as the observables. It is well-known that the net A satisfies only wedge duality. Nevertheless, one very important result of [?] remains true, namely that the restrictions of
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ˆ interpreted A to the charged sectors Hχ which are labeled by the characters χ ∈ G, ∗ as representations of the abstract C -algebra A, satisfy the DHR criterion and are connected to the vacuum by charged fields, i.e. the representation of A in Hχ is of the form πχ (A) = A Hχ ∼ = πχO (A) = ψ A ψ ∗ H0 ,
(6.4)
where ψ ∈ F (O) and αg (ψ) = χ(g)ψ. It was shown in [?, Theorem 3.10] that the dual net in the vacuum sector is given by Ad (O) = AˆL (O) H0 = AˆR (O) H0 ,
(6.5)
AˆL/R (O) = FˆL/R (O)G = FˆL/R (O) ∩ U (G)0 .
(6.6)
where
Here the nonlocal nets FˆL/R (O) are obtained by adjoining to F (O) the disorder operators [?] ULO (G) or URO (G), respectively, which satisfy O O Ad ULO (g) F(WLL ) = αg = Ad URO (g) F (WRR ), O O Ad ULO (g) F(WRR ) = id = Ad URO (g) F (WLL )
(6.7)
and transform covariantly under the global symmetry: O O (h) U (g)∗ = UL/R (ghg −1 ) . U (g) UL/R
(6.8)
For the moment we restrict to the case of abelian groups G. The disorder O (G)00 . On the C ∗ operators commuting with G, AˆL/R (O) is simply A(O) ∨ UL/R ˆ which acts trivially on algebras AˆL and AˆR there is an action of the dual group G A and via O O (g)) = χ(g) UL/R (g) α ˆ χ (UL/R
∀O∈K
(6.9)
on the disorder operators. Since this action commutes with the Poincar´e group and ˆ χ 6= ω0 ∀ χ 6= eGˆ ) it gives rise to inequivalent since it is spontaneously broken (ω0 ◦ α vacuum states on Aˆ via ˆχ . ωχ = ω0 ◦ α
(6.10)
ˆχ,R of πχ to the dual net Ad can now defined using the The extensions π ˆχ,L , π right-hand side of (??) by allowing A to be in AˆL or AˆR . As is obvious from the commutation relation (??) between fields and disorder operators, the extenπχ,R ) is nothing but a soliton sector interpolating between the vacua ω0 sion π ˆχ,L (ˆ and ωχ−1 (ωχ and ω0 ). The moral is that the net Ad , while not having nontrivial localized representations by Theorem ??, admits soliton representations. Furthermore, with respect to Ad , the charged fields ψχ are creation operators for
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solitons since they intertwine the representations of Ad on H0 and Hχ . Due to ULO (g) URO (g) = U (g) and U (g) Hχ = χ(g)1 we have ULO (g) Hχ = χ(g) URO (g −1 ) Hχ ,
(6.11)
so that the algebras AˆL/R (O) Hχ are independent of whether we use the left or right localized disorder operators. In particular, in the vacuum sector ULO (g) and URO (g −1 ) coincide, but due to the different localization properties it is relevant whether ULO (g), considered as an element of Ad , is represented on Hχ by ULO (g) or by χ(g) URO (g −1 ). This reasoning shows that the two possibilities for extending a localized representation of a general non-dual net to a representation of the dual net correspond in the fixpoint situation at hand to the choice between the nets AˆL and AˆR arising from the field extensions FˆL and FˆR . 6.2. General Analysis We begin by first assuming only that π is localizable in wedges. Let O be a double cone and let WL , WR be left and right handed wedges, respectively, containing O. By assumption the restriction of π to A(WL ), A(WR ) is unitarily equivalent to π0 . Choose unitary implementers UL , UR such that Ad UL A(WL ) = π A(WL ) , Ad UR A(WR ) = π A(WR ) .
(6.12)
ˆ R are defined for A ∈ Ad (O) by Then π ˆL, π π ˆ L (A) = UL A UL∗ , π ˆ R (A) = UR A UR∗ .
(6.13)
Independence of these definitions of the choice of WL , WR and the implementers UL , UR follows straightforwardly from wedge duality. We state some immediate consequences of this definition. ˆ R are irreducible, locally normal representations of Proposition 6.2. π ˆL , π ˆL, π ˆ R are normal on left and right handed wedges, Ad and satisfy Haag duality. π respectively. Proof. Irreducibility is a trivial consequence of the assumed irreducibility of π whereas local normality is obvious from the definition (??). Thus, Theorem ?? applies and yields Haag duality in both representations. Normality of, say, πˆL on left handed wedges W follows from the fact that we may use the same auxiliary wedge WL ⊃ W and implementer UL for all double cones O ⊂ W . Clearly, the extensions π ˆL, π ˆ R cannot be normal w.r.t. π0 on right and left wedges, respectively, for otherwise Theorem ?? would imply unitary equivalence to π0 . In general, we can only conclude localizability in the following weak sense. Given
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an arbitrary left handed wedge W , π ˆ L is equivalent to a representation ρ on H0 such that ρ(A) = A ∀ A ∈ A(W ). Furthermore, by duality ρ is an isomorphism of A(W 0 ) onto a weakly dense subalgebra of R(W 0 ) which is only continuous in the norm. In favorable cases like the one considered above this is a local symmetry, acting as an automorphism of A(W 0 ). But we will see shortly that there are perfectly non-pathological situations where the extensions are not of this particularly nice type. In complete generality, the best one can hope for is normality with respect to another vacuum representation π00 . In particular, this is automatically the case if π is a massive one-particle representation [?] which we did not assume so far. If the representation π satisfies the DHR criterion, i.e. is localizable in double cones, we can obtain stronger results concerning the localization properties of the ˆR . By the criterion, there are unitary operators extended representations π ˆL , π O X : Hπ → H0 such that π O (A) ≡ X O π(A) X O∗ = A ∀ A ∈ A(O0 ) .
(6.14)
(By wedge duality, X O is unique up to left multiplication by operators in Ad (O).) Considering the representations O = X Oπ ˆL/R X O∗ π ˆL/R
(6.15)
on the vacuum Hilbert space H0 , it is easy to verify that O O O Ad (WLL ) = id Ad (WLL ), π ˆL
(6.16)
O O O Ad (WRR ) = id Ad (WRR ). π ˆR
(6.17)
O ˜ , the other extension behaving similarly. If A ∈ A(O) We restrict our attention to π ˆL ∗ ˜ Therefore then π ˆL (A) = X Or A X Or whenever Or > O. ∗
O (A) = X O X Or A X Or X O∗ , π ˆL
(6.18)
∗
where the unitary X O X Or intertwines π O and π Or . Associating with every pair (O1 , O2 ) two other double cones by ˆ = sup(O1 , O2 ) , O
(6.19)
ˆ ∩ O10 ∩ O20 O0 = O
(6.20)
(O0 may be empty) and defining ˆ ∩ A(O0 )0 , C(O1 , O2 ) = Ad (O)
(6.21)
we can conclude by wedge duality that ∗
X O X Or ∈ C(O, Or ) .
(6.22)
O ˜ Or )) which already (A) as given by (??) is contained in Ad (sup(O, O, Thus π ˆL O d shows that π ˆL maps the quasilocal algebra A into itself (this does not follow if
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π is only localizable in wedges). Since the double cone Or > O may be chosen arbitrarily small and appealing to outer regularity of the dual net Ad we even have ˜ and thus finally π ˆ O (A) ∈ Ad (sup(O, O)) L
O ˜ ⊂ C(O, O) ˜ . π ˆL (Ad (O))
(6.23)
This result has two important consequences. Firstly, it implies that the representaO tion π ˆL maps the quasilocal algebra into itself: O π ˆL (Ad ) ⊂ Ad .
(6.24)
O O This fact is of relevance since it allows the extensions π ˆ1,L , π ˆ2,L of two DHR representations π1 , π2 to be composed in much the same way as the endomorphisms of A derived from DHR representations in the Haag dual case. In this respect, the extensions π ˆL/R are better behaved than completely general soliton representations as studied in [?]. O O (and π ˆR ), while The second consequence of (??) is that the representations π ˆL still mapping local algebras into local algebras, may deteriorate the localization. We will see below that this phenomenon is not just a theoretical possibility but really occurs. Whereas one might hope that one could build a DHR theory for nondual nets upon the endomorphism property of the extended representations, their ˆR seem to constitute weak localization properties and the inequivalence of π ˆL and π serious obstacles. It should be emphasized that the above considerations owe a lot to Roberts’ local 1-cohomology [?, ?, ?], but (??) seems to be new.
6.3. Fixpoint nets: non-abelian case We now generalize our analysis of fixpoint nets to non-abelian (finite) groups G, P ˜ where the outcome is less obvious a priori. Let Aˆ = g∈G Fg ULO (g) ∈ AˆL (O˜1 ) (Fg must satisfy the condition given in [?, Theorem 3.16]) and let ψi ∈ F (O2 ), where O1 ) be a multiplet of field operators transforming according O2 < O1 (i.e. O2 ⊂ WLL to a finite dimensional representation of G. Then ! X X X X O ∗ ∗ 1 ψi Fg UL (g) ψi = ψi αg (ψi ) Fg ULO1 (g) . (6.25) i
g∈G
g∈G
i
In contrast to the abelian case where ψαg (ψ ∗ ) is just a phase, Og ≡ is a nontrivial unitary operator X Og−1 = Og∗ = αg (ψi )ψi∗
P i
ψi αg (ψi∗ ) (6.26)
i
satisfying αk (Og ) = Okgk−1 .
(6.27)
In particular (??) is not contained in Ad (O1 ) which implies that the map Aˆ 7→ P O2 ˆ ˆ ∗ i ψi A ψi does not reduce to a local symmetry on AL (WRR ). Rather, we obtain
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ˆ and O0 as above we clearly see that a monomorphism into AˆL (WRO2 ). Defining O d ˆ (??) is contained in A (O). Furthermore, due to the relative locality of the net A with respect to Ad and F, (??) commutes with A(O0 ). Thus we obtain precisely the localization properties which were predicted by our general analysis above. We close this section with a discussion of the duality properties in the extended representations π ˆ . In the case of abelian groups G Haag duality holds in all charged sectors since these are all simple. Our abstract result in Theorem ?? to the effect that duality obtains in all locally normal irreducible representations of the dual net applies, of course, to the situation at hand. We conclude that Haag duality also holds for the non-simple sectors which by necessity occur for non-abelian groups G. Since this result is somewhat counterintuitive (which explains why it was overlooked in [?]) we verify it by the following direct calculation. Lemma 6.3. The commutants of the algebras AˆL (O) are given by O O ) ∨ FˆL (WRR ) AˆL (O)0 = AˆL (WLL
∀ O ∈ K.
(6.28)
Proof. For simplicity we assume F to be a local net for a moment. Then AˆL (O)0 = (FˆL (O) ∧ U (G)0 )0 = FˆL (O)0 ∨ U (G)00 = (FL (O) ∨ ULO (G)00 )0 ∨ U (G)00 = (FL (O)0 ∧ ULO (G)0 ) ∨ U (G)00 O O = ((FL (WLL ) ∨ FL (WRR )) ∧ ULO (G)0 ) ∨ U (G)00 O O = (FL (WLL ) ∧ ULO (G)0 ) ∨ FL (WRR ) ∨ U (G)00 O O = AˆL (WLL ) ∨ FˆL (WRR ).
(6.29)
The fourth line follows from the third using the split property. In the last step we have used the identities AˆL (WL ) = AL (WL ) and FL (WR )∨U (G)00 = FˆL (WR ) which hold for all left (right) handed wedges WL (WR ), cf. [?, Proposition 3.5]. Now, if F satisfies twisted duality, (2.23) of [?] leads to F (O) ∨ ULO (G)00 ∼ = F (WRO ) ∨ U (G)00 ⊗ O O 00 0 ∼ O O t F(WL ) and (F(O)∨UL (G) ) = A(WR )⊗F (WRR ) . Using this it is easy to verify that (??) is still true. Proposition 6.4. The net AˆL satisfies Haag duality in restriction to every invariant subspace of H on which AˆL acts irreducibly. Proof. We recall that the representation π of AˆL/R on H is of the form π = ⊕ξ∈Gˆ dξ πξ . Let thus P be an orthogonal projection onto a subspace Hξ ⊂ H on which AˆL acts as the irreducible representation πξ . Since P commutes with AL (O) O ) we have and AL (WLL O O ) ∨ FˆL (WRR )P P AˆL (O)0 P = P AˆL (WLL O O = AˆL (WLL ) ∨ (P FˆL (WRR )P) O O = P AˆL (WLL ) ∨ AˆL (WRR )P ,
(6.30)
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which implies O O ) ∨ AˆL (WRR ) Hξ . (AˆL (O) Hξ )0 = AˆL (WLL
(6.31)
This provides a concrete verification of Theorem ?? in a special, albeit important situation. 7. Conclusions and Outlook We have seen that the combination of Haag duality with the split property for wedges has remarkable unifying power. It implies factoriality of the double cone algebras, n-regularity for all n and irreducibility of time-slice algebras. As a consequence of the minimality of relative commutants of double cone algebras we obtain Haag duality in all irreducible, locally normal representations. The strongest result concerns the absence not only of locally generated superselection (DHR) sectors but also of charges localized in wedges. This in turn implies the uniqueness up to unitary equivalence of soliton sectors with prescribed asymptotic vacua. In the following we briefly relate these results to what is known in concrete models in 1 + 1 dimensions. (a) The free massive scalar field . Since this model is known to satisfy Haag duality and the SPW, Theorem ?? constitutes a high-brow proof of the well-known absence of local charges. Furthermore, there are no non-trivial soliton sectors, since the vacuum representation is unique [?]. Thus, the irreducible representations constructed in [?], which are inequivalent to the vacuum, must be rather pathological. In fact, they are equivalent to the (unique) vacuum only on left wedges. (b) P(φ)2 -models. These models have been shown [?] to satisfy Haag duality in all pure phases, but there is no proof of the SPW. Yet, the split property for double cones, the minimality of relative commutants and strong additivity, thus also the time slice property, follow immediately from the corresponding properties for the free field via the local Fock property. These facts already imply the nonexistence of DHR sectors and Haag duality in all irreducible locally normal sectors. All these consequences are compatible with the conjecture that the SPW holds. There seems, however, not to be a proof of the absence of wedge sectors. (c) The sine-Gordon/Thirring model. For this model neither Haag duality nor the SPW are known. In the case β 2 = 4π, however, for which the SG model corresponds to the free massive Dirac field, there seems to be no doubt that the net Aˆ constructed like in Sec. 6 from the free Dirac field is exactly the local net of the SG model. As shown in [?], also Aˆ satisfies Haag duality and the SPW. Since from the point of view of constructive QFT there is nothing special about β 2 = 4π one may hope that both properties hold for all β ∈ [0, 8π). In view of the results of this paper as well as of [?] it is highly desirable to clarify the status of the SPW in interacting massive models like (b) and (c) as well as that of Haag duality in case (c). (Also the Gross–Neveu model might be expected
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to satisfy both assumptions.) The most promising approach to this problem should be identifying conditions on a set of Wightman (or Schwinger) distributions which imply Haag duality and the SPW, respectively, for the net of algebras generated by the fields. For a first step in this direction see [?, Sec. IIIB]. Acknowledgments I am greatly indebted to K.-H. Rehren for his interest and encouragement, many helpful discussions and several critical readings of the manuscript. Conversations with K. Fredenhagen, J. Roberts, B. Schroer, and H.-W. Wiesbrock are gratefully acknowledged. The work was completed at the Erwin Schr¨odinger Institute, Vienna which kindly provided hospitality and financial support. Last but not least, I thank P. Croome for a very thorough proofreading of a preliminary version. References [1] C. D’Antoni and R. Longo, “Interpolation by type-I factors and the flip automorphism”, J. Funct. Anal. 51 (1983) 361–371. [2] J. J. Bisognano and E. H. Wichmann, “On the duality condition for a Hermitian scalar field”, J. Math. Phys. 16 (1975) 985–1007. [3] D. Buchholz, “Product states for local algebras”, Commun. Math. Phys. 36 (1974) 287–304. [4] D. Buchholz and K. Fredenhagen, “Locality and the structure of particle states”, Commun. Math. Phys. 84 (1982) 1–54. [5] D. Buchholz, G. Mack and I. Todorov, “The current algebra on the circle as a germ of local field theories”, Nucl. Phys. B (Proc. Suppl.) 5B (1988) 20–56. [6] D. Buchholz, “On quantum fields that generate local algebras”, J. Math. Phys. 31 (1990) 1839–1846. [7] R. Dijkgraaf, C. Vafa, E. Verlinde and H. Verlinde, “The operator algebra of orbifold models”, Commun. Math. Phys. 123 (1989) 485–527. [8] S. Doplicher, R. Haag and J. E. Roberts, “Fields, observables and gauge transformations I”, Commun. Math. Phys. 13 (1969) 1–23. [9] S. Doplicher, R. Haag and J. E. Roberts, “Local observables and particle statistics I”, Commun. Math. Phys. 23 (1971) 199–230. [10] S. Doplicher, R. Haag and J. E. Roberts, “Local observables and particle statistics II”, Commun. Math. Phys. 35 (1974) 49–85. [11] S. Doplicher, “Local aspects of superselection rules”, Commun. Math. Phys. 85 (1982) 73–86. [12] S. Doplicher and R. Longo, “Standard and split inclusions of von Neumann algebras”, Invent. Math. 75 (1984) 493–536. [13] S. Doplicher and J. E. Roberts, “Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics”, Commun. Math. Phys. 131 (1990) 51–107. [14] W. Driessler, “Comments on lightlike translations and applications in relativistic quantum field theory”, Commun. Math. Phys. 44 (1975) 133–141; “On the type of local algebras in quantum field theory”, Commun. Math. Phys. 53 (1977) 295–297. [15] W. Driessler, “Duality and absence of locally generated superselection sectors for CCR-type algebras”, Commun. Math. Phys. 70 (1979) 213–220. [16] K. Fredenhagen, “Generalizations of the theory of superselection sectors”, in [?]; “Superselection sectors in low dimensional quantum field theory”, J. Geom. Phys. 11 (1993) 337–348.
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