Mathematical Physics, Analysis and Geometry - Volume 4

Mathematical Physics, Analysis and Geometry 4: 1–36, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands...

Author: V. A. Marchenko | A. Boutet de Monvel | H. McKean (Editors)

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Mathematical Physics, Analysis and Geometry 4: 1–36, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

1

On the Law of Multiplication of Random Matrices VLADIMIR VASILCHUK Université Paris 7 Denis Diderot, Mathématiques, case 7012, Paris, France, Institute for Low Temperature Physics, 47 Lenin ave., 310164 Kharkov, Ukraine (Received: 22 February 2001; in revised form: 2 April 2001) Abstract. We recover Voiculescu’s results on multiplicative free convolutions of probability measures by different techniques which were already developed by Pastur and Vasilchuk for the law of addition of random matrices. Namely, we study the normalized eigenvalue counting measure of the product of two n × n unitary matrices and the measure of the product of three n × n Hermitian (or real symmetric) positive matrices rotated independently by random unitary (or orthogonal) Haar distributed matrices. We establish the convergence in probability as n → ∞ to a limiting nonrandom measure and obtain functional equations for the Herglotz and Stieltjes transforms of that limiting measure. Mathematics Subject Classifications (2000): 15A52, 60B99, 60F05. Key words: random matrices, counting measure, limit laws.

1. Introduction This paper deals with the eigenvalue distribution of products of n×n random matrices. We consider two models: (i) the product of two unitary (resp. orthogonal) matrices and (ii) the product of three Hermitian (or real symmetric) positive matrices. We study the eigenvalue distribution of these two ensembles in the limit n → ∞. Namely, we express the limiting normalized counting measure of eigenvalues of the product via the limits of the same counting measures of the corresponding factors. We assume that these exist and that factors are randomly rotated one with respect to another by a unitary (or orthogonal) random matrix uniformly distributed over the group U(n) (resp. O(n)). In this paper, under weaker assumptions, we obtain the analog of Voiculescu’s results concerning the free multiplicative convolutions of probability measures on the real axis and unit circle. They were studied within the context of free (noncommutative) probability theory introduced by Voiculescu at the beginning of the 90’s (see [2, 8, 10] for results and references). This theory deals with free random variables (operators in von Neumann algebras) that can be modeled by unitary (orthogonal) invariant random matrices [5, 6]. The notion of S-transform introduced in this theory allows one to generalize the functional equations for transforms of limiting counting measures of certain multiplicative unitary invariant models first proposed and studied by Marchenko and Pastur [4].

2

VLADIMIR VASILCHUK

Motivation for studying multiplicative random matrix ensembles is given by the fact that they appear in some physics studies (see, e.g., [3]). We use a simple method of deriving the functional equations for limiting eigenvalue distributions. It is a natural extension of the method proposed in [5] to study an additive analog of our ensembles. The basic idea is the same as in [4]: to study not the counting measure itself but rather some integral transforms that are generating functions of the moments of that measure. We derive functional relations for these transforms using the resolvent identity and differential identities for expectations of smooth functions with respect to the Haar measure of U(n) (or O(n)). The paper is organized as follows. In Section 2, we state and discuss the main results (Theorem 2.2 for unitary matrices and Theorem 2.1 for Hermitian positive matrices). In Section 3, we prove auxiliary Theorems 3.1 and 3.2 concerning Hermitian positive definite matrices under the conditions of uniform in n boundedness of the fourth moment of the normalized counting measure of the factors. In Section 4, we use these results to prove Theorem 2.1. Here the main condition is the uniform boundedness of the second moment of the normalized counting measure of the factors. In Section 5 we prove Theorems 5.1 and 5.2 and then prove Theorem 2.2 giving the solution of the problem for unitary matrices. In Section 6 we prove the auxiliary facts that we need. We also describe generalizations of our results in the cases of orthogonal and real symmetric matrices. 2. Models and Main Results We study two ensembles of n × n random matrices Vn and Hn of the form: Vn = V1,nV2,n ,

(2.1)

where V1,n = Wn∗ Sn Wn ,

V2,n = Un∗ Tn Un

and 1/2

1/2

Hn = H1,n H2,n H1,n ,

(2.2)

where H1,n = Wn∗ An Wn ,

H2,n = Un∗ Bn Un .

We assume that Sn , Tn , An , Bn , Un and Wn are mutually independent. Sn and Tn are random unitary matrices, Un and Wn are unitary (resp. orthogonal) random matrices uniformly distributed over the unitary (orthogonal) group U(n) (resp. O(n)) with respect to the Haar measure. An and Bn are Hermitian positive definite random matrices.

ON THE LAW OF MULTIPLICATION OF RANDOM MATRICES

3

We will restrict ourselves to the case of Hermitian matrices (resp. to the group U(n)). The results for symmetric real matrices (i.e. for the group O(n)) are similar, although their proofs are more difficult (see Section 6). We are interested in the asymptotic behavior, as n → ∞, of the normalized eigenvalue counting measure (NCM) νn of the ensemble (2.1), whose value on any Borel set ⊂ [0, 2π ] is given by νn () =

#{µ(n) i ∈ } , n

(2.3)

where µ(n) are the eigenvalues of Vn . We are also interested in the asymptotic i behavior of the NCM Nn of the ensemble (2.2), whose value on any Borel set ⊂ R is given by Nn () =

#{λ(n) i ∈ } , n

(2.4)

where λ(n) i are the eigenvalues of Hn . The problem was studied recently by Voiculescu [2, 8, 10] within the context of free (noncommutative) probability. Combining Voiculescu’s results on free multiplicative convolution of measures having nonzero first moment [2, 10] with results of asymptotic freeness of n × n Haar distributed unitary matrices and nonrandom diagonal matrices [6, 9, 10], one can easily obtain the following result: PROPOSITION. If the matrices Sn , Tn , An and Bn are nonrandom, if the norms of An and Bn are uniformly bounded in n, i.e. their NCMs N1,n and N2,n have compact support uniformly in n, and if these measures have weak limits as n → ∞ ν1,n → ν1 , N1,n → N1 ,

ν2,n → ν2 , N2,n → N2 ,

(2.5) (2.6)

where ν1,n and ν2,n are the NCM’s of Sn and Tn , then the NCM’s (2.3) and (2.4) of (2.1) and (2.2) converge weakly with probability 1 to nonrandom measures ν and N. Here and below the convergence with probability 1 is understood as that in the natural probability spaces ˜ = ˜ n, n , (2.7) = n

n

where n is the probability space of matrices (2.1), that is the product of respective ñ spaces of Sn and Tn and two copies of the group U(n) for Un and Wn , and where is the probability space of matrices (2.2), that is the product of respective spaces of An and Bn and two copies of the group U(n) for Un and Wn . Besides, according to [10], one can define the S-transforms S1 , S2 and S of the measures ν1 , ν2 and ν, respectively, and the S-transforms S˜1 , S˜2 and S˜ of the

4

VLADIMIR VASILCHUK

measures N1 , N2 and N (see our remark after Theorem 2.2) and one can find the following simple expressions S and S˜ via S1,2 and S˜1,2 S = S1 S2 ,

S˜ = S˜1 S˜2 .

The proof of the asymptotic freeness of n × n Haar distributed unitary matrices and nonrandom diagonal matrices having, uniformly in n, compactly supported NCMs in [6, 8], is based on the asymptotic analysis of the expectations of normalized traces of mixed products of matrices Sn , Tn , Un and Wn and An , Bn , Un and Wn , respectively. It requires a considerable amount of combinatorial analysis, the existence of all moments measures Nr , r = 1, 2, and their rather regular behavior as n → ∞ to obtain the convergence of expectations. In this paper we obtain analogous results under more weak assumptions by a method that does not involve combinatorics. This is because we work with the Stieltjes transforms of the measures (2.4) and (2.6) and with the Herglotz transforms of the measures (2.3) and (2.5). We directly derive functional equations for their limits by using simple identities for expectations of matrix-valued functions with respect to the Haar measure (Proposition 3.2 below) and elementary facts concerning resolvents of Hermitian and unitary matrices. This method was already used in [5] for the additive random ensembles. We list below the properties of the Stieltjes and Herglotz transforms that we will need below (see, e.g., [1]). PROPOSITION 2.1. Let m(dλ) , Im z = 0 s(z) = R λ−z

(2.8)

be the Stieltjes transform of a probability measure m on R, then (i) s(z) is analytic in C \ R and |s(z)| |Im z|−1 . (ii) (iii)

Im s(z)Im z > 0, lim y |s(iy)| = 1.

y→∞

(2.9) Im z = 0.

(2.10) (2.11)

(iv) For any continuous function ϕ with compact support we have the Frobenius– Perron inversion formula 1 φ(λ)m(dλ) = lim φ(λ)Im s(λ + iε). (2.12) ε→0 π R R (v) Conversely, any function satisfying (2.9)–(2.11) is the Stieltjes transform of a probability measure and this one-to-one correspondence between measures and their Stieltjes transforms is continuous for the topology of weak convergence for measures and for the topology of convergence on compact subsets of C \ R for the Stieltjes transforms.


PROPOSITION 2.2. Let 2π iθ e +z µ(dθ), t (z) = eiθ − z 0

|z| < 1

5

(2.13)

be the Herglotz transform of a probability measure µ on [0, 2π ], then (i) t (z) is analytic for |z| < 1 and |t (z) − 1| 2|z|(1 − |z|)−1 , (ii)

t (0) = 1,

Re t (z) > 0,

|z| < 1. |z| < 1.

(2.14) (2.15)

(iii) For any continuous on [0, 2π ] function ϕ we have the inversion formula 2π 2π 1 φ(θ)µ(dθ) = lim φ(θ)Re t (re−iθ ) dθ. (2.16) r→1 2π − 0 0 (iv) Conversely, any function satisfying (2.14)–(2.15) is the Herglotz transform of a probability measure on the unit circle. This one-to-one correspondence between measures and their Herglotz transforms is continuous for the topology of weak convergence for measures and for the topology of convergence on compact subsets of {z ∈ C | |z| < 1} for the Herglotz transforms. Now we state our main results. Since the set of eigenvalues of unitary and Hermitian matrices are unitary invariant, we can replace matrices (2.1) and (2.2) by Vn = Sn Un∗ Tn Un

(2.17)

Hn = An1/2 Un∗ Bn Un An1/2 ,

(2.18)

and

where Sn , Tn , An , Bn and Un are as in (2.1) and (2.2). However, it is useful to keep in mind that the problem is symmetric in Sn and Tn and in An and Bn (as we will see below). THEOREM 2.1. Let Hn be a positive definite random n × n matrix of the form (2.2). Assume that the normalized counting measures N1,n , N2,n of An , Bn converge weakly in probability as n → ∞ to nonrandom probability measures N1 , N2 . We also assume +∞ λ2 E{Nr,n(dλ)} m(2) < ∞, r = 1, 2, (2.19) sup n

0

and

+∞

mr =

λNr (dλ) > 0, 0

r = 1, 2,

(2.20)

6

VLADIMIR VASILCHUK

i.e. the measures N1 , N2 are not concentrated at zero. Then the normalized counting measure Nn of Hn converge in probability to a nonrandom probability measure whose Stieltjes transform +∞ N(dλ) , Im z = 0 (2.21) f (z) = λ−z 0 is the unique solution of the system f (z)(1 + zf (z)) = 1 (z)2 (z), z2 (z) , 1 (z) = f2 1 + zf (z) z1 (z) 2 (z) = f1 1 + zf (z)

(2.22)

in the class of functions f (z), 1,2 (z) which are analytic for Im z = 0 and which satisfy (2.9)–(2.11) and zr (z) = −mr + O(|Im z|−1 ), |Re z| |Im z|, z → ∞.

r = 1, 2, (2.23)

f1 (z), f2 (z) are the Stieltjes transforms of N1 , N2 and E{ · } denotes the expectation with respect to the probability measure generated by An , Bn , Un and Wn . THEOREM 2.2. Let Vn be a random n × n matrix of the form (2.1). Assume that the normalized counting measures ν1,n , ν2,n of Sn , Tn converge weakly in probability as n → ∞ to nonrandom probability measures on the unit circle ν1 , ν2 . Then the normalized counting measure νn of Vn converge in probability to a nonrandom probability measure ν whose Herglotz transform 2π iµ e +z ν(dµ), |z| < 1 (2.24) h(z) = eiµ − z 0 is the unique solution of the system h2 (z) = 1 + 4z1 (z)2 (z), 2z2 (z) , h(z) = h2 1 + h(z) 2z1 (z) h(z) = h1 1 + h(z)

(2.25)

in the class of functions h(z), 1,2 (z) which are analytic for |z| < 1 and which satisfy (2.14)–(2.15) and |1,2 (z)| (1 − |z|)−1 ,

|z| < 1.

h1,2 (z) are the Herglotz transforms of the measures ν1,2 .

(2.26)


7

Both theorems will be proved in Sections 3 and 5. Here we interpret them in terms of S-transform introduced by Voiculescu in the context of C ∗ -algebras. 2.1. VOICULESCU ’ S FORMULATION Consider a probability measure µ on the unit circle and assume that its first moment is nonzero 2π eiθ µ(dθ) = 0. µ1 = 0

Consider the function ϕµ (z) = −

1 + t (z−1 ) 2

where t (z) is the Herglotz transform of µ. Since ϕµ (z) = µ1 + o(1), z → ∞, then, according to the local inversion theorem, there exists a unique inverse function χµ (ϕ) of ϕµ (z), χµ (ϕµ (z)) = z defined and analytic in a neighborhood of −1 and assuming its values in a neighborhood of infinity. On the other hand, for any probability measure m on the real nonnegative semi-axis having nonzero first moment ∞ λm(dλ) > 0, m1 = 0

we can consider the function ϕm (z) = −(1 + z−1 s(z−1 )), where s(z) is the Stieltjes transform of the measure m. Since ϕm (0) = m1 , then, according to the local inversion theorem, ϕm (z) also has a unique inverse function χm (ϕ) defined and analytic in a neighborhood of zero and assuming its values in a neighborhood of zero. Denote Sµ (ϕ) = χµ (ϕ)ϕ −1 (1 + ϕ), Sm (ϕ) = χm (ϕ)ϕ −1 (1 + ϕ) and, following Voiculescu [10], call Sµ (ϕ) the S-transform of the probability measure µ on the unit circle and Sm (ϕ) the S-transform of the probability measure m on the real nonnegative semi-axis. By using the S-transforms S1 , S2 of ν1 , ν2 , we can rewrite (2.25) in the form 1 + ψ(z) 1 + ψ(z) , z1 (z) z2 (z) 1 + ψ(z) , r = 1, 2, Sr (ψ(z)) = − zr (z)

S(ψ(z)) =

(2.27) (2.28)

8

VLADIMIR VASILCHUK

where ψ(z) = ϕ(z−1) = −(1 + h(z))/2 and S(ψ) denotes the S-transform of the limiting normalized counting measure ν of (2.1), whose Herglotz transform is h(z). Then we derive from (2.27) Voiculescu’s very simple expression of S via S1 and S2 S(ψ) = S1 (ψ)S2 (ψ).

(2.29)

It is easy to check that (2.22) leads to the relations (2.27) for ψ(z) = −(1 + zf (z)) and S(ψ) denotes the S-transform of the limiting normalized counting measure N of (2.2), whose Stieltjes transform is f (z). Thus, (2.22) leads to the same expression (2.29) where S1,2 will be the S-transforms of the measures N1,2 . The relation (2.29) was obtained by Voiculescu in the context of C ∗ -algebra studies (see [9, 10] for results and references). 3. Convergence with Probability 1 for Nonrandom An , Bn In this section, we start the proof of Theorem 2.1. As a first step we prove the following theorem: THEOREM 3.1. Let Hn be a positive definite nonrandom n × n matrix of the form (2.2) in which An and Bn are nonrandom Hermitian positive matrices, Un and Wn are random independent unitary matrices distributed according to the Haar measure on U(n). Assume that the normalized counting measures N1,n , N2,n of An and Bn converge weakly as n → ∞ to nonrandom probability measures N1 , N2 , +∞ +∞ λNr,n (dλ) = mr = λNr (dλ) > 0, r = 1, 2, (3.1) lim n→∞

0

+∞

sup n

0

λ4 Nr (dλ) m4 < ∞.

(3.2)

0

Then the normalized counting measure Nn of Hn converges with probability 1 to a nonrandom probability measure N whose Stieltjes transform f (z) (2.21) is the unique solution of (2.22) in the class of functions f (z), 1,2 (z) which are analytic for z ∈ C \ R+ and which satisfy (2.9)–(2.11) and (2.23). We use the technique introduced in [5]. Let us recall its basic means. First we collect elementary facts of linear algebra. PROPOSITION 3.1. Let Mn be the algebra of linear endomorphisms of Cn equipped with the norm induced by the standard Euclidean norm of Cn . Then (i) If {Mj k }nj,k=1 is the matrix of M ∈ Mn in any orthonormal basis of Cn then |Mj k | ||M||.

(3.3)


9

(ii) If Tr M denotes the trace of M ∈ Mn , then |Tr M| n||M||, |Tr M1 M2 |2 Tr M1 M1∗ Tr M2 M2∗ , (3.4) where M ∗ is the adjoint of M. Furthermore, if P ∈ Mn is positive definite, then |Tr MP | ||M||Tr P .

(3.5)

G(z) = (M − z)−1

(3.6)

(iii) Let be the resolvent of M ∈ Mn . It is defined for all nonreal z, Im z = 0 if M is Hermitian. It is defined for all z, |z| = 1 if M is unitary. (iv) If G(z1 ) and G(z2 ) are defined, G(z1 ) − G(z2 ) = (z1 − z2 )G(z1 )G(z2 ).

(3.7)

(v) If M ∈ Mn is Hermitian and Im z = 0, then ||G(z)|| |Im z|−1 .

(3.8)

(vi) If M ∈ Mn is invertible and if {Gj k (z)}nj,k=1 is the matrix of its resolvent G(z) in any orthonormal basis, then |Gj k (z)|

||M −1 || . 1 − |z| ||M −1 ||

(3.9)

(vii) If M1 , M2 ∈ Mn their resolvents G1 (z), G2 (z) satisfy the “resolvent identity” G2 (z) = G1 (z) − G1 (z)(M2 − M1 )G2 (z).

(3.10)

(viii) The differential G (z) of the resolvent G(z) = (M − z)−1 viewed as function of M satisfies G (z) · X = −G(z)XG(z)

(3.11)

for any X ∈ Mn . In particular, ||G (z)|| ||G(z)||2 |Im z|−2 .

(3.12)

Now we present the main technical tool. PROPOSITION 3.2 ([5]). Let .: Mn → C be of class C 1 . Then, for any M ∈ Mn and any Hermitian X ∈ Mn : . (U ∗ MU ) · [X, U ∗ MU ] dU = 0, (3.13) U(n)

where [M1 , M2 ] denotes the commutator M1 M2 − M1 M2 and the integral denotes integration over U(n) with respect to the Haar measure.

10

VLADIMIR VASILCHUK

Proof. cf. [5], Proposition 3.2. The integral .(eiεX U ∗ MU e−iεX )[X, U ∗ MU ] dU = 0 U(n)

does not depend on ε. The derivative at ε = 0 gives (3.13).

✷

PROPOSITION 3.3. The system (2.22) has a unique solution in the class of functions f (z), 1,2 (z) which are analytic for Im z = 0 and which satisfy (2.9)–(2.11) and (2.23). Proof. Assume that there exist two solutions (f , 1,2 ) and (f , 1,2 ) of (2.22). Denote δf = f −f , δ1,2 = 1,2 −1,2 . Then, by using (2.22) and the following relations +∞ λNr (dλ) , r = 1, 2 fr (z) = −z−1 + z−1 λ−z 0 we obtain a linear system for δφ = zδf , δ1 , δ2 : (1 + a1 (z))δφ − b1 (z)δ1 − c1 (z)δ2 = 0, (1 + a2 (z))δφ − b2 (z)δ2 = 0, (1 + a3 (z))δφ − b3 (z)δ1 = 0,

(3.14)

where b1 = z2 , c1 = z1 , a1 = zf + zf , z2 J2 (s2 , s2 ) J2 (s2 , s2 ) , b = z , a2 = 2 (1 + zf )(1 + zf ) 1 + zf +∞ λN2 (dλ) J2 (z , z ) = (λ − z )(λ − z ) 0

(3.15) s2, =

1

z, 2 , + zf , (3.16)

and a3 , b3 can be obtained from a2 and b2 by replacing N2 and 2 by N1 and 1 in the above formulas. For any y0 > 0, consider the domain E(y0 ) = {z ∈ C | |Im z| y0 , |Re z| |Im z|}.

(3.17)

Due to condition (2.23) and the first equation of the system (2.22), we have for z ∈ E(y0 ) 1 + zf , (z) = −z−1 (m1 m2 + O(|Im z|−1 )), Besides, if

t (z , z ) = 0

+∞

λm(dλ) (λ − z )(λ − z )

z → ∞.

(3.18)

11


and m is a probability measure having finite second moment, then we have for z , z ∈ E(y0 ), +∞ λm(dλ)| |z z t (z , z ) − 0 +∞ 2 λ (z + z − λ)m(dλ) = (λ − z )(λ − z ) 0 +∞ λ2 m(dλ), 6y0−1 0

i.e.

+∞

z z t (z , z ) =

λm(dλ) + O(|Im z|−1 ),

0

z , z → ∞, z , z ∈ E(y0 ). Thus, from the relation above, (3.15), (3.16), (3.18) and condition (2.23), we obtain that for z → ∞, z ∈ E(y0 ) sr (z)sr (z)Jr (sr (z), sr (z)) = mr + o(1),

r = 1, 2.

Hence a1 (z) = −2 + o(1), c1 (z) = m1 + o(1),

a2,3 (z) = −1 + o(1), b2 (z) = m1 + o(1),

b1 (z) = m2 + o(1), b3 (z) = m2 + o(1).

Thus the determinant −(1 + a1 )b2 b3 + b1 b2 (1 + a3 ) + c1 b3 (1 + a2 ) of the system (3.15) is asymptotically equal to m1 m2 > 0. We conclude that if y0 is sufficiently large in (3.17), then (3.15) has only the trivial solution, i.e. (2.22) is uniquely soluble. ✷ Proof of Theorem 3.1. Due to the unitary invariance of eigenvalues of Hermitian matrices, we can assume without loss of generality that W = I in (2.2), i.e. we can work with the random matrix (2.18). We will omit below the subindex n in all cases when it will not lead to confusion. Consider the matrices 1/2 1/2 Hˆ n = H2,n H1,n H2,n

H˜ n = H1,nH2,n ,

(3.19)

and their resolvents 1/2 −1/2 ˜ G(z) = (H˜ n − z)−1 = H1,n G(z)H1,n ,

(3.20)

1/2 ˜ −1/2 ˆ G(z) = (Hˆ n − z)−1 = H2,n G(z)H 2,n 1/2

1/2

−1/2

−1/2

= H2,n H1,n G(z)H1,n H2,n ,

(3.21)

12

VLADIMIR VASILCHUK

where G(z) = (Hn − z)−1 is the resolvent of (2.2). Because of the trace property we derive from (3.20) ˜ ˆ n−1 Tr G(z) = n−1 Tr G(z) = n−1 Tr G(z) +∞ Nn (dλ) = gn (z), = λ−z 0

(3.22)

where gn (z) is Stieltjes transform of the NCM of (2.2). Consider the resolvent identity (3.10) for the pair (H˜ , 0): ˜ ˜ G(z)H 1 H2 = zG(z) + I.

(3.23)

Using Proposition 3.2 with .(M) = (H1 M − z)−1 ab , in view of (3.11), we obtain that ˜ 1 [X, H2 ]G)ab = 0. (GH

(3.24)

Choosing the matrix X with only Xab = 0,7 we obtain ˜ ˜ ˜ ˜ (3.25) (G(z)H 1 )aa (H2 G(z))bb = (G(z)H1 H2 )aa Gbb (z). Applying to this quantity n−1 na,b=1 and taking into account (3.23), we obtain the relation δ1,n (z)δ2,n(z) = gn (z) + zgn (z)gn (z),

(3.26)

where −1 ˜ δ1,n (z) = n−1 Tr G(z)H 1 = n Tr G(z)H1 , −1 ˜ ˆ δ2,n (z) = n−1 Tr G(z)H 2 = n Tr G(z)H2 .

(3.27) (3.28)

Introduce now the centered quantities gn◦ (z) = gn (z) − fn (z),

◦ δ2,n (z) = δ2,n (z) − 2,n (z),

(3.29)

where fn (z) = gn (z),

2,n (z) = δ2,n (z).

(3.30)

With these notations, (3.26) becomes (1 + zfn (z))fn(z) = 1,n (z)2,n(z) + r1,n (z),

(3.31)

7 In fact, in this relation (and in similar relations below) we can substitute only Hermitian matrices, e.g. (X(r) )pq = (δa,p δb,q + δa,q δb,p )/2 or (X(i) )pq = −i(δa,p δb,q − δa,q δb,p )/2 (here δ is the Kronecker symbol). Nevertheless, adding the relations obtained, we get relation (3.24) with (X)pq = (X(r) + iX(i) )pq = δa,p δb,q .

13


where ◦ (z)δ2,n(z) − gn◦ (z)zgn(z). r1,n (z) = δ1,n

(3.32)

In the next Theorem 3.2, we show that there exist y0 > 0, and C(y0 ) > 0, both independent of n, such that for z ∈ E(y0 ) (3.17) the variances v1 (z) = (gn◦ (z))2,

◦ v1+r (z) = (δr,n (z))2 ,

◦ v3 (z) = (δ2,n (z))2

satisfy v1 (z)

C(y0 ) , n2

v2 (z)

C(y0 ) , n2

v3 (z)

C(y0 ) . n2

(3.33)

In addition, according to Proposition 3.1, (3.22), (3.27) and (3.23), we have the estimates gn (z) |Im z|−1 ,

δr,n (z) m4 |Im z|−1 , Im z = 0, 1/4

r = 1, 2

and |zgn (z)| 1 + m4 |Im zt|−1 , Im z = 0. 1/2

These bounds and the Cauchy–Schwarz inequality for the average · imply that uniformly in n for z ∈ E(y0 ) |r1,n (z)| v2 v3 + (1 + m4 y0−1 )v1 √ C(y0 ) C(y0 ) 1/2 (1 + m4 y0−1 ) = O(n−1 ). + 2 n n 1/2 1/2

1/2

1/2

(3.34)

Now consider the matrix ∗ ˜ B. Y1 = B −1 U G(z)AU

(3.35)

It is clear that δ1,n (z) = n−1 Tr Y1 . On the other hand, using the resolvent identity ˜ ˜ (3.7) for the pair (G(z), G(0)), we obtain Y1 = B −1 + zZ1 ,

(3.36)

where ∗ ˜ . Z1 = B −1 U G(z)U

(3.37)

By using for the following function of the unitary matrix U ∗ ˜ )ac (B −1 U G(z)U

the analog of (3.13) derived from the left shift invariance of the Haar measure, we obtain ∗ ∗ ∗ ˜ ˜ ˜ ])ac − (H2−1 U G(z)H (B −1 [X, U G(z)U 1 U [H2 , X]U G(z)U )ac = 0.

14

VLADIMIR VASILCHUK

Choosing the matrix X with only Xbc = 0, we get ∗ ∗ ˜ ˜ )cc − (Y1 )ab (U G(z)U )cc (B −1 )ab (U G(z)U −1 ∗ ˜ = (Z1 )ab Icc − (Y1 B )ab (BU G(z)U )cc . Applying the operation n−1 nc=1 to (3.38), we obtain the matrix relation

Z1 = B −1 fn (z) − Y1 gn (z) + Y1 T ∗ δ2,n (z).

(3.38)

(3.39)

Regrouping the terms in (3.39) according to (3.29) and using (3.36), we obtain Z1 =

2 (z) Y1 B −1 + R, 1 + zfn (z)

(3.40)

where ◦ R = (1 + zfn(z))−1 (Y1 B −1 δ2,n (z) − Y1 gn◦ (z)).

(3.41)

Substituting (3.40) in (3.36), we get Y1 (B − z2 (z)(1 + zfn(z))−1 I ) = I + zRB.

(3.42)

Besides, using (3.20), (3.22) and the resolvent identity (3.23), we obtain z2,n (z) = −n−1 Tr H2 + n−1 Tr G(z)H1 H22 H1 , 1/2 2 1/2 ˆ z1,n (z) = −n−1 Tr H1 + n−1 Tr G(z)H 2 H1 H2 1/2

1/2

(3.43) (3.44)

and 1 + zfn (z) 1/2 1/2 = z−1 −n−1 Tr H1 H2 + n−1 Tr G(z)(H1 H2 H1 )2 .

(3.45)

In addition, using Proposition 6.1 with M1 = A and M2 = B, we obtain n−1 Tr H1 H2 = n−1 Tr H1 n−1 Tr H2

(3.46)

2 n−1 Tr (H1 H2 )2 2 n−1 Tr H12 n−1 Tr H2 + 2 + n−1 Tr H1 n−1 Tr H22 − 2 2 + n−1 Tr H1 n−1 Tr H2 + +n−3 Tr H12 n−1 Tr H22 .

(3.47)

and


15

Besides, using (3.5), (3.2), the trace property and (3.46), (3.47), we obtain −1 2 −1 n Tr G(z) H 1/2H2 H 1/2 2 n Tr (H1 H2 ) 1 1 |Im z| 2 8m4 8µ2 , |Im z| |Im z| 1/2 2 1/2 −1 −1

n Tr G(z)H 1/2H 2 H 1/2 n Tr H1 H2 H1 2 1 1 |Im z| −1 n Tr H1 n−1 Tr H22 = |Im z| 3/2

3/4

m µ2 4 |Im z| |Im z| and, similarly, 3/2 3/4 −1 m4 µ2 1/2 2 1/2 n Tr G(z)H ˆ , H H 2 1 1 |Im z| |Im z|

where

µ2 = max sup r=1,2

n

1/2

λ Nr,n (dλ) m4 . 2

From (3.43), (3.44), (3.45) and from the above estimates, we derive for z ∈ E(y0 ) and y0 sufficiently large, uniformly in n, zfn (z) = −1 + O(|Im z|−1 ).

(3.48)

In addition, according to condition (3.1), we have for some n sufficiently large and for all n n |1 + zfn (z)| |z|−1 (m1 m2 )/2, −1 2|z| (z) , r = 1, 2 r,n mr

(3.49)

and mr zr,n (z) Im z , 1 + zfn (z) 2m1 m2 m1 m2 zfn (z) Im z , r = 1, 2. Im r,n (z) 2mr

Im

(3.50)

Thus, the matrix P = B − z2,n (z)(1 + zfn (z))−1 I

(3.51)

16

VLADIMIR VASILCHUK

is uniformly in n invertible for z ∈ E(y0 ) and for y0 sufficiently large, we have ||P −1 ||

2m1 . |Im z|

(3.52)

Thus, we obtain from (3.42) z2 (z) + zRT P −1 , Y1 = G2 1 + zfn(z) where G2 (z) = (B − z)−1 . Applying the operation n−1 Tr to this relation and using (3.41), we obtain z2 (z) + r2,n (z), (3.53) 1 (z) = f2,n 1 + zfn (z) where −1

+∞

f2,n (z) = n Tr G2 (z) = 0

and r2,n (z) =

N2,n (dλ) , λ−z

Im z = 0

−1 z ∗ −1 ◦ ˜ n Tr U G(z)AU P δ2,n (z) − 1 + zf (z) ∗ −1 ◦ ˜ P gn (z) . −n−1 Tr G(z)BU GAU

(3.54)

(3.55)

Using the arguments above for the matrices ∗ ˜ ˜ U (z) = U G(z)U = (U AU ∗ B − z)−1 , G Z2 = U ∗ GU U A Y2 = AU ∗ BGU U A−1 ,

in which the roles A and B are interchanged, we obtain the analogous to (3.53) relation for 2,n (z). Thus, we obtain the system of relations fn (z) + zfn2 (z) = 1,n (z)2,n(z) + r1,n (z), z2,n (z) + r2,n (z), 1,n (z) = f2,n 1 + zfn(z) z1,n (z) + r3,n (z), 2,n (z) = f1,n 1 + zfn(z) where fr,n (z) = n−1 Tr Gr (z) =

+∞ 0

Nr,n (dλ) , λ−z

(3.56)

r = 1, 2.

(3.57)

In addition, according to (3.33), (3.49), (3.52) and the Cauchy–Schwarz inequality, we have for z ∈ E(y0 ) |rs,n (z)|

16 1/4 1/2 1/2 1/2 m4 v5−s + m4 v1 = O(n−1 ), m4−s

s = 2, 3.

(3.58)


17

Besides, for z ∈ C \ R+,ξ where ξ > 0 and R+,ξ is a ξ -neighborhood of the real positive semi-axis R+ , we have the following estimates: |fn (z)| ξ −1 ,

−1 |(n) . 1,2 (z)| m4 ξ 1/4

These estimates imply that {fn (z)} and {(n) 1,2 (z)} are sequences of analytic functions, bounded uniformly in n for z ∈ C \ R+,ξ . Thus these sequences are relatively compact with respect to uniform convergence on any compact subset of C\R+,ξ . In addition, according to the hypothesis of the theorem, the normalized counting measures N1,n , N2,n of H1,n , H2,n converge weakly to limiting measures N1 , N2 . Thus their Stieltjes transforms (3.57) converge uniformly on every compact of E(y0 ), y0 > ξ to the Stieltjes transforms f1,2 (z) of N1,2 . Hence, if y0 is large enough, there exist three functions f , 1 , 2 which are analytic in E(y0 ) and which satisfy the limiting system (2.22) for z ∈ E(y0 ). Its unique solubility in (3.17) where y0 is large enough is proved in Proposition 3.3. Besides, the three functions fn , 1 , 2 are a-priori analytic for z ∈ C \ R+ . Thus, their limits f , 1 , 2 are also analytic for z ∈ C \ R+ . In view of the weak compactness of probability measures and the continuity of the one-to-one correspondence between nonnegative measures and their Stieltjes transforms (see Proposition 2.1(v)), there exists a unique nonnegative measure N such that f admits the representation (2.21). This measure N is a probability measure in view of (3.48). We conclude that the whole sequence {fn(z)} converges uniformly on every compact subset of C \ R+,ξ , ξ > 0 to the limiting function f (z) verifying (2.22). This result, Theorem 3.2 and the Borel–Cantelli lemma imply that the sequence {gn (z)} where gn (z) is defined in (3.22) converge with probability 1 to f (z) for any fixed z ∈ E(y0 ). Since the convergence of a sequence of analytical functions on any countable set having an accumulation point in their common domain of definition implies the uniform convergence of the sequence on any compact of the domain, we obtain the convergence gn (z) to f (z) with probability 1 on any compact of C \ R+,ξ . Due to the continuity of the one-to-one correspondence between probability measures and their Stieltjes transforms, the normalized counting measure (NCM) of the eigenvalues of random matrix (2.2) converge weakly with probability 1 to the non-random measure N whose Stieltjes transform (2.21) satisfies (2.22). Theorem 3.1 is proved. ✷ THEOREM 3.2. Let Hn be a random matrix of the form (2.2) satisfying the conditions of Theorem 3.1. Denote gn (z) = n−1 Tr (Hn − z)−1 = n−1 Tr (H˜ n − z)−1 , δr,n (z) = n−1 Tr Hr,n (H˜ n − z)−1 , r = 1, 2, where H˜ n is defined in (3.19).

(3.59)

18

VLADIMIR VASILCHUK

Then there exist y0 and C(y0 ), both positive and independent of n, such that the variances of the random variables (3.59) satisfy for any z ∈ E(y0 ) C(y0 ) , (3.60) v1 = |gn (z) − gn (z)|2 n2 C(y0 ) , r = 1, 2. (3.61) v1+r = |δr,n (z) − δr,n (z)|2 n2 Proof. We will derive and analyze the system of inequalities vi

3

1/2 1/2

αij (y0 )vi vj

j =1,j =i

+

βi (y0 ) , n2

i = 1, 2, 3.

(3.62)

Below we will use the notations g(z) and δr (z) for gn (z) and δr,n (z), r = 1, 2 and the notations 1 and 2 for two values z1 and z2 of the complex spectral parameter z. We assume that |Im z1,2 | y0 > 0. Consider the matrix ∗ ˜ Q1 = g ◦ (1)z2−1 BU G(2)AU , (3.63) where g ◦ (1) = g(1) − g(1). It follows from (3.23) that n−1 Tr Q1 for z1 = z and z2 = z is the variance (3.60), that we denote v1 (z): v1 (z) = n−1 Tr Q1 |z1 =z,z2 =z = |g ◦ (z)|2.

(3.64)

By using for the following function of the unitary matrix U ∗ ◦ −1 ∗ ˜ ˜ z2 BU G(2)AU U G(1)U cd

aa

that is the analog of (3.13) derived from the left shift invariance of the Haar measure, we obtain ∗ ∗ ˜ ˜ ])aa (BU G(2)AU )cd − z−1 (([X, U G(1)U 2

∗ ∗ ∗ ˜ ˜ ˜ [B, X]U G(1)U )aa (BU G(2)AU )cd + −(U G(1)AU ∗ ◦ ∗ ˜ ˜ )aa (B[X, U G(2)AU ])cd − +(U G(1)U ∗ ◦ ∗ ∗ ˜ ˜ ˜ )aa (BU G(2)AU [B, X]U G(2)U )cd ) = 0. −(U G(1)U

Choosing in the above relation a matrix X with only Xbd = 0, applying to the result the operation n−2 na,d=1 and taking into account that ◦

g (z) = n

−1

n

G◦aa (z),

a=1

we obtain the matrix relation ∗ ˜ ˜ 2 (1)AU ∗ − G ˜ 2 (1)AU ∗ B)+ BU G z2−1 n−2 BU G(2)A(U ∗ ˜ + +Bg ◦ (1)δ1 (2) − g ◦ (1)BU G(2)AU ◦ ∗ ˜ +g (1)BU G(2)AU (1 + z2 g(2))− ∗ ˜ Bδ1 (2) = 0. −g ◦ (1)BU G(2)AU

(3.65)

19


Regrouping the terms in (3.65) according to (3.29) and taking into account that ∗ ∗ ˜ ˜ B − B) = U A−1 G(2)AU B, z2−1 (BU G(2)AU

we obtain Q1 (z2 f (2) − 1 (2)B) ∗ ˜ = −g ◦ (1)g ◦ (2)BU G(2)AU + ◦ ◦ −1 ˜ + g (1)δ1 (2)U A G(2)AU ∗ B + R˜ 1 ,

(3.66)

where ∗ ˜ ˜ 2 (1)AU ∗ − G ˜ 2 (1)AU ∗ B). BU G R˜ 1 = n−2 z2−1 BU G(2)A(U

(3.67)

Besides, according to (3.49) and (3.50), the matrix P2 = (z2 f (2) − 1 (2)B) = 1 (2)(z2 f (2)−1 1 (2) − B) is uniformly in n invertible for z ∈ E(y0 ) and ||P2−1 ||

8 . m2

Multiplying (3.66) by P2−1 from the right and applying the operation n−1 Tr to the result, we obtain in the left-hand side the variance v1 (z) of (3.60). In view of (3.9), (3.49), (3.20) and the trace property, the terms in right-hand side can be estimated for z2 ∈ E(y0 ) as follows 1/2 8m4 −1 ◦ ◦ −1 1/2 1/2 ∗ v1 ; (i) |g (1)g (2)n Tr P2 BU A G(2)A U | m2 y0 (ii) |g ◦ (1)δ1◦ (2)n−1 Tr P2−1 U A−1/2 G(2)A1/2 U ∗ B| (iii)

1/2

8m4 1/2 1/2 v v ; m2 y0 1 2

|z2 |−1 |n−3 Tr P2−1 BU A1/2 G(2)A1/2 (U ∗ BU A1/2 G2 (1)A1/2 U ∗ − −3 −A1/2 G2 (1)A1/2 U ∗ B)| n−2 16m4 m−1 2 y0 .

−1 1/2 to the first inequality of (3.62), in These bounds lead for 8m4 m−1 2 y0 which 1/2

−1 −1 −3 α12 (y0 ) = 16m4 m−1 2 y0 , α13 (y0 ) = 0, β1 = 32m4 m2 y0 . 1/2

(3.68)

To obtain the second inequality of the system, consider the matrix ˜ Q2 = δ1◦ (1)G(2)A.

(3.69)

Applying to Q2 operation n−1 Tr and setting z1 = z, z2 = z, we obtain the variance v2 of (3.61). On the other hand, using Proposition 3.2 for the function ◦ ˜ ˜ .(M) = (G(1)A) aa (G(2)A)cd ,

20

VLADIMIR VASILCHUK

˜ where G(z) = (AU ∗ MU − z)−1 , we obtain relation ˜ ˜ ˜ G(1)A G(2)A+ n−2 (−U ∗ BU G(1)A ˜ ˜ H˜ G(2)A)− ˜ +G(1)A G(1) ◦ ∗ ˜ −δ1 (1)δ1 (2)U BU G(2)A+

˜ ˜ = 0. + δ1◦ (1)G(2)A +δ1◦ (1)g(2)z2 G(2)A

(3.70)

We do this by identifying M and B and performing almost the same procedure as that used in the derivation of (3.65), in particular, choosing for the matrix X the matrix with only the (c, b)th entry nonzero. Regrouping terms in (3.70) according ˜ ˜ = zA−1 G(2) + A−1 , we obtain to (3.29) and taking into account that U ∗ BU G(z) ((1 + z2 f (2))I − z2 1 (2)A−1 )Q2 ˜ ˜ = δ1◦ (1)δ1◦ (2)U ∗ BU G(2)A − δ1◦ (1)g ◦ (2)z2 G(2)A + R˜ 2 ,

(3.71)

where ˜ ˜ ˜ G(1)A G(2)A − R˜ 2 = n−2 (U ∗ BU G(1)A ˜ ˜ ˜ ˜ −G(1)AG(1)H G(2)A).

(3.72)

Multiplying (3.71) by A from the left, we get ˜ − P˜ Q2 = ((1 + z2 f (2))−1 (δ1◦ (1)δ1◦ (2)H˜ G(2)A ◦ ◦ ˜ ˜ −δ1 (1)g (2)z2 AG(2)A + AR2 ),

(3.73)

where P˜ = A−z2 1 (2)(1+z2 f (2))−1 I . It follows from (3.50) that for z2 ∈ E(y0 ) where y0 is sufficiently large, the matrix P˜ is invertible uniformly in n and 2m2 . ||P˜ −1 || |Im z2 | Multiplying (3.73) by P˜ −1 from the left and applying the operation n−1 Tr to the result, we obtain in the left-hand side the variance v2 (z) of (3.61). In view of (3.9), (3.49), (3.20) and the trace property, the terms in right-hand side can be estimated for z ∈ E(y0 ) as follows 1/4 8m4 −1 ◦ ◦ −1 −1 ˜ v2 ; (i) |1 + z2 f (2)| |δ1 (1)δ1 (2)n Tr P H G(2)A| m1 y0 (ii) |1 + z2 f (2)|−1 |δ1◦ (1)g ◦ (2)n−1 Tr P˜ −1 z2 AG(2)A| (iii)

|1 + z2 f (2)|−1 |n−3 Tr P˜ −1 A(H G(1)AG(1)AG(2)− 5/4 −2 −G(1)AG(1)H G(2)A)| n−2 16m4 m−1 1 y0 .

1/2

16m4 1/2 1/2 v v ; m1 2 1

21


−1 These bounds lead for 8m4 m−1 1 y0 1/2 to the second inequality of (3.62), in which 1/4

α21 (y0 ) = 32m4 m−1 1 , 1/2

−2 β2 = 32m4 m−1 1 y0 . 5/4

α23 (y0 ) = 0,

(3.74)

∗ ˜ Using the arguments above for the matrix Q3 = δ2◦ (1)BU G(2)U , we obtain the third inequality of (3.62) where

α31 = α21 = 32m4 m−1 1 , α32 = α23 = 0, 5/4 −2 β3 = β2 = 32m4 m−1 1 y0 . 1/2

(3.75)

Introducing new variables 1/2 1/2

u1 = y0 v1 ,

1/2

u2 = v2 ,

1/2

u3 = v3 ,

we obtain from (3.68), (3.74) and (3.75) the system u2i

3

j =1,j =i

aij ui uj +

γi , n2

i = 1, 2, 3,

(3.76) −1/2

where the coefficients {aij , i = j } have the form aij = y0 bij with bij and γi bounded in y0 and n as y0 → ∞ and n → ∞. By choosing y0 sufficiently large (and then fixing it), we can guarantee that 0 aij 1/4, i = j . Summing the three relations (3.76), we can write the result in the form (cu, u) γ /n2 where γ = γ1 + γ2 + γ3 and the minimum eigenvalue of the matrix c is 1/2. Thus we obtain the bounds (3.60) and (3.61). Theorem 3.2 is proved. ✷

4. Convergence in Probability for Random An and Bn According to Theorem 3.2, the randomness of Un in (2.2) (or (2.18)) already allows us to prove that the variance of the Stieltjes transform of the NCM (2.4) vanishes as n → ∞. Therefore, we have only to prove that the additional randomness due to the matrices An and Bn in (2.2) does not destroy this property. We will prove this fact first for An and Bn whose norms are uniformly bounded in n (see Lemma 4.1 below), and then for the general case of Theorem 2.1 by using some truncation procedure. PROPOSITION 4.1 ([5]). Let {mn } be a sequence of random measures defined on some probability space and {sn } be the sequence of their Stieltjes transforms. Then: (i) The sequence mn converges in probability to a nonrandom probability measure m if and only if the sequence {sn } converges in probability for any fixed z

22

VLADIMIR VASILCHUK

belonging to some compact set K ⊂ {z ∈ C | Im z > 0} to the Stieltjes transform f of the measure m. (ii) We can replace the requirement of their convergence for any z belonging to a certain compact of C± by the convergence for any z belonging to any interval of the imaginary axis, i.e. for z = iy, y ∈ [y1 , y2 ], y1 > 0. (iii) If {mn } is a sequence of random nonnegative measures converging weakly in probability to a nonrandom nonnegative measure m, then the Stieltjes transforms sn of mn and the Stieltjes transform s of m are related by (4.1) lim E sup |sn (z) − s(z)| = 0 n→∞

z∈K

for any compact of C± . Proof. cf. [5], Proposition 4.1.

✷

LEMMA 4.1. Let Hn be the random n×n matrix of the form (2.2) in which An and Bn are random positive definite Hermitian matrices, Un and Vn are random unitary matrices distributed each according to the Haar measure on U(n) and An , Bn , Un and Vn are mutually independent. Assume that the normalized counting measures Nr,n , r = 1, 2 of matrices An and Bn converge weakly in probability as n → ∞ to the nonrandom nonnegative and probability measures Nr , r = 1, 2 respectively and that sup ||An || T < ∞,

sup ||Bn || T < ∞,

n

(4.2)

n

and condition (2.20) of Theorem 2.2 is fulfilled. Then the normalized counting measure Nn of Hn converges weakly in probability to a nonrandom probability measure whose Stieltjes transform (2.21) is a unique solution of the system (2.22) in the class of functions f (z), 1,2 (z) analytic for z ∈ C \ R+ and satisfying conditions (2.9)–(2.11) and (2.23). Proof. In view of Proposition 4.1(ii) it suffices to show that lim E{|gn (z) − f (z)|} = 0

n→∞

for any z belonging to some interval of the imaginary axis, i.e. z = iy,

y ∈ [y1 , y2 ],

0 < y1 < y2 < ∞.

Condition (4.2) implies +∞ λNr,n (dλ) − mr ε → 0, P

r = 1, 2, as n → ∞.

(4.3)

(4.4)

0

Thus, for any sufficiently small η > 0 there exists n(η) such that for all n n(η) +∞ 12 10 mr λNr,n (dλ) mr 1 − η, r = 1, 2. (4.5) P 11 11 0

23


Denote by ∗ the event whose probability is written in the left-hand side of (4.5) and by E∗ { · } the average over product of ∗ and two copies of the group U (n) for Un and Wn . As a result of (4.5), for z ∈ [y1 , y2 ], we have lim E{|gn (z) − f (z)|} 2y1−1 η + lim E∗ {|gn (z) − f (z)|}

n→∞

n→∞

and for all n n(η) and any realization of An and Bn on ∗ , we have +∞ 12 10 mr λNr,n (dλ) mr . 11 11 0

(4.6)

Thus, to complete the proof we have to show that for any z = iy, y ∈ [y1 , y2 ] lim E∗ {|gn (z) − f (z)|} = 0.

(4.7)

n→∞

Since the relation (4.4) and condition (4.2) of the lemma imply evidently the conditions (3.1) and (3.2) of Theorems 3.1 and 3.2, all the results obtained in these theorems are valid in our case for any fixed realization of random matrices An and Bn on ∗ . In addition, all n-independent estimating quantities entering various bounds in the proofs of these theorems and depending on the moments m1 , m2 , m4 in (3.1), (3.2) and on n , ξ and y0 will now depend on m1 , m2 , T and on n(η), ξ and y1 , y2 in (4.3), but not on any particular realization of random matrices An and Bn . Using (3.33), we can write that E∗ {|gn (z) − gn (z)|} E∗ {|v1 (z)|} 1/2

C , n

where the symbol · denotes, as above, the expectation with respect the Haar measure on U (n). Thus, it suffices to show lim E∗ {|gn (z) − f (z)|} = 0,

n→∞

z = iy, y ∈ [y1 , y2 ],

(4.8)

where y1 is big enough. Introduce the quantities γn (y) = iy(gn (iy) − f (iy)),

γr,n (y) = δr,n (iy) − r (iy),

r = 1, 2.

By using the first equation of system (2.22) and first relation of (3.56), we can write the identity γn (y)(1 + iy(gn (iy) + f (iy))) = iy2 (iy)γ1,n (y) + iyδ1,n (iy)γ2,n (y) + iyr1,n (iy).

(4.9)

On other hand, using the integral representation (2.8), from the first two equations of (2.22), we obtain 1 + iyf (iy) = ρ2 (t2 (y)),

(4.10)

24

VLADIMIR VASILCHUK

where

+∞

ρ2 (z) = 1 + zf2 (z) = 0

λN2 (dλ) , λ−z

t2 (y) =

iy2 (iy) . 1 + iyf (iy)

(4.11)

Using (4.10), we obtain γn (y) = [ρ2 (t2,n (y)) − ρ2 (t2 (y))] + ε1,n (y),

(4.12)

where ε1,n (y) = [1 + iygn (iy) − ρ2 (t2,n (y))], iyδ2,n (iy) . t2,n (y) = 1 + iygn (iy)

(4.13)

We have E∗ {|ε1,n (y)|} E∗ {|1 + iygn (iy) − ρ2,n(t2,n (y))|} + +E∗ {|ρ2,n(t2,n (y)) − ρ2 (t2,n (y))|}, where

ρ2,n (z) = 1 + zg2,n (z) = 0

+∞

λN2,n (dλ) λ−z

(4.14)

(4.15)

and where g2,n (z) is the Stieltjes transform of the random NCM N2,n of H2,n (cf. (3.57)) +∞ N2,n (dλ) −1 . g2,n (z) = n Tr G2 (z) = λ−z 0 Besides, by using two first relations of (3.56), we obtain 1 + iygn (iy) = ρ2,n (t2,n (y)) + rˆ2,n (iy)t2,n (y) − r1,n (iy)(1 + gn (iy))−1 , where r2,n (z) =

−1 z ◦ n Tr Y1 P˜ −1 δ2,n (z)− 1 + zgn (z) −n−1 Tr U G(z)AU BU ∗ P˜ −1 gn◦ (z) ,

P˜ −1 = G2 (t2,n (z)), gn◦ (z) = gn (z) − gn (z), ◦ (z) = δr,n (z) − δr,n (z), δr,n

r = 1, 2

are the respective random variables centered by the partial expectations with respect to the Haar measure. Using (4.6) we obtain the analogs of (3.49) and (3.50).


25

In our case, we have for all n n(η) and y ∈ [y1 , y2 ], y1 sufficiently large |1 + iyfn (iy)| y −1 (m1 m2 )/3, 3y , |t2,n (y)| m1 y . Im t2,n (y) 3m1

(4.16)

In addition, we have the analoge of (3.58) C . n This leads to the following bound for the first term in the right-hand side of (4.14): | r2,n (z)|

E∗ {|1 + iygn (iy) − ρ2,n(t2,n (y))|} 1 ∗ 3y2 ∗ E {|r1,n (iy)|} E {|ˆr2,n (iy)|} + m1 m2 C . n As for the second term, using (4.16) we obtain for it the following bound E∗ {|ρ2,n(t2,n (y)) − ρ2 (t2,n (y))|} 3y2 sup E∗ {|g2,n (z) − f2 (z)|} m1 z∈K 3y2 sup E{|g2,n (z) − f2 (z)|}, m1 z∈K

(4.17)

where K is a compact subset of C+ y1 3y2 K = z ∈ C : Im z , |z| . 3m1 m1 The right-hand side of (4.17) tends to zero as n → ∞ in view of the hypothesis of Theorem 2.1 and Proposition 3.3. Thus, there exist 0 < y1 < y2 < ∞ such that for for all y ∈ [y1 , y2 ], lim E∗ {|ε1,n (y)|} = 0.

n→∞

(4.18)

Analogous arguments show that limn→∞ E∗ {|ε2,n(y)|} = 0, ε2,n (y), where ε2,n (y) is defined from (4.13) and (4.11) by interchanging the indexes 1 and 2. Consider now the first term in the right-hand side of (4.12). In view of (4.11) and (4.15), we can write this term in the form [ρ2 (t2,n (y)) − ρ2 (t2 (y))] iy t2,n (y) J2 γn + J2 γ2,n =− 1 + iyf (iy) 1 + iygn (iy) = −a1 γn + b2 γ2,n ,

26

VLADIMIR VASILCHUK

where J2 , a2 and b2 are defined by formulas (3.15) and (3.16), in which we have to replace 2 , 2 , f and f by δ2,n , 2 , gn and f , respectively. Denote by . = {.ij }3i,j =1 the matrix defined by the left-hand side of system (3.15) and by H = {Hi }3i=1 the vector with components H1 = γn , H2 = γ1,n, H3 = γ2,n . It is easy to check using (4.6) that for all n n(η), y ∈ [y1 , y2 ] and y1 sufficiently large, we have m1 m2 det . , 2 i.e. matrix . is uniformly invertible in n and y. Then we have from (4.9) E{|(.H)1 |} E{|yr1,n |}.

(4.19)

Besides, relations (4.12) and (4.18) lead to E{|(.H)2 |} E{|ε1,n|}.

(4.20)

Interchanging indices 1 and 2, in the above arguments we also obtain that E{|(.H)3 |} E{|ε2,n|}.

(4.21)

Denote by || · ||1 the l 1 -norm of C3 and by || · || the induced matrix norm. Then we have E{||H||1 } E{||.−1 .H||} E1/2 {||.−1 ||2 }E1/2 {||.H||21 }.

(4.22)

It follows from our arguments above that all entries of the matrices . and .−1 and all components of the vector H are bounded uniformly in n and in realizations of random matrices An , Bn , Un and Wn in (2.2). Thus, we have ||.−1 ||

3

|(.−1 )ij | C,

||.H||1

i,j =1

3

|.ij ||H|j C.

i,j =1

These bounds and (4.19)–(4.22) imply that E{||H||1 } C 3/2 (E{|r1,n |} + E{|ε1,n|} + E{|ε2,n|})1/2 . In view of (4.18), this inequality imply (4.8), i.e. the assertion of the lemma.

✷

Proof of Theorem 2.1. For any T > 0 introduce the matrices ATn and BnT reT the NCMs of placing eigenvalues An and Bn lying in ]T , ∞[ by T . Denote by Nr,n T T T An and Bn . It is clear that Nr,n converges weakly in probability to NrT as n → ∞ where NrT is defined via Nr by the formula  ⊂ ]−∞, T [,  Nr (), T = {T }, (4.23) Nr () = Nr ([T , ∞[),  0, ⊂ ]T , ∞[,


27

for any Borel set ∈ R. Now denote T = (H1 )1/2H2T (H1 )1/2 , H T = (H2T )1/2 H1 (H2T )1/2, H H T = (H2T )1/2 H1T (H2T )1/2 , where H1T = V ∗ AT V ,

H2T = U ∗ B T U.

nT and H nT − HnT do not exceed It is clear that the dimensions of ranges of Hn − H T T the ranges of H1,n − H1,n and of H2,n − H2,n, respectively, i.e. they do not exceed the number of those [A]i and [B]i which are larger than T . Therefore, the NCMs nT (λ), N nT (λ) and NnT (λ) of matrices Hn , H T , H T and HnT satisfy the Nn (λ), N inequalities +∞ #{|[B]i | T } T = N2,n (dµ), (4.24) |Nn (λ) − Nn (λ)| n T +∞ nT (λ) − NnT (λ)| #{|[A]i | T } = N1,n (dµ). (4.25) |N n T nT , N nT and NnT , respecgnT (z) and gnT (z) be Stieltjes transforms of N Let gnT (z), tively. Because of the trace property, we have nT − z)−1 = n−1 Tr (H nT − z)−1 = gnT (z). gnT (z) = n−1 Tr (H Besides, it follows by definition and from (4.24) and (4.25), that if gn (z) is the Stieltjes transform of Nn , then we have uniformly on z ∈ E(y0 ) gnT (z) − gn (z)| + | gnT (z) − gnT (z)| |gnT (z) − gn (z)| | +∞ +∞ 1 N1,n (dµ) + N2,n (dµ) . y0 T T Hence E{|gnT (z) − gn (z)|} +∞ +∞ 1 E{N1,n (dµ)} + E{N2,n(dµ)} . y0 T T

(4.26)

Since the norms of matrices H1T and H2T are bounded, the results of the Lemma 4.1 is applicable to the function gnT (z), so that, in particular, for n → ∞ it converges in probability to a function f T (z) obeying the system f T (z)(1 + zf T (z)) = T1 (z)T2 (z), zT2 (z) T T , 1 (z) = f2 1 + zf T (z) zT1 (z) . T2 (z) = f1T 1 + zf T (z)

28

VLADIMIR VASILCHUK

T In addition, since E{gnT (z)} and E{δr,n (z)}, r = 1, 2 are bounded uniformly in n and T for z ∈ C \ R+,ξ

1 |E{gnT (z)}| , ξ 1 +∞ T T λE{Nr,n (dλ)} |E{δr,n (z)}| ξ 0 (m(2))1/2 1 +∞ , λE{Nr,n (dλ)} ξ 0 ξ we have |f T (z)|

1 , ξ

|T1,2 (z)|

(m(2) )1/2 . ξ

(4.27)

Besides, as a result of (3.43), (3.44) and (3.45), we have 1 + zf T (z) = −z−1 mT1 mT2 + O(|Im z|−1 ) , zT1,2 (z) = −mT1,2 + O(|Im z|−1 ), where

mTr =

0

+∞

T λE{Nr,n (dλ)},

(4.28)

r = 1, 2.

Besides, for any sequence Tk → ∞, the sequences of analytic functions {f Tk (z)} k (z)} are relatively compact with respect to uniform convergence on any and {T1,2 Tk compact subset of C \ R+,ξ . In addition, according to (4.23), the measures N1,2 converge weakly to the limiting measures N1,2 and mT1,2 → m1,2 ,

T → ∞.

(4.29)

Hence, there exist three analytic functions f (z), 1 (z), 2 (z) verifying (2.22). (4.28) and (4.29) imply that f (z), 1 (z), 2 (z) satisfy the conditions of Proposition 3.3 and, hence, they are uniquely defined. Furthermore, for z ∈ E(y0 ), we have E{|gn (z) − f (z)|} T

T

E{|gn (z) − gn k (z)|} + E{|gn k (z) − f Tk (z)|} + |f Tk (z) − f (z)|, where f Tk (z) denotes a convergent subsequence of f Tk (z). Hence, in view of (4.26), the arguments above, and Lemma 6.1, we conclude that for each z ∈ E(y0 ) lim E{|gn (z) − f (z)|} = 0.

n→∞

In view of Proposition 4.1, we conclude that the NCM of random matrices (2.2) converge weakly in probability to the nonrandom measure N whose Stieltjes transform f (z) satisfies (2.22). Theorem 2.1 is proved. ✷


29

5. Convergence with Probability 1 for Nonrandom Sn , T n and in Probability for Random Sn , T n The proof of Theorem 2.2 follows the scheme of the proof of Theorem 2.1. In this section, we briefly describe the steps that corespond to those used in the previous sections. The first step of the proof of Theorem 2.2 consists in the following statements (cf. Theorem 3.1 and Proposition 3.3). The following Theorem 5.1 generalizes Voiculescu’s result on the multiplicative free convolution of probability measures on the unit circle [8] proved under the condition that the NCMs ν1,2 have nonzero first moments, i.e. 2π eiθ νr (dθ) = 0, r = 1, 2. 0

THEOREM 5.1 ([7]). Let Vn be a random n × n matrix of the form (2.1) in which Sn and Tn are nonrandom unitary matrices, Un and Wn are random independent unitary matrices distributed each according to the Haar measure on U(n). Assume that the normalized counting measures νr,n , r = 1, 2 of Sn and Tn converge weakly as n → ∞ to the nonnegative and probability measures on the unit circle νr , r = 1, 2, respectively. Then the normalized counting measure of νn converges in probability to a nonrandom nonnegative and probability measure ν whose Herglotz transform (2.24) is the unique solution of the system (2.25) in the class of functions h(z), 1,2 (z) analytic for |z| < 1 and satisfying conditions (2.14)–(2.15) and (2.26). PROPOSITION 5.1. System (2.25) has a unique solution in the class of functions h(z), 1,2 (z) analytic for |z| < 1 and satisfying conditions (2.14)–(2.15) and (2.26). Proof. Assume that there exist two solutions (h , 1,2 ) and (h , 1,2 ) of the system. Denote δh = h − h , δ1,2 = 1,2 − 1,2 and δf = f − f , where f , = (h, − 1)/2z. Then, by using (2.24) and the integral representation (2.13) for h1,2, we obtain the linear system for δf , δ1,2 (1 + a1 (z))δf − b1 (z)δ1 − c1 (z)δ2 = 0, a2 (z)δf + δ1 − b2 (z)δ2 = 0, a3 (z)δf − b3 (z)δ1 + δ2 = 0,

(5.1)

where b1 = 2 , c1 = 1 , a1 = z(f + f ), s I2 (s2 , s2 ) I2 (s2 , s2 ) z, , 2 , b = z , s = , a2 = z 2 2 2 1 + zf 1 + zf 1 + zf , 2π ν2 (dθ) I2 (z , z ) = iθ − z )(eiθ − z ) (e 0

(5.2) (5.3)

30

VLADIMIR VASILCHUK

and a3 , b3 can be obtained from a2 and b2 by replacing subindexes 2 by 1 in above formulas. For any 0 < d0 1/4 consider the domain D(d0 ) = {z ∈ C : |z| d0 }.

(5.4)

By using (2.14) and (2.26), we obtain for z ∈ D(d0 ) that , I1,2 (s , s ) 4 s 1/2, 1,2 1,2 1,2 and, hence, ar = o(1),

r = 1, 2, 3,

b2,3 = o(1),

z → 0.

Thus, the determinant 1+ a1 − b2 b3 + b1 a2 + c1 a3 − a1 b2 b3 + b1 b3 a3 + c1 a2 b3 of the system (5.1) is equal asymptotically to 1. We conclude that if d0 in (5.4) is small enough, then system (5.1) has only trivial solution, i.e. system (2.25) is uniquely soluble. ✷ The proof of Theorem 5.1 coincides with the proof of Theorem 3.1 modulo to the substitution of matrices Sn and Tn instead of matrices An and Bn (and V1 and V2 instead of H1 and H2 correspondently). The proof for the unitary case is much simpler, e.g. the bounds analogous to (3.49) and (3.50) will be follows (and also uniform in n) 1 , |fn (z)| 1 − |z| z2,n (z) 1 + zf (z) 1/2, n

|2,n (z)|

1 , 1 − |z|

|z| < 1/4.

Thus, for |z| < 1/4 the matrix T − z2,n(z)(1 + zfn (z))−1 I will be invertible uniformly in n and we need not require the first moments of measures νr,n , r = 1, 2 to be nonzero. The proof is also based on the following properties of variances. THEOREM 5.2 ([7]). Let Vn be the random matrix of the form (2.1) satisfying the condition of Theorem 5.1. Denote gn (z) = n−1 Tr (V − z)−1 ,

δr,n (z) = n−1 Tr Vr (V − z)−1 ,

r = 1, 2. (5.5)

Then there exist d0 and C(d0 ), both independent of n and such that the variances of random variables (5.5) admit the bounds for |z| d0 < 1 C(d0 ) , n2 C(d0 ) = |δr,n (z) − δr,n (z)|2 , n2

v1 = |(gn (z) − gn (z))|2 v1+r

(5.6) r = 1, 2.

(5.7)


31

Proof. The proof follows the scheme of the proof of Theorem 3.2, i.e. we derive and analyze the system of inequalities (3.62) (see [7] for more details). ✷ In addition, using the following relation between Stieltjes and Herglotz transforms h(z) − 1 , f (z) = 2z one can easily establish that (2.25) and (2.22) are equivalent. Now we are going to prove Theorem 2.2. According to Theorem 5.2, the randomness of Un in (2.1) (or (2.17)) an already provides vanishing variance of the gn (z) and, hence, it also provides vanishing of the variance of the Herglotz transform of the NCM (2.3) (see (5.6)). Thus we have only to prove that the additional randomness due to the matrices Sn and Tn in (2.1) does not destroy this property. We use the following analog of Proposition 4.1. PROPOSITION 5.2. Let {µn } be a sequence of random measures on the unit circle defined on a certain probability space and {hn } be the sequence of their Herglotz transforms, then (i) the sequence {µn } converges in probability to a nonrandom probability measure µ on the unit circle if and only if the sequence {hn } converges in probability for any fixed z belonging to a certain compact K ⊂ {z ∈ C | |z| < 1} to the Herglotz transform h of the measure µ; (ii) if {µn } is a sequence of random nonnegative measures on the unit circle converging weakly in probability to a nonrandom nonnegative measure µ and if hn (z) are the Herglotz transforms of µn and h(z) is the Herglotz transform of µ then the functions h(z) − 1 hn (z) − 1 , p(z) = pn (z) = 2z 2z are related as follows: lim E sup |pn (z) − p(z)| = 0 n→∞

z∈K

for any compact of {z ∈ C | |z| < 1}. This proposition can be easily proved by repeating the proof of Proposition 4.1 from [5]. Proof of Theorem 2.2. In view of Proposition 5.2, it suffices to show that lim E{|qn (z) − h(z)|} = 2|z| lim E{|gn (z) − f (z)|} = 0

n→∞

n→∞

for any z belonging to a certain compact D(d0 ) of {z ∈ C : |z| < 1}, where f (z) = (h(z) − 1)/2z and 2π iµ e +z νn (dµ), |z| < 1. qn (z) = eiµ − z 0

32

VLADIMIR VASILCHUK

The results obtained in Theorems 5.1 and 5.2 are valid in our case for any fixed realization of random matrices Sn and Tn . In addition, all n-independent estimating quantities entering various bounds in the proofs of these theorems and depending on d0 will depend now also on d0 but not on any particular realization of random matrices Sn and Tn . We will denote below all these quantities simply by the unique symbol C that may have different value in different formulas. In particular, denoting as above by · the expectation with respect to the Haar measure and using (5.6), we can write that 1/2

E{|gn (z) − gn (z)|} E{|v1 (z)|}

C . n2

Thus, it suffices to show lim E{|gn (z) − f (z)|} = 0,

n→∞

z ∈ D(d0 ),

(5.8)

where d0 is small enough. Introduce the quantities γn (z) = (gn (z) − f (z)),

γr,n (z) = δr,n (z) − r (z),

r = 1, 2. (5.9)

The first equation of system (2.25) and (3.31) leads to the identity γn (z)(1 + z(gn (z) + f (z))) = 2 (z)γ1,n(z) + δ1,n (z)γ2,n(z) + r1,n (z).

(5.10)

By using the second equation of system (2.25), we can write the identity γ1,n (z) = f2 (t2,n (z)) − f2 (t2 (z)) + ε1,n (z),

(5.11)

where 2π h2 (z) − 1 ν2 (dθ) = , iθ − z 2z e 0 ε1,n (z) = δ1,n (z) − f2 (t2,n (z)), zδ2,n (z) z2 (z) , t2 (y) = . t2,n (z) = 1 + zgn (z) 1 + zf (z)

f2 (z) =

(5.12) (5.13) (5.14)

We have E{|ε1,n (y)|} E{|δ1,n (z) − g2,n (t2,n (z))|} + +E{|g2,n (t2,n (z)) − f2 (t2,n (z))|}.

(5.15)

The analogs of (3.53), and (3.54) in our case are r2,n (z), δ1,n (z) = g2,n (zδ2,n (z)(1 + zgn (z))−1 ) +

(5.16)

33


where g2,n (z) is random function defined as follows (cf. (3.54)) g2,n (z) = n−1 Tr G2 (z) =

2π

0

ν2,n(dθ) , eiθ − z

−1 z ◦ n Tr Y1 P −1 δ2,n (z) − r2,n (z) = 1 + zgn (z) −n−1 Tr U G(z)SU T U ∗ P −1 gn◦ (z) , P −1 = G2 (t2,n (z)), and gn◦ (z) = gn (z) − gn (z),

◦ δr,n (z) = δr,n (z) − δr,n (z),

r = 1, 2 (5.17)

are the respective random variables centralized by the partial expectations with respect to the Haar measure. In addition, we have the analog of (3.58) | r2,n (z)|

C . n

This leads to the following bound for the first term in the right-hand side of (5.15): r2,n (iy)|} E{|δ1,n (z) − g2,n (t2,n (z))|} E{|

C . n

The universal bounds |gn (z)|

1 , 1 − |z|

|δr,n |

1 , 1 − |z|

r = 1, 2

are valid on all realizations of random matrices Sn and Tn which imply that for z ∈ D(d0 ), d0 1/4 |tr,n (z)| 1/2,

r = 1, 2.

(5.18)

Thus E{|g2,n (t2,n (z)) − f2 (t2,n (z))|} sup E{|g2,n(ζ ) − f2 (ζ )|}.

(5.19)

|ζ |1/2

The right-hand side of this inequality tends to zero as n → ∞ in view of hypothesis of the Theorem 2.2 and Proposition 5.2. Thus we have limn→∞ E{|ε1,n(z)|} = 0 for all z ∈ D(d0 ). Analogous arguments show that limn→∞ E{|ε2,n (z)|} = 0, where ε2,n (z) is defined from (5.13) and from (5.14) by interchanging the indices 1 and 2. Thus we have lim E{|εr,n (z)|} = 0,

n→∞

r = 1, 2.

(5.20)

34

VLADIMIR VASILCHUK

Consider now the first term in the right-hand side of (5.11). In view of (5.12) we can write this term in the form f2 (t1,n (z)) − f2 (t1 (z)) z zt1,n (z) I2 γ n + I2 γ2,n =− 1 + zf (z) 1 + zf (z) = −a2 γn + b2 γ2,n ,

(5.21)

where I2 , a2 and b2 are defined by formulas (5.3) and (5.3), in which we have to replace 2 , 2 , f and f by δ2,n , 2 , gn and f , respectively. Denote by . = {.ij }3i,j =1 the matrix defined by the left-hand side of system (5.1) and by H = {Hi }3i=1 the vector with components H1 = γn , H2 = γ1,n, H3 = γ2,n . Thus, the rest of the proof of the theorem corresponds line by line to the proof of Lemma 4.1 after (4.19). ✷

6. Appendix PROPOSITION 6.1. Let Mr , r = 1, 2 be nonrandom n × n matrices and Un unitary random matrix uniformly distributed over the unitary group U(n) with respect to the Haar measure. Then (i) n−1 Tr M1 Un∗ M2 Un = n−1 Tr M1 n−1 Tr M2 ; (ii) n−1 Tr (M1 Un∗ M2 Un )2 (1 − n−2 ) 2 2 = n−1 Tr M1 n−1 Tr M22 + n−1 Tr M12 n−1 Tr M2 − 2 2 − n−1 Tr M1 n−1 Tr M2 − − n−3 Tr M12 n−1 Tr M22 ,

(6.1)

(6.2)

where · denotes the average over the unitary group U(n). Proof. (i) Using Proposition 3.2 with .(M2 ) = (M1 U ∗ M2 U )ab we obtain (M1 [X, U ∗ M2 U ])ab = 0. Choosing the matrix X with only (a, b)th nonzero entry and applying the operation n−2 na,b=1 to this relation, we get (6.1). (ii) On the other hand, using Proposition 3.2 with .(M2 ) = (M1 U ∗ M2 U M1 U ∗ M2 U )ab , after the same procedure, we obtain n−1 Tr (M1 Un∗ M2 Un )2 = n−1 Tr M1 n−1 Tr (M1 U ∗ M22 U )+ + n−1 Tr (M12 U ∗ M2 U )n−1 Tr M2 −

− (n−1 Tr (M1 U ∗ M2 U ))2 .

(6.3)

35


Besides, using Proposition 3.2 with .(M2 ) = (M1 U ∗ M2 U )cc (M1 U ∗ M2 U )ab , we obtain (M1 [X, U ∗ M2 U ])cc (M1 U ∗ M2 U )ab + +(M1 U ∗ M2 U )cc (M1 [X, U ∗ M2 U ])ab = 0. Choosing the matrix X with only (a, b)th nonzero entry and applying the operation n−3 na,b,c=1 to this relation, we get (n−1 Tr (M1 U ∗ M2 U ))2 = (n−1 Tr M1 )2 (n−1 Tr M2 )2 + +n−2 n−1 Tr M12 Un∗ M22 Un −

−n−2 n−1 Tr (M1 Un∗ M2 Un )2 . Substituting the relation above in (6.3) and using (6.1) we obtain (6.2).

✷

REMARK 6.1. In this paper we deal with unitary and Hermitian matrices, i.e. we assume that the matrices Sn , Tn , Un and Wn in (2.1) are unitary and An and Bn in (2.2) are Hermitian. It is natural also to consider the case of orthogonal Sn , Tn , Un and Wn and real symmetric An and Bn . This case can be handled by using the analogue of formula (3.13) of the orthogonal group O(n). Indeed, it is easy to see that this analog has the form . (O MO) · [X, O MO] dO = 0, O(n)

where O is the transpose of O and X is a real antisymmetric matrix. By using this formula we obtain instead of (3.25) ˜ 1 )ab (2 G) ˜ bb − (GH ˜ ab ˜ 1 )aa (H2 G) (GH ˜ 1 H2 )ab G ˜ bb − (GH ˜ ab . ˜ 1 H2 )aa G = (GH The second terms in both sides of this formula give two additional terms in (3.32) ˜ 1 ) H2 G ˜ 1 H2 ) G. ˜ − n−2 Tr (GH ˜ n−2 Tr (GH These terms, however, produce the asymptotically vanishing contribution to the remainder (3.32), because, in view of (3.5), (3.8) and (3.9) we have in the case of real symmetric An and Bn −2 n Tr (GH ˜ 1 ) H2 G ˜ 1 H2 ) G ˜ − n−2 Tr (GH ˜ 2 m2/4 ny02 4 and in the case of orthogonal Sn and Tn −2 n Tr (GV1 ) V2 G − n−2 Tr (GV1 V2 ) G

2 . n(1 − d0 )2

36

VLADIMIR VASILCHUK

Similar, and also negligible as n → ∞ terms appear in formulas (3.55), (3.41), (3.67) and (3.72) of the proof of Theorems 3.1 and 3.2. As a result, in this case we obtain the same system (2.25), defining the Herglotz transform of the limiting eigenvalue counting measure of the analogue of (2.1) with the orthogonal Sn and Tn and orthogonal Haar-distributed Un and Wn and the same system (2.22), defining the Stieltjes transform of the limiting eigenvalue counting measure of the analogue of (2.2) with the real symmetric An and Bn and orthogonal Haar-distributed Un and Wn .

Acknowledgements I am thankful to Prof. Anne Boutet de Monvel, Prof. A. Khorunzhy and Prof. L. Pastur for numerous helpful discussions. I also thanks the Ministère des Affaires Etrangères de France for financial support. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Akhiezer, N. I. and Glazman, I. M.: Theory of Linear Operators in Hilbert Space, Frederick Ungar Publishing Co., New York, 1963. Bercovici, H. and Voiculescu, D. V.: Free convolution of measures with unbounded support, Indiana Univ. Math. J. 42 (1993), 733–773. Janik, R. A., Nowak, M. A., Papp, G., Wambach, J., and Zahed, I.: Nonhermitian random matrix models: a free random variable approach, Phys. Rev. E 55 (1997), 4100–4106. Marchenko, V. A. and Pastur, L. A.: Distribution of eigenvalues for some sets of random matrices, Mat. Sb. (N.S.) 72 (114) (1967), 507–536 (Russian). Pastur, L. and Vasilchuk, V.: On the law of addition of random matrices, Comm. Math. Phys. 214 (2000), 249–286. Speicher, R.: Free convolution and the random sum of matrices, Publ. Res. Inst. Math. Sci. 29 (1993), 731–744. Vasilchuk, V.: On the law of multiplication of unitary random matrices, Mat. Fiz. Anal. Geom. 7 (2000), 266–283 (Russian). Voiculescu, D. V.: Limit laws for random matrices and free products, Invent. Math. 104 (1991), 201–220. Voiculescu, D.: A strengthened asymptotic freeness result for random matrices with applications to free entropy, Internat. Math. Res. Notices 1998, No. 1, 41–62. Voiculescu, D. V., Dykema, K. J., and Nica, A.: Free Random Variables, CRM Monograph Series, Amer. Math. Soc., Providence, RI, 1992.


37

Smoothing Properties of the Heat Semigroups Associated to Hamiltonians Describing Point Interactions in One and Two Dimensions A. BEN AMOR and PH. BLANCHARD Universität Bielefeld, Fakultät für Physik, D-33615, Bielefeld, Germany (Received: 30 June 2000; in final form: 12 April 2001) Abstract. Smoothing properties of the heat semigroups associated to Hamiltonians describing point interactions in one and two dimensions are investigated. A construction of Hamiltonians describing point interaction on Lp -spaces is then derived and a full description of their spectra is given. Particularly, we prove the p-independence of their spectra and the exponential growth of the p-norms of such semigroups for large time. Mathematics Subject Classifications (2000): primary 79-XX, secondary 81Q10. Key words: heat semigroup, point interaction, smoothing property, spectrum.

1. Introduction By the heat semigroup we mean the semigroup associated to the equation ∂ψ = −H ψ, ∂t

(1)

where H is a selfadjoint operator and we will denote it by exp(−tH ). The most important case in applications is when H is a generalized Schrödinger operator on Rd , i.e. a perturbation of minus the Laplacian by a suitable measure (for example, in the generalized Kato class) and especially by a potential. Constructions and investigations of the heat semigroups exp(−tH ), where H is a Schrödinger operator, is subject of extensive literature, rich in its methods (analytic, probabilistic, or related to operator theory) and contents. In a survey paper, B. Simon [19] proved the Lp smoothing property of exp(−tH ) with H = −+V for a large class of potentials: If the negative part of the potential is in the Kato class and its positive part is in the local Kato class, then for every t > 0, exp(−tH ) maps continuously Lp into Lq , 1 p q ∞. In the same paper, Lp -growth of these semigroups is also studied. For instance, it is proved that for the class of potentials described above, exp(−tH )p,p has an exponential growth for large t, namely exp(−tH )p,p C exp(αt)

(2)

38

A. BEN AMOR AND PH. BLANCHARD

for large t, where C is a positive constant and α ∈ R. Let us emphasize that for d = 1, 2 and V < 0, and due to the existence of negative eigenvalues of H , one has α > 0. Among other interesting features it is proved in [19] that the spectral bound, inf σ (H ) = − lim t −1 ln exp(−tH )p,p t →+∞

is p-independent. This result is now a well-known fact. For instance, R. Hempel and J. Voigt [15, 16] proved that for certain potentials (including those in the Kato class) the spectrum of H is p-independent as well. A more detailed answer to the question asked by B. Simon about p-independence of spectra of Schrödinger operators can be found in [16]. Later on, Ph. Blanchard and Z. M. Ma [4], proved also the Lp -smoothing property for the heat semigroups associated to H = −+µ on Rd (d 3), where µ is a signed measure whose positive part is smooth and negative part is in the generalized Kato class. In an abstract setting, P. Stollmann and J. Voigt [20] investigated the properties of exp(−tHµ ), Hµ = − + µ on the space L2 (X, m), where m is a measure whose support is X. Recently (cf. [13] and references therein), a particular interest was showed for the study of smoothing properties of exp(−tH ), where H = − 12 +V on the scale of Bessel potential spaces on Rd . In [14] (Theorem 1.1), the authors proved that for V ∈ Kd,loc with negative part V − ∈ Kd , the boundedness of exp(−tH ) from p,s+2 p,s Lp into Lloc is equivalent to V ∈ Lloc , while A. Gulisashvili [13] gave a sharp p,s+2 estimate of the Lp − Lloc norm of exp(−tH ) for V ∈ Lp,s ∩ Kd , where Kd is the Kato class of potentials and Kd,loc the local Kato class of potentials. Hamiltonians describing point interactions, or Schrödinger operators with point interactions, are operators corresponding to perturbations of minus the Laplacian by the linear combination of Dirac measures. The mathematical setting and the description of such Hamiltonians are now quite well-known [3] and a new approach to handling them is given in [5]. It is known [3] that for d = 2, 3, there is only one kind of point interaction that can be denoted by Hα = − + αδ. In contrast, in one dimension there are more possibilities of point interactions. However, we will concentrate here on a one-parameter family corresponding to the δ -interaction [3], which we shall also denote by Hα , −∞ < α +∞. This family is related to selfadjoint extensions of the operator − with its domain the Sobolev space H02 (R \ {0}) and determined by the boundary condition: ϕ (0+ ) = ϕ (0− ),

ϕ(0+ ) − ϕ(0− ) = αϕ (0),

(3)

for every ϕ in the domain of Hα . Hence, up to now operators describing point interactions in Rd do not fit into a standard situation and, in any case, the theory developed in [20] does not include such operators. However, we are saved by the explicit knowledge of the kernels associated to Hα [1]. Our aim in this paper is to use these formulas to establish p, q smoothing properties of Hα on Lp (Rd )

SMOOTHING PROPERTIES OF HEAT SEMIGROUPS

39

for d = 1, 2. As a consequence, we get a construction of the operators Hα on Lp spaces for 1 p < +∞ which we denote by Hα,p (for p = 2 we omit the subscript p). Then combining the techniques used by J. Voigt and P. Stollmann [15, 16] and the new functional calculus developed by E. B. Davies [8], we prove the pindependence of their spectra and derive the exponential growth of the p-norm of exp(−tHα,p ). Such a construction was done by S. Albeverio et al. [2] using the ‘family of pseudo-resolvent’, thereby exploiting the expression of the resolvent kernel of Hα . They conclude the construction for d = 1, p ∈ [1, +∞) or d = 2, p ∈ ]1, +∞[ or d = 3, p ∈ ] 32 , 3[ and the same thing for the C0 -semigroup exp(−tHα ). We will use here the reversed strategy: Using the explicit formula of the heat kernel associated to Hα (cf. [1]) which we denote, as in [1], by P α (t; x, y), we prove that for d = 1, 2, this kernel also defines a bounded linear operator from Lp into Lq for 1 p q +∞. This is the well-known ‘smoothing property’. Moreover, this kernel even defines a strongly continuous semigroup on Lp for 1 p < +∞. Then we denote by −Hα,p the generator of exp(−tHα ) on Lp (1 p < +∞). Using the integral representation of the resolvent function [9], p. 55 (which is the Laplace transform of the semigroup), we get that for k 2 such that Im(k) > 0 and Re(k 2 ) < min(s(Hα ), s(Hα,p )), (Hα,p − k 2 )−1 is a kernel operator whose kernel is Gk , where Gk is the kernel of (Hα − k 2 )−1 . Hence, for d = 1, 2, the construction we propose in this paper includes the one made in [2]. A natural question arises: why is it so? This is related to the properties of the heat kernel. For instance, the heat kernel has better properties than the resolvent kernel. A good example is a comparison between the resolvent and the heat kernel of −. Unfortunately, this method does not work in three dimensions for the reason that we will explain at the end of the paper. For the notations we will adopt those used in [3]. So Hα is the Hamiltoniandescribing interaction placed at the origin in the space L2 , which corresponds in one dimension to the δ -interaction. We shall denote the free Hamiltonian by H0 . The space Lp (Rd ) is denoted simply by Lp for every 1 p +∞. For every linear closed operator T , the spectrum of T is denoted by σ (T ), whereas its spectral bound is denoted by s(T ) and is defined by [9] s(T ) = inf{Re(λ), λ ∈ σ (T )}.

(4)

2. Smoothing Property in Two Dimensions Following the notation of [1], we denote the heat kernel of exp(−tHα ) by P α (t; x, y) for every t > 0. It is given by [1] +∞ 1 e(−αu) α × t u−1 P (t; x, y) = P (t; x, y) + 2π 0 $(u) +∞ |x|2 +|y|2 |x||y| z dz du, (5) (z − 1)u−1 z−u e(−z 4t ) K0 × 2t 1

40


where

1 |x − y|2 P (t; x, y) = exp − 4π t 4t

is the kernel of the free Hamiltonian and K0 is the MacDonald function. We will also denote by P˜α (t; x, y) the difference P α (t; x, y) − P (t; x, y). A way to prove the smoothing property of exp(−tHα ) has been developed in [19]: first prove the boundedness from L∞ into L∞ and the boundedness from L1 into L∞ and then use the Riesz–Thorin convexity theorem [18] to conclude. The first step is given by this lemma. LEMMA 2.1. For every t > 0, we have P α (t; x, y) dy < +∞. sup x∈R2

(6)

R2

Proof. Since P (t; x, y) dy = 1, sup x∈R2

R2

we just have to prove that P˜α (t; x, y) dy < +∞. sup x∈R2

(7)

R2

Set

I (x) =

and

P˜α (t; x, y) dy R2

J (x) =

2 +|y|2 4t

(−z |x|

e R2

Then 1 I (x) = 2π

+∞

t 0

u−1

)

K0

e−αu $(u)

|x||y| z dy. 2t

+∞

(z − 1)u−1 z−u J (x) dz du.

1

Let us make a suitable estimate for J (x). Using polar coordinates, we get +∞ |x|2 r2 r|x| r dr. e(−z 4t ) K0 z J (x) = 2π e(−z 4t ) 2t 0 With the change of variable s = z(r|x|/2t) one get t 2 2t 2 +∞ (− 2z|x| 2s ) e sK0 (s) ds J (x) = 2π z|x| 0

(8)

41


which is equal to (cf. [12], p. 717, 3) 1 2 1 z|x|2 z|x|2 2 ( z|x|2 ) (−z |x| ) 4t 8t , e W− 1 ,0 2π e 2 2 t 4t

(9)

where Wχ,µ is the Whittaker function. On the other hand, one has (cf. [17], p. 305) 1 z W− 1 ,0 (z) = z 2 e( 2 ) which gives 2

2π t J (x) = √ . 2z Thus

+∞ −αu +∞ 1 ue t (z − 1)u−1 z−u−1 dz du sup I (x) = √ $(u) 2 2 0 1 x∈R 1 = √ ν(te−α ), 2

(10)

where the function ν is defined by (cf. [10]): +∞ xs ds. ν(x) = $(s + 1) 0 From this follows PROPOSITION 2.1. For every t > 0 and every p such that 1 p +∞ the operator exp(−tHα ): Lp → Lp

(11)

is bounded and defines a strongly continuous semigroup for 1 p < +∞. Proof. First let us denote by C(t) = 1 + √12 ν(te−α ). For 1 < p +∞, let q be the conjugate of p: p −1 + q −1 = 1. Then, by Hölder inequality, we have for every f ∈ Lp p1 p p α p q |exp(−tHα )f (x)| (C(t)) P (t; x, y)|f (y)| dy , (12) R2

thereby

p exp(−tHα f (x) dx R2 p p q (C(t)) |f (y)| R2

α

P (t; x, y) dx dy.

Now using the symmetry property of the heat kernel, we get p p |exp(−tHα )f (x)| dx (C(t)) |f |p dy. R2

(13)

R2

R2

(14)

42


To prove that it defines a strongly continuous semigroup, let us denote by S(t) the operators whose kernel is P˜ α , then we have S(t)p,p √12 ν(te−α ). By the properties of the function ν [10], p. 219, we have limt →0 ν(t) = 0, we then get lim exp(−tHα ) = I.

(15)

t →0

Now defining Tα (t) by Tα (t) = exp(−tHα ) for t > 0 and Tα (0) = I , we get a strongly continuous semigroup which we shall denote by exp(−tHα ). For p = 1, the proof is straightforward. ✷ We are now in a position to prove boundedness from L1 into L∞ . LEMMA 2.2. For every t > 0 we have sup P α (t; x, y) < +∞.

(16)

x,y∈R2

Proof. To achieve this goal, we are going to use the construction of the Hamiltonian Hα via Dirichlet forms as done in [3] which we recall here. We omit the case α = ∞, which corresponds to the free Hamiltonian. Let ϕα be the following function ϕα (x) = H0(1)(2ie−2πα+0(1) |x|),

x ∈ R2 \ {0},

(17)

where H0(1) is the Hankel function. Denote by Hϕα the operator associated to the local positive Dirichlet form ¯ α2 dx. ∇f ∇gϕ (18) Eϕα : D(Eϕα ) ⊂ L2 (ϕα2 ), Eϕα (f, g) = R2

Then the Hamiltonian Hα is related to this Dirichlet form by [3] Hα = ϕα [Hϕα − βI ]ϕα −1 ,

(19)

where β = 4e2(−2πα+0(1)) . Clearly exp(−tHα ) = eβt ϕα exp(−tHϕα )ϕα −1 . Now if we denote by q α (t; x, y) the heat kernel of the operator exp(−tHϕα ), then P α (t; x, y) = eβt ϕα (x)ϕα −1 (y)q α (t; x, y). Now since the Dirichlet form Eϕα is local and positive, then the kernel q α (t; x, y) is Markovian [11], hence 0 < q α (t; x, y) 1. Now we have on the diagonal the following estimate: P α (t; x, x) eβt .

(20)

Using the Chapman–Kolmogorov equation α P α (s; x, z)P α (t; z, y) dz, P (t + s; x, y) =

(21)

R2

we get P α (t; x, y) eβt which completes the proof. We now formulate the following theorem:

✷


43

THEOREM 2.1. For every t > 0 and every 1 p q ∞, the operator exp(−tHα ): Lp → Lq

(22)

is bounded. Theorem 2.1 gives a variety of properties of the heat semigroup exp(−tHα ) known for a large class of Schrödinger operators, at least for operators H = − + V where V is in the Kato class. For every 1 p < +∞, let us denote by −Hα,p the generator of exp(−tHα ) on Lp which will be denoted exp(−tHα,p ). For p = 2, we omit the index p. Interpretation of point interaction on Lp spaces as an extension of − on D0 = {f ∈ C0∞ (Rd ), f (0) = 0} was done in [6]. There the authors show that point interaction can be defined on Lp for d = 1, 2 and 1 < p < +∞ or d = 3 and 32 < p < 3 as negative generators of analytic semigroups. Their construction is essentially based on estimations of the resolvent of Hα with α = 1. We here emphasize that for d = 1, 2 we have fewer restrictions on p than in [2] or in [6]. Now the question of the p-independence of the spectra of Hα arises. Let us note that for p = +∞ one cannot hope to get the inclusion σ (Hα ) ⊂ σ (Hα,p ). For the eigenfuction of Hα which is equal to i (1) H (2ieβ |x|), x = 0, 4 0 where β = −2π α + 0(1), is unbounded.

(23)

PROPOSITION 2.2. For every t > 0 and p such that 1 p < +∞ we have (i) The spectrum of Hα,p is p-independent. (ii) Every isolated eigenvalue of Hα of algebraic multiplicity m is an isolated eigenvalue of Hα,p with the same multiplicity and conversely. (iii) The spectral bound of Hα,p satisfies − lim t −1 ln exp(−tHα,p )p,p = −4 exp(2(2α + 0(1))). t →∞

(24)

Proof. The proof of assertion (ii) is as in [15], (iii) follows from (i) and the characterization of the spectral bound of generators associated to strongly continuous semigroups [9], p. 299. So the important point is to prove (i). Following R. Hempel and J. Voigt [15], we are going to prove first that σ (Hα ) ⊂ σ (Hα,p ).

(25)

A crucial argument to prove (25) is that, for every 1 p q < +∞ and every t > 0, the operator exp(−tHα,p ) maps continuously Lp into Lq , for exp(−tHα,p ) has the same kernel as exp(−tHα ). Once this is observed, one can continue the proof of (25) as in [20]. Let us prove now the reversed inclusion. We may suppose that 2 p +∞, and then conclude by duality. Given ξ ∈ ρ(Hα ) and f ∈ Lp ∩ L2 , then in fact

44


exp(−tHα )f = exp(−tHα,p )f , now by the integral representation of the resolvent function [9], we have for every ξ such that Re(ξ ) < min(s(Hα ), s(Hα,p )), Rξ (Hα )f = Rξ (Hα,p )f which implies that Hα f = Hα,p f . Since ρ(Hα ) is connected, we get, for every ξ ∈ ρ(Hα ) and every f ∈ Lp ∩ L2 , (Hα − ξ I )−1 (Hα,p − ξ I )f = f.

(26)

It is now sufficient to prove that T (ξ ) = (Hα − ξ I )−1 is bounded as an operator on Lp . Indeed, from the kernel formula of (Hα − ξ I )−1 [3], we get that T (ξ ) is a closed operator in L∞ whose domain is the whole space L∞ , hence by the Banach theorem we conclude that T (ξ ) is bounded on L∞ . On the other hand, it is bounded on L2 , thus by the Riesz–Thorin convexity theorem, we get the boundedness of ✷ T (ξ ) on Lp for every 2 p +∞ which implies the result. Remark 2.1. From Proposition (2.2), we get exp(−tHα,p )p,p ∼ exp(4t exp(2α + 0(1))),

(27)

for large t which expresses the exponential growth of the p-norm of the operator exp(−tHα,p ), while for small t we have exp(−tHα,p )p,p 1 +

C . |log(t)|

(28)

In [4] it is proved that for 1 p < +∞ and every f ∈ Lp the function exp(−tH ) tends to zero at infinity. The same phenomenon occurs in our situation. PROPOSITION 2.3. For 1 p < +∞ and f ∈ Lp we have lim |exp(−tHα,p )f (x)| = 0.

(29)

|x|→∞

Proof. We give the proof for p = 2, for p = 2, the proof is more or less the same. The proof is based on the explicit formula of the heat kernel [1], Equation (3.16), namely,

A

P (t; x, y) = P (t; x, y) +

e− t

α

+∞

×

r

(4π t|x||y|) u−1 1

0

(r + 1)u+ 2

where (|x| + |y|)2 A= 4

A 0 e− t r K

and

0 (z) = K

1 2

0

+∞ u −αu

t e × $(u)

|x||y| (r + 1) dr du, 2t

2z exp(z)K0 (z). π

(30)

45


Let us recall that [1] 0 (r)| = M < +∞. sup |K

(31)

r0

Taking into account that lim exp(−tH0 )f (x) = 0,

|x|→+∞

we should just prove that +∞ u −αu +∞ t e r u−1 1 × 1 1 $(u) 0 (4π t|x| 2 ) 0 (r + 1)u+ 2 A e− t A 0 |x||y| (r + 1) f (y) dy dr du e− t r K × 2t R2 (4π t|y|)

(32)

tends to zero for |x| → ∞. We denote by A(x) the last term in Equation (32), then 1 − (|x|+|y|)2 4t e |f (y)| dy A(x) M 1 R2 |y| 2 for which, by Hölder inequality, we obtain 12 +∞ 1 s2 − e 2t ds . A(x) 2π 2 Mf L2

(33)

0

This yields |exp(−tHα,p )f (x)| |exp(−tH0 )f (x)| + C

ν(te−α ) 1

|x| 2

f L2 ,

where C > 0, and this completes the proof.

(34) ✷

3. Smoothing Property in One Dimension In one dimension, the Hamiltonian Hα corresponds for α = 0 to the free Hamiltonian, so in this section we will omit the case α = 0. For α = 0, the heat kernel of Hα is given by [1] (Formula 3.4), sgn(xy) − (|x|+|y|)2 4t e + P α (t; x, y) = P (t; x, y) + √ 4π t sgn(xy) +∞ − 2 u − (|x|+|y|+u)2 4t e α e du, +2 √ α 4π t 0 where P (t; x, y) = √

1 4π t

e−

|x−y|2 4t

(35)

46


is the heat kernel of the free one-dimensional Hamiltonian. The arguments used to prove the p, q-smoothing property of the heat semigroup in one dimension are quite similar to those used in the previous section. Indeed, we have the following easy lemma: LEMMA 3.1. For every t > 0, we have sup |P α (t; x, y)| < +∞. x∈R

Proof. A direct computation shows that, for every α > 0, we have √ sup |P α (t; x, y)| C1 (t) = 2 + 2 4π t. x∈R

(36)

R

(37)

R

While for α < 0, the following estimate holds true: 2 √ 4t α π t exp 2 . sup |P (t; x, y)| 2 + |α| α x∈R R

(38) ✷

LEMMA 3.2. For every t > 0, we have 1 C1 + C2 t − 2 ; α > 0, α sup |P (t; x, y)| 4t − 12 C1 + C2 t + exp( α 2 ); α < 0, x,y∈R with positive constants Cj and Cj . Now using Lemmas 3.1, 3.2, and the Riesz–Thorin theorem, one can easily establish the following theorem: THEOREM 3.1. For every t > 0 and every 1 p q ∞, the operator exp(−tHα ): Lp → Lq

(39)

is bounded and, for p = q < +∞, it defines a strongly continuous semigroup. Now, as in the first section, we denote by −Hα,p the generator of the operator exp(−tHα ) on the space Lp and for p = 2 we will omit the subscript p. The spectral properties of the operators Hα,p can be easily investigated using the smoothing properties of their semigroups. PROPOSITION 3.1. For every 1 p < +∞ we have (i) σ (Hα,p ) is p-independent. (ii) Every isolated eigenvalue of Hα of algebraic multiplicity m is an isolated eigenvalue of Hα,p with the same multiplicity 4 and, conversely. ; α < 0, (iii) lim t −1 ln exp(−tHα,p )p,p = α 2 t →+∞ 0; α > 0.


47

We shall prove only assertion (i). To this aim we use a new method which relies on the new functional calculus introduced by E. B. Davies [8]. The application of this calculus requires an estimate for the resolvent function and the the spectra of Hα,p must be real. Thanks to the explicit formula of the kernel of the operator (Hα − k 2 )−1 , one can prove a suitable estimate for the resolvent functions. Let us first recall that for k 2 ∈ ρ(Hα ), Im k > 0 the kernel of (Hα − k 2 )−1 is given by [3], p. 92, i α sgn(xy) ik(|x|+|y|) e . (40) Gk (x, y) = eik|x−y| + 2k 2(−ikα + 2) It is obvious that Gk (x, y) is also the kernel of (Hα,p −k 2 )−1 for every k 2 ∈ ρ(Hα )∩ ρ(Hα,p ) with Im(k) > 0. LEMMA 3.3. For every k 2 ∈ ρ(Hα ) such that Im k > 0, we have 4 . sup |Gk (x, y)| dy |Im(k 2 )| x∈R R

(41)

Proof. We have

−Im(k)(|x|+|y|) 1 −Im(k)|x−y| α e e + , |Gk (x, y)| |2k| 2(−ikα + 2) which yields |α| 1 1 + . |Gk (x, y)| dy |k|Im(k) |(−ikα + 2)| Im(k) R Observing that 1 |k|Im(k) |Re(k)|Im(k) = |Im(k 2 )| 2 and that 2 − ik |Re(k)|, α we get the desired estimate.

(42)

(43)

✷

Using estimate (41) we conclude that for every k 2 ∈ ρ(Hα ) with Im k > 0 the operator whose kernel is Gk defines a bounded operator on Lp for 1 p +∞. This operator is nothing else but (Hα,p − k 2 )−1 , thereby giving that C \ R ⊂ ρ(Hα,p ).

(44)

Thus, σ (Hα,p ) ⊂ R. Now the operators Hα,p satisfies all hypotheses (especially H 1) required by the functional calculus [8]. Proof of Proposition 3.1. Using Lemma 4 in [7] and the fact that for every ξ ∈ ρ(Hα ) ∩ ρ(Hα,p ) and every f ∈ Lp ∩ Lq , we have Rξ (Hα )f = Rξ (Hα,p )f , we get the spectral p-independence. ✷

48


Remark 3.1. From Proposition 3.1(iii), we observe that for α < 0 the Lp norm of exp(−tHα,p ) has an exponential growth for large t, this is, however, similar to the case of perturbations of the Laplacian by negative potentials. While for α > 0, exp(−tHα,p )p,p behaves like exp(−tH )p,p for large t. So that the operator exp(−tHα,p ) in one dimension behaves somewhat different as in two dimensions where the behavior does not depend on the sign of α. We close this section with the following proposition: PROPOSITION 3.2. For 1 p < +∞ and f ∈ Lp we have lim | exp(−tHα,p )f (x)| = 0.

(45)

|x|→∞

Remark 3.2. We here studied only the case of a one-parameter family of point interaction, corresponding to δ -interaction. However, the same method applies for the four-paramater family of self-adjoint extensions. The method we give here does not work for d = 3 for the simple reason that for each α = +∞ we have P α (t; x, y) dy = +∞. sup x∈R3

R3

In fact, for α 0, we have [1] P α (t; x, y)

2t P (t; |x| + |y|), |x||y|

(46)

where P (t; |x| + |y|) =

1 (4π t)

3 2

e−

(|x|+|y|)2 4t

.

Hence,

|x|2 4π 2 t |x| α ˜ D−2 √ e− 2t , P (t; x, y) dy |x| t R3

(47)

where Dν is the cylindrical hypergeometric function [17]. This implies that P˜ α (t; x, y) dy = +∞. lim x→0

R3

Similarly, one can prove the same result for positive α. References 1.

Albeverio, S., Brze´zniak, Z. and D¸abrowski, L.: Fundamental solution of the heat and Schrödinger equations with point interaction, J. Funct. Anal. 130(1) (1995), 220–254.


2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

49

Albeverio, S., Brze´zniak, Z. and D¸abrowski, L.: The heat equation with point interaction in Lp spaces, Integral Equations Operator Theory 21(2) (1995), 127–138. Albeverio, S., Gesztezy, F., Høegh-Krohn, R. and Holden, H.: Solvable Models in Quantum Mechanics, Texts and Monogr. Phys., Springer-Verlag, New York, 1988. Blanchard, Ph. and Ma, Z. M.: Semigroup of Schrödinger operators with potentials given by Radon measures, In: Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988), World Scientific, Teaneck, NJ, 1990, pp. 160–195. Caspers, W. and Clément, P.: A different approach to singular solutions, Differential Integral Equations 7(5–6) (1994), 1227–1240. Caspers, W. and Clément, Ph.: Point interactions in Lp , Semigroup Forum 46(2) (1993), 253– 265. Davies, E. B.: Lp spectral independence and L1 analyticity, J. London Math. Soc. (2) 52(1) (1995), 177–184. Davies, E. B.: The functional calculus, J. London Math. Soc. (2) 52(1) (1995), 166–176. Engel, K. J. and Nagel, R.: One-Parameter Semigroups of Linear Evolution Equations, Springer, New York, 2000. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.: Tables of Integral Transforms, Vol. 3, McGraw-Hill, New York, 1954. Fukushima, M.: Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, 1980. Gradshteyn, I. S. and Ryzhik, I. M.: Table of Integrals, Series, and Products, Academic Press, New York, 1965. Gulisashvili, A.: Sharp estimates in smoothing theorems for Schrödinger semigroups, J. Funct. Anal. 170(1) (2000), 161–187. Gulisashvili, A. and Kon, M. A.: Exact smoothing properties of Schrödinger semigroups, Amer. J. Math. 118(6) (1996), 1215–1248. Hempel, R. and Voigt, J.: The spectrum of a Schrödinger operator in Lp (Rν ) is p-independent, Comm. Math. Phys. 104(2) (1986), 243–250. Hempel, R. and Voigt, J.: On the Lp -spectrum of Schrödinger operators, J. Math. Anal. Appl. 121(1) (1987), 138–159. Magnus, W., Oberhettinger, F. and Soni, R. Pal.: Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, New York, 1966. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis and Self-adjointness, Academic Press, New York, 19xx. Simon, B.: Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7(3) (1982), 447–526. Stollmann, P. and Voigt, J.: Perturbation of Dirichlet forms by measures, Potential Anal. 5(2) (1996), 109–138.


51

On the Tidal Motion Around the Earth Complicated by the Circular Geometry of the Ocean’s Shape Without Coriolis Forces Dedicated to Professor L. V. Ovsyannikov on the occasion of his 80th birthday RANIS N. IBRAGIMOV Department of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada. e-mail: [email protected] (Received: 10 February 2000; in revised form: 29 March 2001) Abstract. The Cauchy–Poisson free boundary problem on the stationary motion of a perfect incompressible fluid circulating around the Earth is considered in this paper. Rotation plays a significant role in the early stages of the formation of solitary waves. However, these effects are less important on the solitary waves once they are formed. Therefore, for simplicity, rotation is not included for these simulations. The main concern is to find the inverse conformal mapping of the unknown free boundary in the hodograph plane onto some fixed mapping in the physical domain. The approximate solution to the problem is derived as the application of such a method. The behaviour of tidal waves around the Earth is discussed. It is shown that one of the features of the positively curved bottom is that the problem admits two different higher-order systems of shallow water equations, while the classical problem for the flat bottom admits only one system. Mathematics Subject Classification (2000): 30C20. Key words: free boundary, inverse conformal mapping.

1. Introduction The Cauchy–Poisson problem on the stationary motion of a perfect fluid which has a free boundary and has a solid bottom represented by a circle with a sufficiently large radius, is considered in this paper. For simplicity, the fluid is not considered in the real, rotating reference frame of the Earth. However, we note that the presence of Coriolis effects do not change the qualitive analysis of the presented method. We have shown in [4] that such a problem can be associated with a two-dimensional model to an oceanic motion around the Earth, since we consider strictly longitudinal flow. Since the problem is a free boundary problem, the analysis is rather difficult. Permanent water waves have been considered in a large number of papers. However, most researchers are concerned with fluid motion which is infinitely deep

52

RANIS N. IBRAGIMOV

and extends infinitely both rightward and leftward (see Crapper [1], Stoker [9] or Stokes [10] for the history). Such problems are usually called Stokes’s problem if the surface tension is neglected and Wilton’s problem if the surface tension is taken into account (see [6] for more details). We consider water waves for which the ratio of the depth of fluid above the circular bottom to the radius of the circle is small (shallow water). Our primary concern is to find the conformal mapping (for the Stokes problem) of the unknown free boundary onto fixed mapping. The resulting Dirichlet problem can be solved numerically using Okamoto’s method [3]. A more detailed structure of the bifurcation of solutions for the related problem was numerically computed by Fujita et al. [3]. The existence of nontrivial solutions for the analogously reduced Dirichlet problem can be found in [4] and [7] as well as in the classical literature (see, e.g., [8] or [9]). Higher-order shallow water equations in the nonstationary case are derived in this paper. It is shown that the present problem admits two different systems of shallow water equations, while the classical problem for the flat bottom admits only one system (see [2]). We note that papers [3, 6, 7] are concerned with fluid whose surface tension is taken into account. In fact, the surface tension plays the role of a ‘regulator’ of the problem which substantially simplifies the analysis. Furthermore, the nature of the problem requires that the surface tension should be neglected. Thus, this paper represents a more systematic approach to the problem. The paper aims to investigate the problem by using a conformal mapping which distinguishes it from [3, 4, 6, 7]. 2. Basic Equations The analysis of this problem is performed in the following notation: R is the radius of the circle, r is a distance from the origin, θ is a polar angle, h0 is the undisturbed level of the liquid above the circle, and h = h (θ) is the level of the disturbance of the free boundary above the circle. For the sake of simplicity, we assume that the pressure is constant on the free boundary. The stream function ψ = ψ(r, θ) defines the velocity vector, i.e., 1 v r = − ψθ , r

v θ = ψr .

Hence, irrotational motion of an ideal incompressible fluid of the constant pressure in the homogeneous gravity field g = const is described by the stream function ψ in the domain h = {(r, θ) : 0 θ 2π, R r R + h0 + h(θ)}, which is bounded by the bottom R = {(r, θ) : r = R, θ ∈ [0, 2π ]} and the free boundary with equation h = {(r, θ) : r = R + h0 + h(θ), θ ∈ [0, 2π ]}. Note that

53

TIDAL MOTION AROUND THE EARTH

ψ is an harmonic function in h , since we assumed that the flow is irrotational. More specifically, we assume that the fluid is incompressible and inviscid and that the flow is stationary. Then the problem is to find the function h (θ) and the stationary, irrotational flow beneath the free boundary r = R + h0 + h (θ) given by the stream function ψ which satisfy the following differential equations:

ψ = 0(in h ),

ψ =0

|∇ψ|2 + 2gh = constant 1 2

2π

ψ =a

(on R ),

(on h ),

(1) (2)

(on h ),

(R + h0 + h(θ))2 dθ = π(R + h0 )2 ,

(3)

0

where a = const denotes the flow rate. Equations (1) to (3) represent the free boundary Cauchy–Poisson problem in which the boundary h is unknown as well as the stream function. 3. The Inverse Transforms Principle 3.1. CONSTANT FLOW The exact solution h≡0

and

ψ = ψ0 = a log r

(4)

of Equations (1)–(3) corresponds to the constant flow with an undisturbed free boundary. The trivial solution (4) represents a flow whose streamlines are concentric circles with the common center at the origin. The following nondimensional quantities are introduced: √ h0 gh0 h0 F = , h = h0 h , ψ = aψ , ε= , r = R + h0 r , R a where F is a Froude number and R is used as a vertical scale. We consider ε as the small parameter of the problem. After dropping the prime, Equations (1)–(3) are written by ψ , h , and (r , θ) as follows: (ε) ψ = 0

(5)

(in h ),

ψ = 0 (on R ),

(6)

ψ = 1 (on h ),

(7)

∇(ε)ψ 2 + 2F −2 h = constant 1 2

0

2π

(8)

(on h ),

(1 + ε + εh(θ))2 dθ = π(1 + ε)2

(on h ).

(9)

54

RANIS N. IBRAGIMOV

Here the Laplace and gradient operators are given by ε∂θ 2 2 , ∂r , (ε) = (ε∂θ ) + [(1 + εr)∂r ] , ∇(ε) = (1 + εr) where the subscripts imply differentiation. We further consider the complex potential ω(ζ ) = ϕ + iψ, where ζ = (1 + εr)eiθ is the independent complex variable and ϕ(ζ ) is the velocity potential which is characterized by the analyticity of ϕ + iψ, i.e., ϕr =

εψθ , (1 + εr)

εϕθ = −ψr . (1 + εr)

We note that the complex velocity dω/dζ is a single-valued analytic function of ζ , although ω is not single-valued. In fact, when we turn around the bottom r = 1 2π once, ϕ increases by − 0 ψr (1, θ)dθ which has a positive sign by the maximum principle (Hopf’s lemma). Hence, if we remove the width of annulus region θ = 0, r ∈ [1, 1 + ε], then at every point (r, θ), ω(ζ ) is single-valued analytic function which maps the rectangular (in the ω(α)-hodograph plane) domain with 2π and ψ ∈ [0, 1] ϕ ∈ 0, − log(1 + ε) as coordinates onto the annulus h0 = {(r, θ) : 1 < r < 1 + ε, θ ∈ [0, 2π ]}. We represent the constant flow (4) by ω(ζ ) = ϕ + iψ =

i log(1 + εr) − θ , log(1 + ε)

(10)

where r = ξ0 (ψ) and θ = η0 (ϕ) transform the rectangular domain in the hodograph plane into h0 . Consequently, each conformal mapping by the function ω(ζ )

Figure 1.


55

between hodograph and physical planes represents an irrotational flow in the physical ζ plane. Furthermore, Equations (10) implies that (11) η0 (ϕ) = −ϕ log(1 + ε) and ξ0 (ψ) = ε −1 [1 + ε]ψ − 1 .

3.2. REDUCTION ONTO THE BOUNDARY Now a two-dimensional infinitesimal disturbance ξ and η is superimposed on ξ0 and η0 . Then the resulting transform components are ξ = ξ0 + ξ ,

η = η0 + η .

The perturbed quantities ξ and η are assumed to be small quantities so that the nontrivial solution is close to the trivial one. Then, with Equation (11), the inverse transform can be combined to form εξ + iη , log ζ = ψ log(1 + ε) − iϕ log(1 + ε) + log 1 + 1 + εξ0 since log(1 + εξ0 ) = ψ log(1 + ε). Consequently, the polar angle θ and radius r are given by the equations εξ −1 ψ r=ε (1 + ε) 1 + −1 . θ = −ϕ log(1 + ε) + η , 1 + εξ0 Since the motion is irrotational, we can decrease the dimension of the problem by one. In other words, we introduce the boundary value for the function ξ and reduce the basic equations to the quantities which arise from the condition on the free boundary ψ = 1. To this end, we introduce the regular function f (ω) = α +iβ such that 1 (λϕ , −ηϕ ). (α, β) = 2 (λ + η2 )ϕ Then the nonlinear boundary condition (8) can be reduced to the differential equation of the conservation form for f (ω) by virtue of the following lemma: LEMMA 1. Let the function λ (ϕ, ψ) defined by 1+

εξ = eλ(ϕ,ψ) 1 + εξ0

(12)

is differentiable at least once. We introduce the derivative operator ∂n along the normal to ψ = 1 by ∂n µ = λψ |ψ=1 , where µ(ϕ) = λ(ϕ, 1). Then, on the free boundary, the following three relations hold:

1 + ε 2 2µ e , (13) βϕ = −∂n α = −∂ϕ ∂n τ + 2 where τ = (b − µF −2 )(1 + ε)2 , and b = const is the Bernoulli constant.

56

RANIS N. IBRAGIMOV

Proof. Since the velocity potential ω defined by Equation (10) is an analytic function, ξ and η are single-valued and they satisfy the Cauchy–Riemann equations ε ξϕ = ηψ , 1 + εξ

ε ε 2 (1 + εξ )−1 ξψ − ξ0ψ = −ηϕ 1 + εξ (1 + εξ0 )

which can be simplified as λϕ = ηψ ,

λψ = −ηϕ .

(14)

From Equations (11), (14) and presentation ζ = (1 + εξ )eiη , it follows that 2 dζ = ε 2 (ε −1 + 1)2 e2λ λ2 + (1 + εξ )2 η2 , (15) ϕ ϕ dϕ since ξ0 = 1 on the free boundary. By virtue of Equations (14), (15) and the presentation ηϕ = − log(1 + ε) − ∂n µ, the Bernoulli equation (8) takes the form 1 2λ 2 2 2 (1 + εξ ) (log(1 + ε) + ∂n µ) − τ e λϕ + (1 + ε)2

1 2

= 0,

which can be transformed in to the conservation law, (1 + ε 2 ) 2µ ∂ τ+ e . α(ϕ, 1) = ∂ϕ 2

(16)

Thus, the first equation of Equations (13) holds due to the analyticity of the function f (ω) and the second equation of Equations (13) follows from the definition of the normal derivative operator ∂n . Finally, the last equation is the consequence of the ✷ changing of the order of differentiation ∂ϕ ∂n µ = ∂n µϕ . Note that the function µ(ϕ) is found through the analyticity of f (ω) and thus transformation ξ(ϕ, ψ) is determined by definition (12) as 1 + ξ0 (ψ) (eλ(ϕ,ψ) − 1). ξ = ε

3.3. SOLUTION TO THE DIRICHLET PROBLEM IN A FIXED DOMAIN In view of Lemma 1, it follows from the definition for the function α(ϕ, 1) and Equation (16) that integrating Equations (13) over ϕ along ψ = 1 leads to the following equation on the free boundary: 4τ e2µ [log(1 + ε) + ∂n µ] + ∂n ([2τ + 1 + ε 2 ]e2µ ) − εδ0 = 0,

57


where δ0 is the constant of integrating which represent the horizontal impulse flow. Finally, simplifying the last equation, we arrive at the Dirihlet problem in the fixed domain λϕϕ + λψψ = 0 λ(ϕ, 0) = 0,

(0 < ψ < 1),

(17)

λ(ϕ, 1) = µ,

(18)

ε 1 −2 ∂n µ + − µ∂n µF − b + 1 − F 4 2 δe−2µ log(1 + ε) (µF −2 − b) + = 0, + 2 (1 + ε)2 −2

(19)

where we denote δ = δ0 /8. Now the problem (Equations (17) to (19)) is reduced to finding one function µ(ϕ) since if the function µ is known, then function λ(ϕ, ψ) is defined as the solution of the mixed problem for the Laplace equation (17) and the boundary conditions (18). In particular, λψ |ψ=1 can be considered as the result of the action of the operator ∂n on the function µ. Namely, we represent λ (ϕ, ψ) by the Fourier series (see, for example, [8] or [9]). Then the dependence between the nth Fourier coefficients of functions λ, µ and ∂n µ is given by [λ(ϕ, ψ)]n =

sinh nψ µn , sinh n

[µ(ϕ)]n = µn ,

[∂n µ(ϕ)]n = µn cot n, (20)

in which µ(ϕ) = µn einϕ (summation is assumed). Thus, problem (17)–(19) is written in terms of [µ(ϕ)]n only. Since [∂n µ]n are given by Equation (20), we can represent the disturbance µ(ϕ) by the expansion in series with respect to parameter ε (see also [8]). Consequently, we apply the stretching transformation and expansion ∞ ε i ε µ¯ i , ε(∂¯n µ) ¯ i (µ, ∂n µ) =

(i = 0, ∞)

(21)

i=0

which is characteristic of shallow water. Substitution of representation (21) into Equation (19) and elimination mod ε 3 (neglecting of the terms with ε m , m 4) yields the approximate solution of the form [µi ]n = [µi (b, F )]n as follows:

b F −2 1 b −2 µ1 − [1 − F + 4b](∂n µ)1 − 2δµ1 + δ − + ε −2δ + + 2 4 2 4

F −2 1 b F −2 µ1 + µ2 + (∂n µ)1 − + ε 3δ − − 6 4 4 2 2

58

RANIS N. IBRAGIMOV

1 −2 2 − [1 − F + 4b](∂n µ)2 + 4δµ1 + 2δµ1 − 2δµ2 + µ1 (∂n µ)1 + 4

F −2 F −2 1 b F −2 µ1 − µ2 + µ3 + (∂n µ)2 − + ε −4δ + + 8 6 4 2 2 3

−

1 [1 − F −2 + 4b](∂n µ)3 − 4

4 − 6δµ1 − 4δµ21 − δµ31 + 4δµ2 − 2δµ3 + 4δµ1 µ2 + µ1 (∂n µ)2 + 3 (22) + (∂n µ)1 µ2 + o(ε 4 ). Thus, in view of Equations (20), Equation (22) represents the recurrent system of algebraic equations for determination of all [µi ]n , where the horizontal impulse flow has asymptotic δ = b/2. The shape of the free boundary h (θ) can be determined numerically using Okmaoto’s method [3]. The existence of exact solution (ψ, h) can be established analytically by the Fixed Point Theorem (see, for example, [7] or [4]). 4. Behavior of Tides Waves 4.1. EXISTENCE OF STATIONARY WAVES The main concern of this section is the evolution of tides around the Earth in time t. In order to bring out the essential parameters of the problem, the dimensional fundamental equations, Equations (1) to (2), are written, in the nonstationary case, as follows: (ψ) = 0 (in h ), ψθ = 0 (on R ), rht + ψθ + ψr hθ = 0 (on h ), r ψθ2 2 + ψr + rghθ = 0 (on h ), −hθ ψt θ + r ψt r + 2 r2 θ 2

where = (∂θθ + r 2 ∂rr + r∂r ). The perturbed quantities h and ψ are interrelated as h = h ,

ψ =−

γ log r + ψ , 2π


59

where γ is the intensity of the vortex. For the small disturbances, we obtain the linear problem in the domain D0 = {(r, θ) : R r R + h0 , 0 θ 2π } as follows:

(ψ ) = 0 (in D0 ),

(23)

ψθ = 0 (on R ),

(24)

hθ +

γ hθ ψθ − = 0 (on h0 ), r 2π r 2

r 2 ψtr −

γ ψ + rghθ = 0 (on h0 ). 2π rθ

(25) (26)

Since Equations (23)–(26) are linear, the method of superposition is applicable (see also Friedrichs [2]). Hence it is sufficient to look for periodic solutions of the form (h , ψ ) = (H, )(r)) exp{i(kθ − wt)}

(27)

in which the wave number k is a given real quantity and eigenvalues w give the different modes of the tide’s wave propagation. Substitution of representation (27) into Equations (23)–(26) leads to the expression )(r) = c(r k − R 2k r −k ) and to the equations kc kγ − (r k − R 2k r −k ) = 0, H w+ 2 2π r r kγ kg H − w+ (kr k−1 + kR 2k r −k−1 ) = 0 cr 2π r 2 in which c is a constant of integration. Consequently, the determinantal equation for the longitudinal tide wave is as follows:

kg (R + h0 )k−1 − R 2k (R + h0 )−k−1 kγ (R+h0 ) − . (28) w=± (R + h0 )k−1 + R 2k (R + h0 )−k−1 2π(R + h0 )2 Thus surface tide waves (on the constant flow) are dispersive with two different modes of propagation. Simplification of relation (28) shows that the tide wave is propagated with a speed γ w tanh [k ln(1 + ε)] = ± gh0 − . a0 = 2 k kRh0(1 + ε) 2π R (1 + ε)2

60

RANIS N. IBRAGIMOV

Hence, the condition of the existence of stationary tide waves (a0 = 0) is 1 |γ | 2π Rε − 2 (1 + ε)2 gh0 .

4.2. SPLITTING PHENOMENA FOR SHALLOW WATER EQUATIONS We suppose that the parameter ε is infinitesimally small. So we consider R as the natural physical scale. Note that kinematic condition can be written as the mass balance equation. Namely, R+h −1 v θ dr = 0, (29) ht + (R + h) ∂θ R

since the radial velocity component is given by R+h vθθ dr. v r = −r −1 R

Hence, the mass balance equation (29) takes the form rht + ∂θ (uh) = 0 (on h ), where the average velocity u (θ, t) is defined by the relation R+h −1 v θ (r, θ, t)dr. u(θ, t) = h R

To go further, it is better to introduce an nondimensionalization here. We put t=

R t, U

ψ = h0 U ψ ,

u = U u ,

where U is a unit of velocity. Hereafter, the prime will be omitted. Then the impulse equation is written as 1 ε 2 ψθ2 ε 2 hθ ψt θ 2 ∂θ + ψt r + + ψr + − (1 + εh)2 2(1 + εh) (1 + εh)2 hθ = 0. (30) + (1 + εh) We represent function ψ by the Lagrangian expansion (see also [8] thei stream (i) ε ψ . Then the Laplace equation takes the form (mod ε 2 ) or [2]) ψ = ∞ i=0 (1) (0) (0) + ε ψrr + 2rψrr + ψr(0) + ψrr (0) (2) (1) (0) + ψrr + 2rψrr + r 2 ψrr + ψr(1) + rψr(0) = 0. (31) + ε 2 ψθθ Equation (31) represents the recurrent system of differential equations for the determination of ψ (i) as the solution of the Cauchy problem with boundary conditions


61

ψ(0, θ, t) = 0, ψ(1 + h, θ, t) = uh for ψ (0) and zero boundary conditions for ψ (1) and ψ (2) . Hence, the function ψ (mod ε 2 ) is as follows: 2 r r + ψ = ur + ε u − uh 2 2 r2 r3 2 2r 2r + ε −uθθ + uh + uθθ h − uh . (32) 6 4 6 4 We use the Tailor expansion (1 + εh)−1 = 1 − εh + (εh)2 + · · ·

(33)

to write Equation (30) as ψt r + 12 (ε 2 ψθ2 + ψr2 )θ − ε 2 hθ ψt θ + 1 + (εh − 1) εh(ψr2 )θ + εhh θ + hθ = 0. 2

(34)

Let us multiply Equation (30) by (1 + εh)2 and then use expansion (33). Then Equation (30) becomes ψt r + 12 (ε 2 ψθ2 + ψr2 )θ − ε 2 hθ ψt θ + 1 + εh ψt r (2 + εh) + (ψr2 )θ + hθ + hθ = 0. 2

(35)

If we substitute ψ defined by Equation (32) into (34), we obtain the following equation of the shallow water theory: h u 2 ut + uuθ + hθ + ε ut − ht + u hθ − hhθ + 2 2 3 h h2 h + ε 2 hhθ uθt − uθθt + ut + uθθ ht + hhθ u2θ + 3 4 3 2 3 1 1 2 2 (36) + h uθ uθθ + 4 uuθ + hu hθ − 3 uθ uθθ − 3 uuθθθ + h hθ = 0. Consequently, the substitution of ψ (32) into (35) yields h u h u2 3 ut + uuθ + hθ + ε ut − ht + uut + hθ + huuθ + 2 2 2 2 2 2 h 9 h + ε 2 − hhθ uθt − uθθt + h2 ut + ht uθθ − uhht + 3 4 3 3 1 1 + hhθ u2θ + h2 uθ uθθ + h2 uuθ + u2 hhθ − uθθθ + 4 4 3 1 uh 3 hθ = 0. + h2 uθ + uhθ + 4 2 2

(37)

62

RANIS N. IBRAGIMOV

Equations (36) and (37) supplied with the kinematic condition ∂t εh2 + 2h + 2∂θ (uh) = 0

(38)

represent two systems of the shallow water equations. To verify that these two systems are different, we compute their first integrals in the stationary case as follows: 1 2 1 3 2 2 2 2 h + ε 2c h − h + ε 3c h − h = J1 , 2 3 1 2 17 2 2 1 2 2 1 2 c h − c h − ch = J2 , h + εc h + ε 2 2 4 2 where c, J1 , J2 are constants of integrating. Obviously, J1 = J2 . At first sight, it seems that the problem (23)–(26) does not have a unique solution because of that fact. However, it can be shown that the solution of the problem is invariant with respect to the decomposition of the function which represents the free boundary. Since it is not difficult, the proof is omitted. Acknowledgements I wish to express my gratitude to Prof. N. Makarenko of Novosibirsk State University, Russia, for his helpful comments and advice throughout this study. In addition, I would like to thank Prof. L.V. Ovsyannikov for valuable discussions on this subject at seminars at Lavryentiev’s Institute of Hydrodynamics, Russia. I am grateful to Prof. H. Okamoto of the Research Institute of Mathematical Sciences (Kyoto University) for arranging my visit to Japan and for encouraging discussions. Also, I am grateful to Professor K. Lamb of the University of Waterloo, Canada, for valuable discussions and support. References 1. 2. 3. 4. 5. 6. 7.

Crapper, G. D.: Introduction to Water Waves, Ellis Horwood, London, 1984. Friedrichs, K. O. and Hyers, D. H.: The existence of solitary waves, Comm. Pure. Appl. Math. 7 (1954). Fujita, H., Okamoto, H. and Shoji, M.: A numerical approach to a free boundary problem of a circulating perfect fluid, Japan J. Appl. Math. 2 (1985), 197–210. Ibragimov, R. N.: Stationary surface waves on circular liquid layer, Quaest. Math. 23(1) (2000). Levi-Civita, T.: Determination rigoureuse des ondes permanents d’ampleur finie, Math. Ann. 93 (1925), 264–314. Okamoto, H.: Nonstationary free boundary problem for perfect fluid with surface tension, J. Math. Soc. Japan 38(3) (1986). Okamoto, H. and Shoji, M.: On the existence of progressive waves in the flow of perfect fluid around a circle, In: T. Nishida, M. Mimura and H. Fujii (eds), Patterns and Waves, 1986, pp. 631–644.


8. 9. 10. 11.

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Ovsjannikov, L. V. and Makarenko, N. I.: Nonlinear Problems of Surface and Internal Waves Theory, Nauka, Novosibirsk, 1985. Stoker, J. J.: Water Waves, Interscience Publishers, New York, 1957. Stokes, G. G.: On the theory of oscillatory waves, Trans. Cambridge Philos. Soc. 8 (1847), 441–455. Lamb, K. G.: Are solitary waves solitons? Stud. Appl. Math. 101 (1998), 298–308.


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From the Solution of the Tsarev System to the Solution of the Whitham Equations TAMARA GRAVA Department of Mathematics, University of Maryland, College Park 20742-4015, U.S.A. and Department of Mathematics, Imperial College, London SW7 2BZ, U.K. e-mail: [email protected]. (Received: 24 October 2000; in final form: 30 May 2001) Abstract. We study the Cauchy problem for the Whitham modulation equations for increasing smooth initial data. The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems. This collection of systems is enumerated by the genus g = 0, 1, 2, . . . of the corresponding hyperelliptic Riemann surface. Each of these systems can be integrated by the socalled hodograph transformation introduced by Tsarev. A key step in the integration process is the solution of the Tsarev linear overdetermined system. For each g > 0, we construct the unique solution of the Tsarev system, which matches the genus g + 1 and g − 1 solutions on the transition boundaries. Mathematics Subject Classifications (2000): 35Q53, 58F07. Key words: Whitham equations, hyperelliptic Riemann surfaces, linear overdetermined systems of Euler–Poisson Darboux type.

1. Introduction The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems of the form [1–3] ∂ui ∂ui − λi (u1 , u2 , . . . , u2g+1 ) = 0, ∂t ∂x x, t, ui ∈ R, i = 1, . . . , 2g + 1, g = 0, 1, 2, . . . ,

(1.1)

with the ordering u1 > u2 > · · · > u2g+1 . For a given g, the system (1.1) is called g-phase Whitham equations. For g > 0, the speeds λi (u1 , u2 , . . . , u2g+1 ), i = 1, 2, . . . , 2g + 1, depend through u1 , . . . , u2g+1 on complete hyperelliptic integrals of genus g. For this reason, the g-phase system is also called a genus g system. The zero-phase Whitham equation has the form ∂u ∂u − 6u = 0, (1.2) ∂t ∂x where we use the notation u1 = u. Equations (1.1) were found by Whitham [1] in the single-phase case g = 1 and more generally by Flaschka, Forest and McLaughlin [2] in the multi-phase case.

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The Whitham equations were also found in [3] when studying the zero dispersion limit of the Korteweg–de Vries equation. The hyperbolic nature of the equations was found by Levermore [4]. In this paper we study the initial-value problem of the Whitham equations for increasing smooth (C ∞ ) initial data u(x, t = 0) = u0 (x). The initial-value problem consists of the following. We consider the evolution on the x − u plane of the initial curve u(x, t = 0) = u0 (x) according to the zerophase equation (1.2). The solution u(x, t) of (1.2), with the initial data u0 (x), is given by the characteristic equation x = −6tu + f (u),

(1.3)

where f (u)|t =0 is the inverse function of u0 (x). The solution u(x, t) in (1.3) is globally well defined only for 0 t < t0 , where t0 = 16 minu∈R [f (u)] is the time of gradient catastrophe of (1.3). Near the point of gradient catastrophe and for a short time t > t0 , the evolving curve is given by a multivalued function with three branches u1 (x, t) > u2 (x, t) > u3 (x, t), which evolve according to the one-phase Whitham equations (see Figure 1). Outside the multivalued region, the solution is given by the zero-phase solution u(x, t) defined in (1.3). On the phase transition boundary, the zero-phase solution and the one-phase solution are C 1 -smoothly attached (see Figure 1). Since the Whitham equations are hyperbolic, other points of gradient catastrophe can appear in the branches u1 (x, t) > u2 (x, t) > u3 (x, t) themselves or in u(x, t). In general, for t > t0 , the evolving curve is given by a multivalued function with an odd number of branches u1 (x, t) > u2 (x, t) > · · · > u2g+1 (x, t), g 0.

(a)

(b)

Figure 1. In picture (a), the dashed line represents the formal solution of the zero-phase equation and the continuous line represents the solution of the one-phase equations. The solution (u1 (x, t) , u2 (x, t), u3 (x, t)) of the one-phase equations and the position of the boundaries x − (t) and x + (t) are to be determined from the conditions u(x − (t), t) = u1 (x − (t), t), ux (x − (t), t) = u1x (x − (t), t), u(x + (t), t) = u3 (x + (t), t), ux (x + (t), t) = u3x (x + (t), t), where u(x, t) is the solution of the zero-phase equation.

67

THE TSAREV AND WHITHAM EQUATIONS

These branches evolve according to the g-phase Whitham equations. The g-phase solutions for different g must be glued together in order to produce a C 1 -smooth curve in the (x, u) plane evolving smoothly with t (see Figure 1b). The initialvalue problem of the Whitham equations is to determine, for almost all t > 0 and x, the phase g(x, t) 0 and the corresponding branches u1 (x, t) > u2 (x, t) > · · · > u2g+1 (x, t) from the initial data x = f (u)|t =0 . For generic initial data, it is not known whether the solution of the Whitham equations has a finite genus. Some results in this direction have been obtained in [5, 6]. Using the geometricHamiltonian structure [7] of the Whitham equations, Tsarev [8] showed that these equations can be locally integrated by a generalization of the method of characteristic. Namely, he proved that if the functions wi = wi (u1 , u2 , . . . , u2g+1 ), i = 1, . . . , 2g + 1, solve the linear over-determined system 1 ∂λi ∂wi = [wi − wj ], ∂uj λi − λj ∂uj

i, j = 1, 2, . . . , 2g + 1, i = j,

(1.4)

where λi = λi (u1 , u2 , . . . , u2g+1 ), i = 1, . . . , 2g + 1, are the speeds in (1.1), then the solution u(x, t) = (u1 (x, t), u2 (x, t), . . . , u2g+1 (x, t)) of the so-called hodograph transformation x = −λi (u)t + wi (u),

i = 1, . . . , 2g + 1,

(1.5)

satisfies system (1.1). Conversely, any solution (u1 (x, t), u2 (x, t), . . . , u2g+1 (x, t)) of (1.1) can be obtained in this way. Furthermore, the solution wi (u), i = 1, . . . , 2g + 1, g 0, of (1.4) must satisfy some natural matching conditions which guarantee that the g-phase solutions of (1.1) for different g are glued together in order to produce a C 1 -smooth curve in the (x, u) plane. Tsarev theorem relies on two factors: (a) the existence of a solution of the linear over-determined system (1.4); (b) the existence of a real solution u1 (x, t) > u2 (x, t) > · · · > u2g+1 (x, t) of the hodograph transformation (1.5). In this paper, we investigate problem (a). We construct a new expression for the solution of the Tsarev system. This construction enables us to extend the solution of the Tsarev system, for g > 1, from analytic initial data with polynomial or exponential growth at infinity [15] to any smooth initial data. In addition, this new expression has the advantage that: (i) it is possible to evaluate explicitly the Jacobian of the hodograph transformation (see (5.45)). It turns out that the determinant of the Jacobian is propor2g+1 tional to the product j =1 g (ui ; u), where the function g (r; u) can be explicitly obtained from the initial data; (ii) it is simpler to study the hodograph transformation near the phase transition boundary.

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TAMARA GRAVA

The investigation of the initial value problem of the Whitham equations was initiated by Gurevich and Pitaevskii [9]. In the case g 1, they solved Equations (1.1) for step-like initial data and studied numerically the case of cubic initial data. Krichever [10] introduced an algebro-geometric procedure to integrate (1.4). Based on this procedure, Potemin [11] obtained the explicit solution of the system (1.4) for cubic initial data. Kudashev and Sharapov in [12] and Gurevich, Krylov and El in [13] connected the solution of the Tsarev system (1.4) to the solution of some linear overdetermined systems of the Euler–Poisson–Darboux type introduced in [14]. The structure of the solution of the systems of the Euler–Poisson–Darboux type was in depth investigated by Tian in [5], where he obtained for g 1 the solution of the Tsarev system from the solution of the systems of the Euler–Poisson–Darboux type for any smooth monotone initial data. Furthermore, he partly solved problem (b) proving the solvability of the hodograph transformation for g 1. The structure of the multiphase solutions has been investigated by Tian [15] and later by El [16]. In [15], Tian built the solution of the Tsarev system for g > 1 for generic polynomial initial data. Again, such a solution is constructed in terms of solutions of the linear overdetermined systems of the Euler–Poisson–Darboux type. Tian’s formula is not a-priori generalizable to analytic initial data u0 (x) which are bounded at infinity or to any smooth initial data. In this paper, we obtain a new expression for the solution wi (u), i = 1, . . . , 2g + 1, of the Tsarev system for polynomial initial data. Such an expression enables us to generalize Tian’s result to any increasing smooth initial data. Furthermore, we prove that the solution obtained is unique. This paper is organized as follows. In Section 2 we give some background to Abelian differentials on hyperelliptic Riemann surfaces. We describe the Whitham equations in Section 3 where we show that the solution of the Tsarev system with some given natural matching conditions is unique. In Section 4, we construct a new formula for the solution of the Tsarev system (1.4) for polynomial initial data. Then we show that the formula obtained can be extended to any smooth increasing initial data. In Section 5, we prove that such formula guarantees that the g-phase solutions of (1.1) for different g are glued together in order to produce a C 1 -smooth multivalued curve in the x − u plane evolving smoothly with time. Our conclusions are drawn in Section 6. 2. Riemann Surfaces and Abelian Differentials: Notations and Definitions Let

Sg := P = (r, µ), µ2 =

2g+1

(r − uj ) ,

(2.1)

j =1

be the hyperelliptic Riemann surface of genus g 0 with real branch points g u1 > u2 > · · · > u2g+1 . We choose the basis {αj , βj }j =1 of the homology group H1 (Sg ) so that αj lies fully on the upper sheet and encircles clockwise the

69


interval [u2j , u2j −1 ], j = 1, . . . , g, while βj emerges on the upper sheet on the cut [u2j , u2j −1 ], passes anti-clockwise to the lower sheet through the cut (−∞, u2g+1 ] and return to the initial point through the lower sheet. The one-forms that are analytic on the closed Riemann surface Sg except for a finite number of points are called Abelian differentials. We define on Sg the following Abelian differentials [17]: (1) The canonical basis of holomorphic one-forms or Abelian differentials of the first kind φ1 , φ2 . . . φg : φk (r) =

r g−1 γ1k + r g−2 γ2k + · · · + γgk µ(r)

k = 1, . . . , g.

dr,

The constants γik are uniquely determined by the normalization conditions φk = δj k , j, k = 1, . . . , g.

(2.2)

(2.3)

αj

We remark that the holomorphic differential having all its α-periods equal to zero is identically zero [17]. g

(2) The set σk , k 0, g 0, of the Abelian differentials of the second kind with a pole of order 2k + 2 at infinity, with asymptotic behavior 1 3 g (2.4) σk (r) = r k− 2 + O(r − 2 ) dr for large r and normalized by the condition g σk = 0, j = 1, . . . , g.

(2.5)

αj

We use the notation g

σ0 (r) = dp g (r),

g

12σ1 (r) = dq g (r), g

g 0.

(2.6)

g

In the literature, the differentials dp (r) and dq (r) are called quasi-momentum g and quasi-energy, respectively [7]. The explicit formula for the differentials σk , k 0, is given by the expression g

Pk (r) dr, µ(r) g k , Pk (r) = r g+k + c1k r g+k−1 + c2k r g+k−2 + · · · + cg+k g

σk (r) =

(2.7)

where the coefficients cik = cik (u), u = (u1 , u2 , . . . , u2g+1 ), i = 1, . . . , g + k, are uniquely determined by (2.4) and (2.5). (3) The Abelian differential of the third kind ωqq0 (r) with first-order poles at the points Q = (q, µ(q)) and Q0 = (q0 , µ(q0 )) with residues ±1, respectively. Its periods are normalized by the relation ωqq0 (r) = 0, j = 1, . . . , g. (2.8) αj

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TAMARA GRAVA

2.1. RIEMANN BILINEAR RELATIONS Let ω1 and ω2 be two Abelian differentials on the Riemann surface Sg . We suppose that ω1 has some poles with nonzero residue so that the integral d−1 ω1 has logarithm singularities on Sg . Let s be the path connecting the singular points of d−1 ω1 . We have the following relation:

g ω1 ω2 − ω2 ω1 + $(d−1 ω1 )ω2 j =1

αj

= 2π i

βj

αj

βj

s

Res[(d−1 ω1 )ω2 ],

(2.9)

Sg −s

difference of the values of d−1 ω1 on the two sides of the where $(d−1 ω1 ) is the cut s and the quantity Sg −s Res[(d−1 ω1 )ω2 ] is the sum of the residues of the differential (d−1 ω1 )ω2 on the cut surface Sg − s. This formula is known as the Riemann bilinear period relation [18]. Assuming ω1 = ωqq0 and ω2 = ωpp0 in (2.9) we obtain q p ωqq0 = ωpp0 . (2.10) p0

q0

Differentiating the above expression, with respect to p and q we obtain the identity dq [ωqq0 (p)] = dp [ωpp0 (q)],

(2.11)

where dq and dp denote differentiation with respect to q and p, respectively. In the following we mainly use the normalized Abelian differential of the third g kind ωz (r) which has simple poles at the points Q± (z) = (z, ±µ(z)) with residue ±1 respectively. g The differential ωz (r) is explicitly given by the expression dr µ(z) − = φk (r) µ(r) r − z k=1 g

ωzg (r)

αk

dt µ(z) , µ(t) t − z

(2.12)

where φk (r), k = 1, . . . , g, is the normalized basis of holomorphic differentials. g Using the explicit expression of the φk (r)’s in (2.2), we write ωz (r) in the form dr µ(z) r g−j − dr, = Nj (z, u) µ(r) r − z j =1 µ(r) g

ωzg (r)

(2.13)

where Nj (z, u) =

g k=1

γjk αk

dt µ(z) µ(t) t − z

(2.14)

71


and the coefficients γjk have been defined in (2.2). In order to provide a more useful expression for the Nj ’s, we apply the Riemann bilinear relation (2.9) to g g the differentials σm (r) and ωz (r), getting Q+ (z) σmg (ξ ) = − Res [ωzg (r) d−1 σmg (r)], m = 0, . . . , g, r=∞ Q− (z)

g 4 = − −µ(z)*mg + Nj (z)+m+1−j . (2.15) 2m + 1 j =1 In the above formula, *mg = 1 for m = g and zero otherwise, and the +l ’s are the coefficients of the expansion for ξ → ∞ of +2 +1 +l 1 −g− 12 =ξ + 2 + ··· + l + ··· . (2.16) +0 + µ(ξ ) ξ ξ ξ We define +k = 0 for k < 0. Solving (2.15) for Nj (z, u), we obtain   +˜ N1 (z, u) 0  N2 (z, u)   +˜ 1     ···  =  ···  N (z, u)   +˜ g−1 g −µ(z) +˜ g 

0 +˜ 0 ··· +˜ g−2 +˜ g−1

0 0 ··· ··· ···

Q+ (z) g   − 14 Q− (z) σ0 (ξ ) 0  + (z) g   0   − 3 QQ− (z) σ1 (ξ )  4   ··· ,  · Q·+· (z)    g 2g−1 0 − Q− (z) σg−1 (ξ )  4 Q+ (z) g +˜ 0 − 2g+1 Q− (z) σg (ξ ) 4 (2.17)

··· ··· ··· +˜ 0 +˜ 1

where the +˜ k ’s are the coefficients of the expansion for ξ → ∞ of +˜ 1 +˜ 2 +˜ l g+ 21 ˜ + 2 + ··· + l + ··· . +0 + µ(ξ ) = ξ ξ ξ ξ

(2.18)

From the relation (2.17), we obtain the identity which will be useful later Q+ (z) g+1 1 g ˜ (2k − 1)+g+1−k σk−1 (ξ ). µ(z) = 4 k=1 Q− (z)

(2.19)

The next proposition is also important for our subsequent considerations. g

PROPOSITION 2.1. The Abelian differentials of the second kind σk (r), k 0, defined in (2.4) satisfy the relations 1 1 1 1 g dr Res ωrg (z) zk+ 2 , (2.20) σk (r) = Res ωzg (r) zk− 2 dz = − 2 z=∞ 2k + 1 z=∞ g

g

where ωz (r) has been defined in (2.12), ωr (z) is the normalized Abelian differential of the third kind with simple poles at the points Q± (r) = (r, ±µ(r)) with residue ±1, respectively, and dr denotes differentiation with respect to r.

72

TAMARA GRAVA g

1

g

Proof. The differential Resz=∞ [ωz (r) zk− 2 dz] is normalized because ωz (r) is a normalized differential. From (2.13) and (2.17) it can be easily shown that 1 1 3 Res ωzg (r) zk− 2 dz = r k− 2 dr + O(r − 2 ) dr

z=∞

g

for r → ∞.

1

Therefore, Resz=∞ [ωz (r) zk− 2 dz] coincides with the normalized Abelian differeng tial of the second kind σk (r). For proving the second equality in (2.20) we consider the integral in the z variable 1 g g k+ 12 k+ 21 g dz (ωz (r) z ) = z (dz ωz (r)) + (k + 12 )(ωz (r) zk+ 2 ), 0= C∞

C∞

C∞

(2.21) where C∞ is a close contour around the point at infinity. From (2.11), we obtain g g the identity dz ωz (r) = dr ωr (z). Substituting the above identity in the right-hand side of (2.21), we obtain the second relation in (2.20). ✷

3. Preliminaries on the Theory of the Whitham Equations The speeds λi (u1 , u2 , . . . , u2g+1 ) of the g-phase Whitham equations (1.1) are given by the ratio [1, 2]: dq g (r) , i = 1, 2, . . . , 2g + 1, (3.1) λi (u) = dp g (r) r=ui where dp g (r) and dq g (r) have been defined in (2.6). In the case g = 0 dp 0 (r) = √

dr , r −u

12r − 6u dq 0 (r) = √ dr, r −u

(3.2)

so that one obtains the zero-phase Whitham equation (1.2). For monotonically increasing smooth initial data x = f (u)|t =0 , the solution of the zero-phase equation (1.2) is obtained by the method of characteristic [1] and is given by the expression x = −6tu + f (u).

(3.3)

The zero-phase solution is globally well-defined only for 0 t < t0 , where t0 = 16 minu∈R [f (u)] is the time of gradient catastrophe of the solution (3.3). The breaking is caused by an inflection point in the initial data. For t t0 , we expect to have single, double and higher phase solutions. For higher genus the Whitham equations can be locally integrated using a generalization of the characteristic equation (3.3). We have the following theorem of Tsarev [8]

73


THEOREM 3.1. If wi (u) solves the linear over-determined system ∂wi = aij [wi − wj ], ∂uj 1 ∂λi , aij = λi − λj ∂uj

i, j = 1, 2, . . . , 2g + 1, i = j, (3.4)

then the solution (u1 (x, t), u2 (x, t), . . . , u2g+1 (x, t)) of the hodograph transformation x = −λi (u) t + wi (u),

i = 1, . . . , 2g + 1,

(3.5)

satisfies system (1.1). Conversely, any solution (u1 , u2 , . . . , u2g+1 ) of (1.1) can be obtained in this way in a neighborhood (x0 , t0 ) where the uix ’s are not vanishing. To guarantee that the g-phase solutions for different g are attached continuously, the following natural matching conditions involving f (u) must be imposed on wi (u1 , u2 , . . . , u2g+1 ), i = 1, . . . , 2g + 1, g > 0. When ul = ul+1 , 1 l 2g, g

wl (u1 , . . . , ul−1 , ul , ul , ul+2 , . . . , u2g+1 ) g = wl+1 (u1 , . . . , ul−1 , ul , ul , ul+2 , . . . , u2g+1 )

(3.6)

and, for 1 i 2g + 1, i = l, l + 1, g

wi (u1 , . . . , ul−1 , ul , ul , ul+2 , . . . , u2g+1 ) g−1

= wi

(u1 , . . . , uˆ l , uˆ l , . . . , u2g+1 ).

(3.7)

The superscript g and g − 1 in the wi ’s specify the corresponding genus and the hat denotes the variable that have been dropped. When g = 1 we have that w11 (u1 , u1 , u3 ) = w21 (u1 , u1 , u3 ),

w21 (u1 , u3 , u3 ) = w31 (u1 , u3 , u3 ) (3.8)

and w31 (u1 , u1 , u3 ) = f (u3 ),

w11 (u1 , u3 , u3 ) = f (u1 ),

(3.9)

where f (u) is the initial data. In the following, we will sometimes omit the superscript g when we are referring to genus g quantities. We remark that the λi (u)’s in (3.1) satisfy the matching conditions (3.6)–(3.8) and, for g = 1, we have λ1 (u1 , u3 , u3 ) = 6u1 ,

λ3 (u1 , u1 , u3 ) = 6u3 .

The matching conditions (3.6)–(3.9) guarantee that the solution of the Tsarev system (3.4) is unique.

74

TAMARA GRAVA g

THEOREM 3.2. If the initial data f (u) ≡ 0, then the solution wi (u) of (3.4) with matching conditions (3.6)–(3.9) is identically zero for 1 i 2g + 1, for any g 0 and for all u1 > u2 > · · · > u2g+1 . Proof. The proof is obtained by induction on g. The statement is satisfied for g = 0. For g = 1, we repeat the arguments of [5]. We fix u2 and we consider Equation (3.4) with matching conditions (3.8)–(3.9), namely ∂w11 = a13 [w11 − w31 ], ∂u3

∂w31 = a31 [w31 − w11 ], ∂u1

w11 (u1 , u2 , u2 ) = f (u1 ) ≡ 0,

w31 (u2 , u2 , u3 ) = f (u3 ) ≡ 0.

We can regard each of the above equations as a first-order linear ordinary differential equation with a nonhomogeneous term. Integrating them, we obtain a couple integral equation. By the standard contraction mapping method, it can be shown that when f (u) ≡ 0, this system has only the zero solution, i.e. w11 ≡ w31 ≡ 0 for (u1 , u3 ) satisfying u1 > u2 > u3 . Because of the arbitrariness of u2 , w11 and w31 vanish as a function of (u1 , u2 , u3 ) and, therefore, by (3.4), so does w21 (u). Now we suppose the theorem true for genus g − 1 and we prove it for genus g. We fix g g u2 > u3 > · · · > u2g and we consider Equation (3.4) for w1 and w2g+1 with the matching conditions (3.6)–(3.7), namely g ∂ g g w1 = a1(2g+1) w1 − w2g+1 , ∂u2g+1 g ∂ g g w2g+1 = a(2g+1)1 w2g+1 − w1 , (3.10) ∂u1 g g−1 w1 (u1 , u2 , . . . , u2g , u2g ) = w1 (u1 , u2 , . . . , u2g−1 , uˆ 2g , uˆ 2g ) ≡ 0, g

g−1

w2g+1 (u2 , u2 , . . . , u2g , u2g+1 ) = w2g+1(uˆ 2 , uˆ 2 , u3 , . . . , u2g , u2g+1 ) ≡ 0. Repeating the arguments developed for genus g = 1, we may conclude that g g w1 (u) ≡ w2g+1(u) ≡ 0, for arbitrary u1 > u2 > · · · > u2g+1 . We then repeat the above argument fixing u1 > u3 > · · · > u2g−1 > u2g+1 and considering g g Equation (3.4) for w2 (u) and w2g (u) with the matching conditions (3.6)–(3.7), namely g ∂ g g w2 = a2(2g) w2 − w2g , ∂u2g g ∂ g g w2g = a(2g)2 w2g − w2 , ∂ug 2 (3.11) w2 (u1 , u2 , . . . , u2g−1 , u2g+1 , u2g+1 ) g−1

(u1 , u2 , . . . , u2g−1 , uˆ 2g+1 , uˆ 2g+1 ) ≡ 0, g g−1 w2g (u1 , u1 , u3 , . . . , u2g , u2g+1 ) = w2g (uˆ 1 , uˆ 1 , u3 , . . . , u2g , u2g+1 ) = w2

g

g

≡ 0.

It can be easily shown that also w2 (u) ≡ w2g (u) ≡ 0 for arbitrary u1 > u2 > · · · > u2g+1 . Repeating these arguments some other g − 2 times, we conclude that


75

g

wi (u) ≡ 0 for 1 i g, g + 2 i 2g + 1 and for arbitrary u1 > u2 > · · · > u2g+1 . Applying (3.4) and the matching conditions (3.6)–(3.7), we can prove that g ✷ also wg+1 (u) is identically zero. The theorem is then proved. The solution of the Tsarev system (3.4) with the matching conditions (3.6)– (3.9) has been obtained in [15, 10] for monotonically increasing polynomial initial data of the form x = fa (u) = c0 + c1 u + · · · + ck uk + · · · ,

(3.12)

where we assume that only a finite number of ck is different from zero. For such initial data, the wi (u)’s which satisfy (3.4) and the matching conditions (3.6)–(3.9) are given by the expression [15] ds g (r) , i = 1, . . . , 2g + 1. (3.13) wi (u) = dp g (r) r=ui The differential ds g (r) in (3.13) is given by ds (r) = g

∞ k=0

2k k! g ck σk (r), (2k − 1)!!

(3.14)

g

and the differentials σk (r), k 0 have been defined in (2.7). For the initial data (3.12), Tian [15] has reduced the expression of the wi (u)’s in (3.13) to the form

wi (u) =

g ∂ j qk γk ∂ui k=1 j

∂γ1 ∂ui

,

i = 1, 2 . . . , 2g + 1, 1 j g,

(3.15)

j

where the γk ’s are the normalization constants of the holomorphic differential φj (r) in (2.2) and the functions qk = qk (u), k = 1, . . . , g, solve the linear overdetermined system of the Euler–Darboux–Poisson type [15] ∂ 2 qk (u) ∂qk (u) ∂qk (u) = − , ∂ui ∂uj ∂ui ∂uj i, j = 1, . . . , 2g + 1, k = 1, . . . , g, g−k 2g−1 1 d 1 u−k+ 2 ug− 2 fa(k−1) (u) , qk (u, u, . . . , u) = (2g − 1)!! du 2(ui − uj )

(3.16)

2g+1

where fa(k−1)(u) is the (k − 1)th derivative of the polynomial initial data fa (u) defined in (3.12). Thus, the solution of the Tsarev system is reduced to the solution of the above systems of equations. The systems (3.16) can be integrated for any smooth initial data [5].

76

TAMARA GRAVA

THEOREM 3.3. Let f (u) be a smooth function with domain (a, b), −∞ a < b +∞. The initial value problem ∂ 2 qk (u) ∂qk (u) ∂qk (u) = − , ∂ui ∂uj ∂ui ∂uj i = j, i, j = 1, . . . , 2g + 1, g > 0, qk (u, u, . . . , u) = Fk (u), 2(ui − uj )

(3.17) (3.18)

2g+1

Fk (u) =

g−k 2(g−1) 1 d 1 u−k+ 2 g−k ug− 2 f (k−1)(u) , (2g − 1)!! du

(3.19)

with the ordering b > u1 > u2 > · · · > u2g+1 > a, has one and only one solution. The solution is symmetric and is given by 1 1 1 1 qk (u) = ··· dξ1 dξ2 . . . dξ2g (1 + ξ2g )g−1 × C −1 −1 −1 × (1 + ξ2g−1 )g− 2 . . . (1 + ξ3 ) 2 (1 + ξ1 )− 2 × 3

×

Fk (

1

1+ξ2g 2 1+ξ1 (. . . ( 1+ξ ( 2 u1 2 2

+

1

1−ξ1 u2 ) 2

+

1−ξ2 u3 ) 2

+ · · ·) +

(1 − ξ1 )(1 − ξ2 ) . . . (1 − ξ2g )

k = 1, . . . , g, 2g where C = m=1 Cm , 1 m (1 + µ) 2 −1 dµ, Cm = √ 1−µ −1

1−ξ2g u2g+1 ) 2

,

(3.20)

m > 0.

(3.21)

Proof. To prove the theorem we follow the procedure in [5]. We start with the following lemma. LEMMA 3.4 [5]. The system 2(z − y)hzy = hz − ρhy , h(z, z) = s(z),

ρ > 0,

(3.22)

has, for any smooth initial data s(z), one and only one solution. Moreover, the solution can be written explicitly 1 1+µ s( 2 z + 1−µ y) ρ 1 2 (1 + µ) 2 −1 dµ, (3.23) h(z, y) = √ Cρ −1 1−µ where Cρ has been defined in (3.21). Using the above lemma, the linear over-determined systems (3.17) can be integrated for any smooth initial data in the following way. Suppose that qk (u1 , u2 , . . . , u2g+1 ) is the solution of (3.17)–(3.19).

77


Clearly Ak (u1 , u2g+1 ) = qk (u1 , u1 , . . . , u1 , u2g+1 ) 2g

satisfies ∂ 2 Ak ∂Ak ∂Ak = − 2g , ∂u1 ∂u2g+1 ∂u1 ∂u2g+1 Ak (u, u) = Fk (u), 2(u1 − u2g+1 )

(3.24)

which, by Lemma 3.4, implies that 1 1+ξ 1−ξ Fk ( 2 2g u1 + 2 2g u2g+1 ) 1 (1 + ξ2g )g−1 dξ2g . (3.25) Ak (u1 , u2g+1 ) = C2g −1 1 − ξ2g For each fixed u2g+1 , the function Bk (u1 , u2g , u2g+1 ) = qk (u1 , . . . , u1 , u2g , u2g+1 ) 2g−1

satisfies ∂ 2 Bk ∂Bk ∂Bk = − (2g − 1) , ∂u1 ∂u2g ∂u1 ∂u2g Bk (u, u, u2g+1 ) = Ak (u, u2g+1 ).

2(u1 − u2g )

(3.26)

Again using Lemma 3.4, we obtain 1 1 1 3 dξ2g dξ2g−1 (1 + ξ2g )g−1 (1 + ξ2g−1)g− 2 × Bk (u1 , u2g , u2g+1 ) = C2g C2g−1 −1 −1 ×

Fk (

1+ξ2g 1+ξ2g−1 ( 2 u1 2

1+ξ

+ 22g−1 u2g ) 1 − ξ2g 1 − ξ2g−1

1−ξ2g u2g+1 ) 2

.

(3.27)

Continuing the process of integration, we obtain the solution (3.20). The uniqueness follows from Lemma 3.4 and the above arguments. The boundary conditions (3.18)–(3.19) are clearly satisfied. The symmetry property is obtained considering the solution hk (u) = qk (u) − qk (P (u)), where P (u) is any reordering of the ui ’s. Such a solution satisfies (3.17) with Fk (u) = 0 and therefore, by construction, equals zero. ✷ The formula for the wi ’s obtained in (3.13) is valid only for a Taylor series with an infinite radius of convergence. This corresponds to an increasing initial data which is the sum of exponentials, sine, cosine, and polynomials. Therefore, it is not obvious that formula (3.15) can be extended to analytic initial data which have a different behavior at infinity or to any smooth initial data.

78

TAMARA GRAVA

In the next section we provide a new expression for the wi (u)’s equivalent to (3.15) which enables us to make such an extension. Namely, we show that these wi (u)’s are the unique solution of the Tsarev system (3.4) with matching conditions (3.6)–(3.9) for any monotonically increasing smooth initial data. Furthermore, this new expression enables us to evaluate the Jacobian of the hodograph transformation (3.5) very easily. Namely, the Jacobian turns out to be propor2g+1 tional to the product j =1 g (ui ; u), where the function g (r; u) satisfies a linear overdetermined system of the Euler–Poisson–Darboux type. 4. Solution of the Tsarev System In this section, we build the solution of the Tsarev system (3.4) with matching conditions (3.6)–(3.9) for monotonically increasing smooth initial data. We consider initial data of the form x = f (u)|t =0 , where f (u) is a monotonically increasing function. The domain of f is the interval (a, b), where −∞ a < b +∞, and the range of f is the real line (−∞, +∞). In order to obtain such a solution, we need the following technical lemma: LEMMA 4.1. The differential ds g (r) defined in (3.14) can be written in the form

2g+1 R g (r) dr, (4.1) ∂uk 5 g (r; u) dr + ds g (r) = 2µ(r) ∂r 5 g (r; u) + µ(r) k=1 where

! F (z) dz , 5 (r; u) = − Res z=∞ 2µ(z)(z − r) k = 1, . . . g, z fa (ξ ) dξ, F (z) = √ z−ξ 0 g

2g+1

R (r) = 2 g

(4.3)

2g+1

∂uk qg (u)

g

(r − un ) +

n=1,n=k

k=1

+

! zg−k F (z) dz qk (u) = − Res , z=∞ 2µ(z) (4.2)

k g qk (u) (2n − 1)+˜ k−n Pn−1 (r),

k=1

(4.4)

n=1

g the polynomials Pn (r), n 0, have been defined in (2.7), the +˜ k ’s have been defined in (2.18) and fa (ξ ) is the analytic initial data (3.12). Proof. Using the second identity in (2.20), we rewrite the differential ds g (r) defined in (3.14) in the form # " (4.5) ds g (r) = −dr Res[ωrg (z)F (z)] , z=∞

79

THE TSAREV AND WHITHAM EQUATIONS g

where ωr (z) has been defined in (2.12) and F (z) is the Abel transform defined in (4.3) of the analytic initial data (3.12). The identity (4.5) can be checked in a g straightforward manner. Using the explicit expression of ωr (z) in (2.13), we obtain ds (r) = 2dr (µ(r)5 (r; u)) + g

g

g

qk (u)

k=1

k

g (2n − 1)+˜ k−n σn−1 (r),

(4.6)

n=1

where 5 g (r; u) and qk (u) have been defined in (4.2). From (4.2) we get the relations 5 g (r; u) 5 g (ui ; u) − = 2∂ui 5 g (r; u), r − ui r − ui and for g = 0 we define u1 = u and

2∂ui qg (u) = 5 g (ui ; u)

(4.7)

2∂u q0 (u) := 5 0 (u; u) = fa (u). Using (4.7) we transform the expression for ds g (r) in (4.6) to the form (4.1).

✷

The relation (4.1) enables us to write the quantities ds g (r) , i = 1, . . . , 2g + 1, wi (u) = dp g (r) r=ui

in (3.13) in the form

2g+1 1 2∂ui qg (u) (ui − un ) + wi (u) = g P0 (ui ) n=1,n=i +

g k=1

k g qk (u) (2n − 1)+˜ k−n Pn−1 (ui ) .

(4.8)

n=1

We observe that in the formula (4.8) all the information on the initial data is contained in the functions qk (u). The functions qk = qk (u), k = 1, . . . , g, solve the linear over-determined system (3.16). THEOREM 4.2 (Main Theorem). Let f (u) be a smooth monotonically increasing function with domain (a, b), −∞ a < b +∞ and range (−∞, +∞). If qk = qk (u1 , u2 , . . . , u2g+1 ), 1 k g, is the symmetric solution of the linear over-determined system (3.17)–(3.19), with the ordering b > u1 > u2 > · · · > u2g+1 > a, then wi (u), i = 1, . . . , 2g + 1, defined by 2g+1 1 2∂ui qg (u) (ui − un ) + wi (u) = g P0 (ui ) n=1,n=i

g k g qk (u) (2n − 1)+˜ k−n Pn−1 (ui ) , (4.9) + k=1

n=1

solves the Tsarev system (3.4) with matching conditions (3.6)–(3.9).

80

TAMARA GRAVA

Proof. We consider the nontrivial case where qk (u) ≡ 0, k = 1, . . . , g, and ∂uj qg (u) ≡ 0, j = 1, . . . , 2g + 1. The proof of the theorem consists of two parts. (a) The wi (u)’s defined in (4.9) satisfy (3.4). Using the definition of wi (u) in (4.9) we have the following relation: 2g+1 2g+1 n=1, n=i (ui − un ) n=1, n=i (ui − un ) ∂uj ∂ui qg (u) − 2 ∂ui qg (u) × ∂uj wi (u) = 2 g g P0 (ui ) P0 (ui ) g ∂uj P0 (ui ) 1 × + + g ui − uj P0 (ui ) g

g g Pn−1 (ui ) (2n − 1) g qk (u)+˜ k−n , + ∂ uj P (u ) i 0 k=n n=1 i = j, i, j = 1, . . . , 2g + 1.

(4.10)

From (3.4), we obtain g g g 1 ∂λi Pk (ui ) Pk (uj ) ∂ Pk (ui ) = − , ∂uj P0g (ui ) λi − λj ∂uj P0g (ui ) P0g (uj ) i = j, i, j = 1, . . . , 2g + 1, k 1

(4.11)

and, from [15], we have g

∂uj P0 (ui ) 1 1 ∂λi 1 − =− , g λi − λj ∂uj 2 ui − uj P0 (ui ) i = j, i, j = 1, . . . , 2g + 1,

(4.12)

g

where λi (u) has been defined in (3.1) and Pk (r) has been defined in (2.4). Using the definition of +˜ k in (2.18), we get k 1 ∂ +˜ k =− . +˜ k−m um−1 j ∂uj 2 m=1

(4.13)

It is easy to verify that the functions Fk (u) and the solutions qk (u), k = 1, . . . , g, of (3.17)–(3.19) satisfy the relations 2g + 1 Fk+1 (u) + u∂u Fk+1 (u), k = 1, . . . , g − 1, 2 1 ∂ui qk (u) = qk+1 (u) + ui ∂ui qk+1 (u), 2 i = 1, . . . , 2g + 1, k = 1, . . . , g − 1, g > 0.

∂u Fk (u) =

g > 0, (4.14)

Repeatedly applying the relations (4.14), we obtain the following expression for ∂uj qk (u): 1 g−k qm+k (u)um−1 + uj ∂uj qg (u), j 2 m=1 g−k

∂uj qk (u) =

k = 1, . . . , g − 1.

(4.15)

81


From (3.17) and (4.11)–(4.15) we can write ∂uj wi (u), i = j , in (4.10) in the form 2g+1 n=1,n=i (ui − un ) ∂ui qg (u) − ∂uj qg (u) − ∂uj wi (u) = g ui − uj P0 (ui ) 2g+1 g ∂uj P0 (ui ) 1 n=1,n=i (ui − un ) ∂ui qg (u) + −2 + g g ui − uj P0 (ui ) P0 (ui ) g−1 g−k g g Pn−1 (ui ) 1 ˜ +k−n (2n − 1) g qm+k (u)um−1 + + j 2 P (u ) i 0 k=n n=1 m=1

g g−k +˜ k−n uj ∂uj qg (u) + + k=n

g g g k−n Pn−1 (ui ) 1˜ m−1 (2n − 1) g qk (u) − +k−n−m uj + + 2 P (u ) i 0 n=1 k=n m=1 g (2n − 1) + n=1

1 ∂λi λi − λj ∂uj

g g g Pn−1 (ui ) Pn−1 (uj ) − g qk (u)+˜ k−n . g P0 (ui ) P0 (uj ) k=n (4.16)

Simplifying, we obtain 2g+1 ∂uj wi (u) = −

− un ) ∂u qg (u) g uj )P0 (ui ) j

n=1,n=i (ui

(ui − 2g+1

n=1,n=i (ui − g P0 (ui )

+

un )

1 ∂λi ∂ui qg (u) +2 λi − λj ∂uj g g Pn−1 (ui ) ∂λi 1 − (2n − 1) + g λi − λj ∂uj n=1 P0 (ui )

+

g g Pn−1 (uj ) qk (u)+˜ k−n + − g P0 (uj ) k=l g

P (ui ) g−k (2n − 1) n−1 +˜ k−n uj ∂uj qg (u), + g P0 (ui ) k=n n=1 g

g

i = j, i, j = 1, . . . , 2g + 1. Adding and subtracting the quantity ∂λi 1 wj λi − λj ∂uj

(4.17)

82

TAMARA GRAVA

to (4.17), we can reduce it to the form ∂λi 1 [wi − wj ] λi − λj ∂uj 2g+1 2 ∂λi n=1,n=j (uj − un ) − = ∂uj qg (u) g λi − λj ∂uj P0 (uj )

2g+1 g g g Pn−1 (ui ) n=1,n=i (ui − un ) g−k . (4.18) + (2n − 1) g +˜ k−n uj − g (ui − uj )P0 (ui ) P (u ) i 0 n=1 k=n

∂ uj w i −

The term in parentheses in the right-hand side of (4.18) does not depend on the initial data f (u). It is identically zero for the initial data (3.12) because, in such a case, the wi ’s satisfy (3.4) [15]. Therefore we conclude that ∂ uj w i −

1 ∂λi [wi − wj ] = 0 λi − λj ∂uj

(4.19)

for any smooth monotonically increasing initial data x = f (u)|t =0 . (b) The wi (u)’s satisfy the matching conditions (3.6)–(3.9). In the following, we use the superscript g to denote the corresponding genus of the quantities we are referring to. We have the following relations. When ul = ul+1 = v for 1 l 2g, the +˜ k ’s defined in (2.18) satisfy g +˜ k (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ) g−1 = +˜ k (u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 )−

− v +˜ k−1 (u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 ), g−1

k 1, g > 1,

(4.20)

and the qk (u)’s defined in (3.20) satisfy g

qk (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 )− g

− vqk+1 (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ) g−1

= qk (u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 ), k = 1, . . . , g − 1, g > 1.

(4.21)

For ui = ul = ul+1 = v, we have g−1

∂ui qg−1 (u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 )− 1 − qgg (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ) 2 = (ui − v)∂ui qgg (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ),

(4.22)

which follows from (4.14). g g Next we study the behavior of the polynomials Pk (r) = Pk (r; u), k 0, when √ two branch points become coincident. For this purpose, let us consider ul = v+ *,

83


√

g

ul+1 = v − *, where 0 < * 1. When l is odd, the polynomial Pk (r; u) defined in (2.7) satisfies the following expansion [20]: √ √ g Pk (r; u1 , . . . , ul−1 , v + *, v − *, ul+2 , . . . , u2g+1 ) g−1

= (r − v)Pk

(r)+

g−1 * g−1 g−1 ˜ − (r − v) r g−1−k ∂v Nk (v) + O(* 2 ). (4.23) + σk (v) ∂v µ(v) 2 k=1

In the above formula, the normalized Abelian differential of the second kind g−1 Pk (r) g−1 dr, σk (r) = µ(r) ˜ with pole at infinity of order 2k + 2 and with asymptotic behavior (2.4) is defined on the Riemann surface µ˜ 2 = (r − u1 )(r − u2 ) . . . (r − ul−1 )(r − ul+2 ) . . . (r − u2g+1 ),

(4.24)

the quantity g−1

g−1

σk

(v) =

Pk (v) µ(v) ˜

g−1

and the Nk (v)’s have been defined in (2.17). When l is even, we have [20] √ √ g Pk (r; u1 , . . . , ul−1 , v + *, v − *, ul+2 , . . . , u2g+1 ) Q+ (v) (r − v) g−1 g−1 g−1 µ(r)ω ˜ σk (ξ ), (r − v)Pk (r) − v (r) log * Q− (v)

(4.25)

where ωvg−1(r) is the normalized Abelian differential of the third kind defined on the Riemann surface (4.24) and with simple poles at the points Q± (v) = (v, ±µ(v)) ˜ with residue ±1, respectively. From (4.23) and (4.25), we have that for ui = ul = ul+1 = v g

Pk (ui ; u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ) g−1

= (ui − v)Pk

(ui ),

k 0.

(4.26)

Using the relations (4.20)–(4.22) and (4.26), we obtain for i = l, l + 1, i = 1, . . . , 2g + 1, g

wi (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ) 2g+1 k=1,k=i, l, l+1 (ui − uk ) g−1 ∂ui qg−1 + =2 g−1 P0 (ui )

84

TAMARA GRAVA g−1

+

(2m − 1)

m=1 g

+

Pm−1 (ui ) g−1

g−1

g−1

(ui ) k=m

P0

g−1

(2m − 1)

m=1

Pm−1 (ui ) g−1 P0 (ui )

g−1

qk

g−1 +˜ g−m

g−1 +˜ k−m +

2g+1 −

k=1,k=i, l, l+1 (ui g−1 P0 (ui )

− uk )

qgg (u)|ul =ul+1 , (4.27)

with g−1 g−1 +˜ k = +˜ k (u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 ),

k0

and g−1

g−1

= qk

qk

(u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 ),

k = 1, . . . , g − 1.

The above reduces to the form g

wi (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ) g−1

= wi

+

(u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 )+

2g+1 g g−1 Pm−1 (ui ) g−1 k=1,k=i, l, l+1 (ui − uk ) +˜ g−m − qgg (u)|ul =ul+1 . (2m − 1) g−1 g−1 P (u ) P (u ) i i 0 0 m=1 (4.28)

Using (2.19), the term in parentheses in the right-hand side of (4.28) turns out to be identically zero. Therefore, the boundary conditions (3.7) are satisfied for any smooth monotonically increasing initial data. For studying the boundary conditions √ (3.6), we need to√distinguish between l odd or even. Let us define ul = v + *, ul+1 = v − *, where 0 < * 1. g From (4.23) when l is odd√ the polynomial Pk (r; u) defined in (2.7) has the following expansion at r = v ± *: √ √ √ g Pk (v ± *; u1 , . . . , ul−1 , v + *, v − *, ul+2 , . . . , u2g+1 ) √ g−1 g−1 g−1 ˜ + 2∂v Pk (v)) + O(* 2 ), = ± * Pk (v) + *(σk (v)∂v µ(v) k 0, (4.29) so that √ √ √ g Pk (v ± *; u1 , . . . , ul−1 , v + *, v − *, ul+2 , . . . , u2g+1 ) lim g √ √ √ *→0 P (v ± *; u1 , . . . , ul−1 , v + *, v − *, ul+2 , . . . , u2g+1 ) 0 g−1

=

Pk

(v)

g−1 P0 (v)

.

(4.30)

85

THE TSAREV AND WHITHAM EQUATIONS g−1

g−1

We remark that here and below Pk (v) = Pk (v; u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 ). When l is even, we have from (4.25) √ √ √ g Pk (v ± *; u1 , . . . , ul−1 , v + *, v − *, ul+2 , . . . , u2g+1 ) Q+ (v) √ g−1 1 g−1 µ(v) ˜ σk (ξ ), (4.31) ± * Pk (v) − − log * Q (v) so that

√ √ √ g Pk (v ± *; u1 , . . . , ul−1 , v + *, v − *, ul+2 , . . . , u2g+1 ) lim g √ √ √ *→0 P (v ± *; u1 , . . . , ul−1 , v + *, v − *, ul+2 , . . . , u2g+1 ) 0 Q+ (v) g−1 (ξ ) Q− (v) σk = Q+ (v) g−1 . (ξ ) Q− (v) σ0

(4.32)

Using (4.20), (4.21) and (4.30) we deduce that for l odd and ul = ul+1 = v g

wl (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ) g−1 g−1 k 2µ˜ 2 (v) ∂ g g−1 g−1 Pn−1 (v) ˜ = g−1 qg (u) + qk (2n − 1)+k−n g−1 + P0 (v) ∂ul P0 (v) ul =ul+1 k=1 n=1 +

qgg (u)|ul =ul+1

g g−1 g−1 Pn−1 (v) ˜ (2n − 1)+g−n g−1 , P0 (v) n=1

(4.33)

where g−1

qk

g−1

= qk

(u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 )

and g−1 g−1 +˜ k = +˜ k (u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 ).

Since the functions qk (u) in (3.20) are symmetric with respect to u1 , u2 , . . . , u2g+1 , we immediately obtain that ∂ ∂ g g qg (u) = qg (u) . (4.34) ∂ul ∂ul+1 ul =ul+1 ul =ul+1 Therefore, combining (4.33) and (4.34), we deduce that g

wl (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ) g = wl+1 (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ),

l odd.

When l is even, using (4.20), (4.21) and (4.32), we obtain g

wl (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ) g = wl+1 (u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 )

(4.35)

86

TAMARA GRAVA

=

g−1 k=1

+

g−1 qk

k

(2n −

n=1

qgg (u)|{ul =ul+1 =v}

Q+ (v) g−1 Q− (v) σn−1 (ξ ) g−1 1)+˜ k−n Q+ (v) g−1 + (ξ ) Q− (v) σ0 g

(2k −

k=1

Q+ (v) g−1 Q− (v) σk−1 (ξ ) g−1 1)+˜ g−k Q+ (v) g−1 . (ξ ) Q− (v) σ0

(4.36)

We conclude from (4.35) and (4.36) that the boundary condition (3.6) is satisfied. When g = 1 we deduce from (4.35) w11 (u1 , u1 , u3 ) = w21 (u1 , u1 , u3 ) and from (4.9) and (4.26) w31 (u1 , u1 , u3 ) = 2(u3 − u1 )∂u3 q1 (u1 , u1 , u3 ) + q1 (u1 , u1 , u3 ). From (3.17) and (3.20) we get the relation q1 (u1 , u1 , u3 ) = f (u3 ) + 2(u1 − u3 )∂u3 q1 (u1 , u1 , u3 ) so that w31 (u1 , u1 , u3 ) = f (u3 ). An analogous result can be obtain when u2 = u3 , so that the boundary conditions (3.8)–(3.9) are satisfied. The theorem is then proved. ✷ Remark 4.3. Expression (3.15) obtained by Tian [15] for the polynomial initial data (3.12) is equivalent to (4.9) for any smooth monotonically increasing initial data. Indeed, (3.15) can be reduced to (4.9) using (4.14) and the identity    γ j  0 0 ··· 0 +0 1  γ2j   +1 + 0 · · · 0  0  ∂   ···    ··· ··· ··· ···  ···     j ∂ui  + · · · +0 0   γg−1  g−2 +g−3 j +g−1 +g−2 · · · +1 +0 γg   σ0 (r)φj (r) Resr=ui [ dr ] σ (r)φ (r)   3 Resr=ui [ 1 dr j ]  1 ,  (4.37) =  ··· σg−2 (r)φj (r)  4  ] (2g − 3) Resr=ui [ dr σ (r)φ (r) (2g − 1) Resr=ui [ g−1 dr j ] where the σk (r)’s are the Abelian differentials of the second kind defined in (2.4), φj (r) is the j th holomorphic differential defined in (2.2) and Resr=ui is the residue evaluated at r = ui . The +k ’s are the coefficients of the expansion defined in (2.16). The above identity can be obtained from Proposition 5.4 (see below). Formula (3.15) is not effective for proving Theorem 4.2.

87


5. C 1 -Smoothness of the Solution of the Hodograph Transformation In the following, we show that the g-phase solutions of the Whitham equations u1 (x, t) > u2 (x, t) > · · · > u2g+1 (x, t) for different g 0 describe a C 1 -smooth multivalued curve in the x–u plane evolving smoothly in time. The matching conditions (3.6)–(3.9) guarantee that the C 1 -smoothness is preserved on the phase transition boundary where two Riemann invariants coalesce. THEOREM 5.1 ([8]). Let us suppose that the g-phase solution defined by (3.5) exists for some x and t > 0. Then on the solution of (3.5) (∂x ui (x, t))−1 = ∂ui (−λi (u)t + wi (u)),

i = 1, . . . , 2g + 1.

Proof. Deriving with respect to x Equation (3.5), we obtain ∂uj (−λi (u)t + wi (u))∂x uj . 1 = ∂ui (−λi (u)t + wi (u))∂x ui +

(5.38)

(5.39)

j =i

By (1.4) we have ∂uj (−λi (u)t + wi (u)) = −∂uj λi (u)t + ∂uj wi (u), i = j, i, j = 1, . . . , 2g + 1 ∂λi 1 (−λi (u)t + wi (u)) − (−λj (u)t + wj (u)) = λi − λj ∂uj = 0.

(5.40)

Thus (5.39) becomes 1 = ∂ui (−λi (u)t + wi (u))∂x ui ,

i = 1, . . . , 2g + 1.

(5.41) ✷

In the next theorem, we evaluate explicitly ∂ui (−λi (u)t + wi (u)), i = 1, . . . , 2g + 1. THEOREM 5.2. On the solution of the hodograph transformation (3.5), the following relation is satisfied: 2g+1 m=1 (ui − um ) ∂ m=i g (ui ; u), (−λi (u)t + wi (u)) = g ∂ui P0 (ui ) i = 1, . . . , 2g + 1, g > 0, (5.42) where

2g+1

g (r; u) = ∂r 5 g (r; u) +

k=1

∂uk 5 g (r; u)

(5.43)

88

TAMARA GRAVA

and 5 g (r; u) satisfies the linear overdetermined system ∂ g ∂ ∂2 5 (r; u) − 5 g (r; u) = 2(ui − uj ) 5 g (r; u), ∂ui ∂uj ∂ui ∂uj i, j = 1, . . . , 2g + 1, ∂ ∂2 ∂ g 5 (r; u) − 2 5 g (r; u) = 2(r − uj ) 5 g (r; u), ∂r ∂uj ∂r∂uj

i = j,

(5.44)

j = 1, . . . , 2g + 1,

2g f (g)(r). 5 g (r; r, . . . , r ) = (2g + 1)!! 2g+1

Remark 5.3. For polynomial initial data, the solution of the above initial value problem coincides with the function 5 g (r; u) defined in (4.2). The integration procedure of (5.44) is analogous to the one illustrated in the proof of Theorem 3.17. The function 5 g (r; u) that solves (5.44) satisifes the relations (4.7). We observe that the determinant of the Jacobian of the hodograph transformation can be obtained easily from (5.42), namely 2g+1 2g+1 m=1 (ui − um ) m=i det(Jacobian) = . (5.45) g (ui ; u) g P (u ) i 0 i=1 Therefore, the above determinant is nondegenerate if g (ui ; u) = 0, i = 1, . . . , 2g + 1. Proof of Theorem 5.2. From a generalization of a result in [4], we obtain g g 1 ∂ Pk (r) ∂ Pk (ui ) = , (5.46) ∂ui P0g (ui ) 2 ∂r P0g (r) r=ui so that ∂ (−λi (u)t + wi (u)) ∂ui 2g+1 2g+1 g k=1 m=1 (ui − uj ) ∂ P1 (r) k=i m=k,i ∂ui qg (u)+ + 2 = −6t g ∂r P0g (r) r=ui P0 (ui ) 2g+1 2g+1 m=1 (ui − um ) m=1 (ui − um ) m=i m=i g 2 (∂ui ) qg (u) − 2 ∂ui qg (u)∂ui P0 (ui )+ +2 g g 2 P0 (ui ) (P0 (ui )) g g g ∂ Pn−1 (r) 1 (2n − 1) qm (u)+˜ m−n + + 2 ∂r P g (r) n=1

(2n − g

+

n=1

r=ui m=n 0 g g P (ui ) 1) n−1 ∂u (qm (u)+˜ m−n ). g P0 (ui ) m=n i

(5.47)

89


In the above relation we need to compute the following derivatives: g−m ∂ui (qm (u)+˜ m−n ) = ui +˜ m−n ∂ui qg (u),

(5.48)

which are obtained from (2.18) and (4.14) and g

g

g

∂ui P0 (ui ) = ∂r P0 (r)|r=ui + ∂ui P0 (r)|r=ui , g

where P0 (r) = r g + α10 r g−1 + · · · + αg0 . In order to obtain the derivative of the normalization constants α10 , α20 , . . . , αg0 , we need the following proposition. PROPOSITION 5.4 ([19]).√Let be ω1 (r) and ω2 (r) two normalized Abelian differentials on Sg . Let ξ = 1/ r be the local coordinate at infinity and ak1 ξ k dξ, ω2 = ak2 ξ k dξ. ω1 = k

k

Define the bilinear product Vω1 ω2 =

1 a−k−2 ak2 k0

k+1

,

then ω1 (r)ω2 (r) ∂ , Vω1 ω2 = Res r=ui ∂ui dr

i = 1, . . . , 2g + 1,

(5.49)

where Resr=ui (ω1 (r)ω2 (r))/dr is the residue of the differential (ω1 (r)ω2 (r))/dr evaluated at r = ui . Applying the above proposition to σ0 and σk , k = nontrivial simplifications we obtain  1   0  α0  0 0 1 0 ui  1 0 0 α2     1 ∂    1 ··· ···  ···  = −  ...  −  ··· ∂ui  α 0  2  ug−1  2  ug−3 ug−4 · · · i i i g−1 g−1 g−2 g−3 αg0 ui ui ··· ui  +˜ 0 0 +˜ 0 +˜ 1 g  P (ui ) 1  ··· ··· + 2g+1 0  2 k=1,k=i (ui − uk )  +˜ ˜ + g−2 g−3 +˜ g−1 +˜ g−2   1 g 3P1 (ui )     ... × , g  (2g − 3)P (u )  i g−2 g (2g − 1)Pg−1 (ui )

0, . . . , g − 1 and after · · · 0   α10  · · · 0   α20    ··· ···  ···  +   0 0 0 αg−1  1 0 αg0  0 ··· 0 0 ··· 0  ··· ··· ··· × · · · +˜ 0 0  · · · +˜ 1 +˜ 0

(5.50)

90

TAMARA GRAVA

where the +˜ k ’s have been defined in (2.18). From the above formula, we obtain g g g ∂ui P0 (ui ) = ∂r P0 (r)|r=ui + ∂ui P0 (r)|r=ui 1 g = ∂r P0 (r)|r=ui + 2 g g g P (ui ) 1 g g−m (2n − 1)Pn−1 (ui ) ui +˜ m−n . + 2g+1 0 2 k=1,k=i (ui − uk ) n=1 m=n (5.51) Using relations (5.48) and (5.51), we simplify (5.47) to the form ∂ (−λi (u)t + wi (u)) ∂ui 2g+1 ∂ui (2 m=1 (ui − um )∂ui qg (u)) m=i + = g P0 (ui ) g g g g P (r) P (r) 1 + (2n − 1) n−1 qm +˜ m−n + + ∂r −12t 1g g 2 P0 (r) n=1 P0 (r) m=n 2g+1

m=1 (ui − um ) m=i ∂ui qg (u) . +2 g P0 (r) r=ui

(5.52)

The above expression can be written in the form ∂ (−λi (u)t + wi (u)) ∂ui g g ∂ −xP0 (r) − 12tP1 (r) + R g (r) + = g ∂r 2P0 (r) r=ui 2g+1

2g+1 m=1 (ui − um ) ∂u qg (u) − ∂u qg (u) m=i i k 2(∂ui )2 qg + , + g u − u P0 (ui ) i k k=1

(5.53)

k=i

where the polynomial R g (r) has been defined in (4.4). We remark that such a polynomial has been defined for the analytic initial data (3.12), but it can be naturally extended to any smooth initial data. Applying the relations (3.17) and (4.7) to the last term of the above expression we obtain ∂u qg (u) − ∂u qg (u) i k u − u i k k=1

2g+1

2(∂ui ) qg + 2

k=i

2g+1

= 2(∂ui ) qg + 2 2

k=1 k=i

∂ui ∂uk qg

91

THE TSAREV AND WHITHAM EQUATIONS 2g+1 ∂uk 5 g (ui ; u) = (∂r + ∂ui )5 (r; u) r=ui + g

k=1 k=i

= g (ui ; u), where the function g (r; u) has been defined in (5.43). Substituting the above relation in (5.53), we obtain ∂ (−λi (u)t + wi (u)) ∂ui g g ∂ −xP0 (r) − 12tP1 (r) + R g (r) + = g ∂r 2P0 (r) r=ui 2g+1 m=1 (ui − um ) m=i + g (ui ; u). g P0 (ui )

(5.54)

We observe that g

g

−xP0 (r) − 12tP1 (r) + R g (r) ≡ 0,

g > 0,

on the solution of the Whitham equations. Indeed the hodograph transformation (3.5) is equivalent to imposing 2g + 1 zeros on the above polynomial which has degree 2g. Consequently, such a polynomial is identically zero. Therefore, we can simplify (5.54) to the form (5.42) when u1 > u2 > · · · > u2g+1 satisfy the g-phase Whitham equations, g > 0. ✷ From (5.42) it is clear that the ui ’s evolve in such a way that the graph of their curve is certainly C 1 -smooth in the x − u plane for u1 > u2 > · · · > u2g+1 . Indeed, g (r; u) is a smooth function of r and u when the initial data (5.44) is smooth. The next lemma shows that this property is preserved when two Riemann invariants coalesce. LEMMA 5.5. When ul = ul+1 the following relations are satisfied: ∂ui (−λi (u)t + wi (u))|{ul =ul+1 =v} g−1

= ∂ui (−λi

g−1

t + wi

),

i = l, l + 1,

(5.55)

where g−1

λi

g−1 wi

g−1

= λi =

(u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 ) g−1 wi (u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 ).

Regarding ∂x ul and ∂x ul+1 , we have lim

ul →ul+1

and

! 1 1 − = 0. (5.56) ∂ul (−λl (u)t + wl (u)) ∂ul+1 (−λl+1 (u)t + wl+1 (u))

92

TAMARA GRAVA

Proof. We analyze the behavior of the derivatives ∂ui (−λi (u)t + wi (u)), i = 1, . . . , 2g + 1 when ul = ul+1 = v. We first prove (5.55). From (4.23) and (4.25) we have that, for ui = ul = ul+1 = v , g

g−1

Pk (ui ; u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 ) = (ui − v)Pk g ∂r Pk (r; u1 , . . . , ul−1 , v, v, ul+2 , . . . , u2g+1 )|r=ui g−1 g−1 = P (ui ) + (ui − v)∂r P (r) , k

k

(ui ),

k 0, (5.57)

r=ui

where g−1

Pk

g−1

(r) = Pk

(r; u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 ),

k 0.

Applying (4.20), (4.21) and (5.57) to expression (5.52), we obtain g g ∂u (−λ (u)t + w (u)) {ul =v i

i

i

ul+1 =v}

! g−1 g−1 (r)|r=ui 1 g−1 ∂r P0 + + = − −λi t + wi g−1 2(ui − v) 2P0 (ui ) 2 k=i (ui − uk ) ∂ q g (u)|{ul =ul+1 =v} + + 2(ui − v) g−1 2 g ∂u P0 (ui ) i

4 k=i (ui − uk ) k=i m=k,i (ui − um ) + 2(ui − v) × + g−1 g−1 P0 (ui ) P0 (ui ) × ∂ui qgg (u){ul =u =v} + l+1 g g qg (u)|{ul =ul+1 =v} g−1 g−1 (2n − 1)+˜ g−n ∂r Pn−1 (r)|r=ui + + g−1 2P0 (ui ) n=1

g 1 g−1 g−1 (2n − 1)Pn−1 (ui )+˜ g−n + + (ui − v) n=1

g−1 g−1 1 g−1 g−1 g−1 ˜ g−1 ∂r (2n − 1)Pn−1 (r) qm +m−n − 12tP1 (r) + + g−1 2P0 (ui ) r=u i m=n n=1 g−1

g−1 1 g−1 g−1 g−1 (2n − 1)Pn−1 (ui ) qmg−1 +˜ m−n − 12tP1 (ui ) , + g−1 2(ui − v)P0 (ui ) n=1 m=n (5.58) where

2g+1

=

k=1 k=l,l+1

and

2g+1

=

m=1 m=l,l+1

.

93


Using the relation (2.19), we obtain g

g−1 g−1 (2n − 1)+˜ g−n ∂r Pn−1 (r)r=ui = 2

k=i (ui

n=1

− uk )

(5.59)

(ui − uj )

j =i

and g

g−1 g−1 (2n − 1)+˜ g−n Pn−1 (ui ) =

n=1

k=i

(ui − uk ).

(5.60)

Substituting (5.59) and (5.60) in (5.58) and applying relation (4.22), we can easily obtain (5.55). For proving (5.56), we consider only l even. Analogous considerations can be done for l odd. Using expansion (4.25) and relation (2.13), we have √ √ g ∂r Pk (r; u1 , . . . , ul−1 , v + *, v − *, ul+2 , . . . , u2g+1 )) g−1

Pk + so that

g−1

(r) + (r − v)∂r Pk (r)+ g−1 g−1 ∂r ((r − v) j =1 Nj (v)r g−1−j ) log *

Q+ (v) Q− (v)

g−1

σk

(ξ ),

k 0, (5.61)

√ √ *, v − *, ul+2 , . . . , u2g+1 )|r=v±√* √ g P0 (v ± *) Q+ (v) g−1 g−1 g−1−j g−1 (ξ ) v σk (v) Q− (v) σk g−1 − Q+ (v) g−1 Nj (v)± − log * Q+ (v) g−1 µ(v) ˜ σ (ξ ) σ (ξ ) j =1 − − g

∂r Pk (r; u1 , . . . , ul−1 , v +

Q (v)

0

σ0 (v)σk (v)

Q (v)

0

√

* log2 *. ± Q+ (v) g−1 2 ( Q− (v) σ0 (ξ ))

(5.62)

Substituting (4.32), (4.36) and (5.62) in (5.56) and using the explicit expression (5.38) of the derivative, we obtain g g ∂u (−λ (u)t + w (u)) {ul =v+√* l

l

l

√ ul+1 =v− *}

g−1 √ σ0 (v) 1 2 × log * − * log * Q+ (v) g−1 2 (ξ ) Q− (v) σ0 Q+ (v) g−1 ! g−1 (ξ ) Z g−1 (v) σ0 (v) Q− (v) Z − g−1 × Q+ (v) g−1 Q+ (v) g−1 σ0 (v) σ (ξ ) σ (ξ ) − − 0 0 Q (v) Q (v)

and

g g ∂ul+1 (−λl+1 (u)t + wl+1 (u))

√ {ul =v+ * √ ul+1 =v− *}

(5.63)

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TAMARA GRAVA

g−1 √ 1 σ0 (v) × log * + * log2 * Q+ (v) g−1 2 σ (ξ ) − Q (v) 0 Q+ (v) g−1 ! g−1 (ξ ) Z g−1 (v) σ0 (v) Q− (v) Z − g−1 , × Q+ (v) g−1 Q+ (v) g−1 σ0 (v) (ξ ) Q− (v) σ0 (ξ ) Q− (v) σ0

(5.64)

where g−1

Z g−1 (r) = −12tσ1

(r) +

g−1 g−1 g−1 g−1 (2n − 1)σn−1 (r) qmg−1 +˜ m−n + m=n

n=1

2g+1

+ 2µ(r) ˜ dr

∂uk qgg (u){ul =u

l+1 =v}

.

k=1

In the above formulas, Z g−1 (v) =

Z g−1 (r) , dr r=v

qmg−1 = qmg−1 (u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 )

and +˜ mg−1 = +˜ mg−1 (u1 , . . . , ul−1 , ul+2 , . . . , u2g+1 ). Substituting (5.63) and (5.64) into (5.56), it is easy to verify that the limit (5.56) is satisfied. ✷ Combining Theorem 5.2 and Lemma 5.5, we conclude that the g-phase solutions u1 (x, t) > u2 (x, t) > · · · > u2g+1 (x, t) for different g 0 are glued together in order to produce a C 1 -smooth multivalued curve in the x−u plane evolving smoothly with time. 6. Conclusion In this work, we have obtained a new formula for the solution wi (u), i = 1, . . . , 2g + 1, of the Tsarev system with matching conditions (3.6)–(3.9) for any increasing smooth initial data. As in [15], this new formula is expressed through some auxiliary functions which solve the linear overdetermined systems of the Euler–Poisson–Darboux type. We have shown that the matching conditions (3.6)– (3.9) for the wi (u)’s guarantee that the solution of the Tsarev system is unique and that the g-phase solutions of the Whitham equations for different g, are glued together in order to produce a C 1 -smooth multivalued curve in the (x, u) plane evolving smoothly with time. The new formula (4.9) enabled us to obtain a simple expression for the Jacobian of the hodograph transformation. We believe that this formula will be useful in the future to investigate the solvability of the hodograph transformation for g > 1. Indeed, it is still an open problem to prove the existence


95

of a real solution u1 (x, t) > u2 (x, t) > · · · > u2g+1 (x, t), g > 1, of the hodograph transformation (1.5) for all x and t > 0 and for generic smooth initial data.

Acknowledgements I am indebted to Professor Boris Dubrovin who posed me the problem of this work and gave me many hints on how to reach the solution. I am grateful to Professor Sergei Novikov for his suggestions during the preparation of the manuscript. This work was partially supported by a CNR grant 203.01.70, by a grant of S. Novikov, and by a Marie Curie Fellowship. I wish to thank the anonymous referee for the suggested improvements to the manuscript. I am also grateful to T. Liverpool for carefully reading the manuscript.

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Whitham, G. B.: Linear and Nonlinear Waves, Wiley, New York, 1974. Flaschka, H., Forest, M. and McLaughlin, D. H.: Multiphase averaging and the inverse spectral solution of the Korteweg–de Vries equations, Comm. Pure Appl. Math. 33 (1980), 739–784. Lax, P. D. and Levermore, C. D.: The small dispersion limit of the Korteweg–de Vries equation, I, II, III, Comm. Pure Appl. Math. 36 (1983), 253–290, 571–593, 809–830. Levermore, C. D.: The hyperbolic nature of the zero dispersion KdV limit, Comm. Partial Differential Equations 13 (1988), 495–514. Fei Ran Tian: Oscillations of the zero dispersion limit of the Korteweg–de Vries equations, Comm. Pure Appl. Math. 46 (1993), 1093–1129. Grava, T.: Existence of a global solution of the Whitham equations, Theoret. Math. Phys. 122(1) (2000), 46–58. Dubrovin, B. and Novikov, S. P.: Hydrodynamic of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Russian Math. Surveys 44(6) (1989), 35–124. Tsarev, S. P.: Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Soviet Math. Dokl. 31 (1985), 488–491. Gurevich, A. G. and Pitaevskii, L. P.: Nonstationary structure of collisionless shock waves, JEPT Lett. 17 (1973), 193–195. Krichever, I. M.: The method of averaging for two dimensional integrable equations, Funct. Anal. Appl. 22 (1988), 200–213. Potemin, G. V.: Algebraic-geometric construction of self-similar solutions of the Whitham equations, Uspekhi Mat. Nauk 43(5) (1988), 211–212. Kudashev, V. R. and Sharapov, S. E.: Inheritance of KdV symmetries under Whitham averaging and hydrodynamic symmetry of the Whitham equations, Theoret. Math. Phys. 87(1) (1991), 40–47. Gurevich, A. V., Krylov, A. L. and El, G. A.: Evolution of a Riemann wave in dispersive hydrodynamics, Soviet Phys. JEPT 74(6) (1992), 957–962. Eisenhart, L. P.: Ann. of Math. 120 (1918), 262. Fei Ran Tian: The Whitham type equations and linear over-determined systems of Euler– Poisson–Darboux type, Duke Math. J. 74 (1994), 203–221. El, G. A.: Generating function of the Whitham–KdV hierarchy and effective solution of the Cauchy problem, Phys. Lett. A 222 (1996), 393–399. Springer, G.: Introduction to Riemann Surfaces, Addison-Wesley, Reading, Mass., 1957.

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Rodin, Yu. L.: The Riemann Boundary Value Problem on Riemann Surfaces, Math. Appl. Soviet Ser., D. Reidel, Dordrecht, 1987. Dubrovin, B.: Lectures on 2-D Topological Field Theory, Lecture Notes in Math. 1620, Springer-Verlag, Berlin, 1996. Fay, J.: Theta Functions on Riemann Surface, Lecture Notes in Math. 352, Springer-Verlag, Heidelberg, 1973.

19. 20.


97

A General Framework for Localization of Classical Waves: I. Inhomogeneous Media and Defect Eigenmodes ABEL KLEIN1 and ANDREW KOINES2, 1 University of California, Irvine, Department of Mathematics, Irvine, CA 92697-3875, U.S.A.

e-mail: [email protected] 2 Texas A&M University, Department of Mathematics, College Station, TX 77843-3368, U.S.A. (Received: 7 February 2001) Abstract. We introduce a general framework for studying the localization of classical waves in inhomogeneous media, which encompasses acoustic waves with position dependent compressibility and mass density, elastic waves with position dependent Lamé moduli and mass density, and electromagnetic waves with position dependent magnetic permeability and dielectric constant. We also allow for anisotropy. We develop mathematical methods to study wave localization in inhomogeneous media. We show localization for local perturbations (defects) of media with a spectral gap, and study midgap eigenmodes. Mathematics Subject Classifications (2000): 35Q60, 35Q99, 78A99, 78A48, 74J99, 35P99, 47F05. Key words: wave localization, inhomogeneous media, defects, midgap eigenmodes.

1. Introduction We provide a general framework for studying localization of acoustic waves, elastic waves, and electromagnetic waves in inhomogeneous media, i.e., the existence of acoustic, elastic, and electromagnetic waves such that almost all of the wave’s energy remains in a fixed bounded region uniformly over time. Our general framework encompasses acoustic waves with position dependent compressibility and mass density, elastic waves with position dependent Lamé moduli and mass density, and electromagnetic waves with position dependent magnetic permeability and dielectric constant. We also allow for anisotropy. In this first article we develop mathematical methods to study wave localization in inhomogeneous media. As an application we show localization for local perturbations (defects) of media with a gap in the spectrum, and study midgap eigenmodes. In the second article [17], we use the methods developed in this article This work was partially supported by NSF Grant DMS-9800883. Current address: Orange Coast College, Department of Mathematics, Costa Mesa, CA 92626,

U.S.A.

98

ABEL KLEIN AND ANDREW KOINES

to study wave localization for random perturbations of periodic media with a gap in the spectrum. Our results extend the work of Figotin and Klein [7–11, 16] in several ways: (1) We study a general class of classical waves which includes acoustic, electromagnetic and elastic waves as special cases. (2) We allow for more than one inhomogeneous coefficient (e.g., electromagnetic waves in media where both the magnetic permeability and the dielectric constant are position dependent). (3) We allow for anisotropy in our wave equations. (4) In [17] we prove strong dynamical localization in random media, using the recent results of Germinet and Klein [14] on strong dynamical localization and of Klein, Koines and Seifert [18] on a generalized eigenfunction expansion for classical wave operators. Previous results on localization of classical waves in inhomogeneous media [7– 11, 16, 3] considered only the case of one inhomogeneous coefficient. Acoustic and electromagnetic waves were treated separately. Elastic waves were not discussed. Our approach to the mathematical study of localization of classical waves, as in the work of Figotin and Klein, is operator theoretic and reminiscent of quantum mechanics. It is based on the fact that many wave propagation phenomena in classical physics are governed by equations that can be recast in abstract Schrödinger form [21, 8, 16]. The corresponding self-adjoint operator, which governs the dynamics, is a first order partial differential operator, but its spectral theory may be studied through an auxiliary self-adjoint, second order partial differential operator. These second-order classical wave operators are analogous to Schrödinger operators in quantum mechanics. The method is particularly suitable for the study of phenomena historically associated with quantum mechanical electron waves, especially Anderson localization in random media [7, 8, 11, 16] and midgap defect eigenmodes [9, 10]. Physically interesting inhomogeneous media give rise to nonsmooth coefficients in the classical wave equations, and hence in their classical wave operators (e.g., a medium composed of two different homogeneous materials will be represented by piecewise constant coefficients). Thus we make no assumptions about the smoothness of the coefficients of classical wave operators. Since we allow two inhomogeneous coefficients, we have to deal with domain questions for the quadratic forms associated with classical wave operators. We must also take into account that many classical wave equations come with auxiliary conditions, and the corresponding classical wave operators are not elliptic (e.g., the Maxwell operator – see [21, 20, 8, 11, 16]). This paper is organized as follows: In Section 2 we introduce our framework for studying classical waves. We discuss classical wave equations in inhomogeneous media and wave localization. We define first and second order classical wave operators, and use them to rewrite the wave equations in abstract Schrödinger form. We state our results on wave localization created by defects. In Section 3 we study classical wave operators, obtaining the technical tools that are needed for proving localization in inhomogeneous and random media. We study finite volume classi-

A GENERAL FRAMEWORK FOR LOCALIZATION OF CLASSICAL WAVES: I

99

cal wave operators, discuss interior estimates, give an improved resolvent decay estimate in a gap, prove a Simon–Lieb-type inequality and an eigenfunction decay inequality. In Section 4 we study periodic classical wave operators, and prove a theorem that gives the spectrum of a periodic classical wave operator in terms of the spectra of its restriction to finite cubes with periodic boundary condition. In Section 5 we study the effect of defects on classical wave operators, and give the proofs and details of the results on defects and wave localization stated in Section 2.

2. The Mathematical Framework 2.1. CLASSICAL WAVE EQUATIONS Many classical wave equations in a linear, lossless, inhomogeneous medium can be written as first order equations of the form: ∂ ψt (x) = D∗ φt (x), ∂t ∂ R(x)−1 φt (x) = −Dψt (x), ∂t K(x)−1

(2.1)

where x ∈ Rd (space), t ∈ R (time), ψt (x) ∈ Cn and φt (x) ∈ Cm are physical quantities that describe the state of the medium at position x and time t, D is an m×n matrix whose entries are first-order partial differential operators with constant coefficients (see Definition 2.1), D∗ is the formal adjoint of D, and, K(x) and R(x) are n × n and m × m positive, invertible matrices, uniformly bounded from above and away from 0, that describe the medium at position x (see Definition 2.3). In addition, D satisfies a partial ellipticity property (see Definition 2.2), and there may be auxiliary conditions to be satisfied by the quantities ψt (x) and φt (x). The physical quantities ψt (x) and φt (x) then satisfy second-order wave equations, with the same auxiliary conditions: ∂2 ψt (x) = −K(x)D∗ R(x)Dψt (x), ∂t 2 ∂2 φt (x) = −R(x)DK(x)D∗ φt (x). ∂t 2

(2.2) (2.3)

Conversely, given (2.2) (or (2.3)), we may write this equation in the form (2.1) by introducing an appropriate quantity φt (x) (or ψt (x)), which will then satisfy Equation (2.3) (or (2.2)). The medium is called homogeneous if the coefficient matrices K(x) and R(x) are constant, i.e., they do not depend on the position x. Otherwise the medium is said to be inhomogeneous.

100


EXAMPLES. Electromagnetic waves: Maxwell equations are given by (2.1) with d = n = m = 3, ψt (x) the magnetic field, φt (x) the electric field, D = D∗ the curl, Dφ = ∇ × φ, K(x) =

1 I3 µ(x)

and

R(x) =

1 I3 , ε(x)

with µ(x) the magnetic permeability and ε(x) the dielectric constant. (By Ik we denote the k × k identity matrix.) The auxiliary conditions are ∇ · µψt = 0 and ∇ · εφt = 0. Acoustic waves: The acoustic equations in d dimensions may be written as (2.1), with n = 1, m = d, ψt (x) the pressure, φt (x) the velocity, D the gradient, Dφ = ∇φ, D∗ ψ = −∇ · ψ,

K(x) =

1 I1 κ(x)

and

R(x) =

1 Id , (x)

with κ(x) the compressibility and (x) the mass density. The auxiliary condition is ∇ × φt = 0. The usual second-order acoustic equation for the pressure is then given by (2.2). Elastic waves: The equations of motion for linear elasticity, in an isotropic medium, can be written as the second-order wave equation ρ(x)

∂2 ψt (x) = − ∇[λ(x) + 2µ(x)]∇ ∗ + ∇ × µ(x)∇× ψt (x), 2 ∂t

(2.4)

where x ∈ R3 , ψt (x) is the medium displacement, ρ(x) is the mass density, and, λ(x) and µ(x) are the Lamé moduli. It is of the form of Equation (2.2), with n = 3, D the differential operator given by Dψ = (∇ ∗ ψ) ⊕ (∇ × ψ)

(a 4 × 3 matrix),

K(x) =

1 I3 , ρ(x)

and R(x) = (λ(x) + 2µ(x))I1 ⊕ µ(x)I3

(a 4 × 4 matrix).

2.2. WAVE EQUATIONS IN ABSTRACT SCHRÖDINGER FORM The wave equation (2.1) may be rewritten in abstract Schrödinger form [21, 8, 16]: d t = Wt , dt ψ where t = φ t and −i

W=

(2.5)

t

0 iR(x)D

−iK(x)D∗ 0

.

(2.6)

101


The (first-order) classical wave operator W is formally (and can be defined as) a self-adjoint operator on the Hilbert space (2.7) H = L2 Rd , K(x)−1 dx; Cn ⊕ L2 Rd , R(x)−1 dx; Cm , where, for a k × k positive invertible matrix-valued measurable function S(x), we set L2 Rd , S(x)−1 dx; Ck

= f : Rd → Ck ; f, S(x)−1 f L2 (Rd ,dx;Ck ) < ∞ . The auxiliary conditions to the wave equation are imposed by requiring the solutions to Equation (2.5) to also satisfy ⊥ t , t = PW

(2.8)

⊥ denotes the orthogonal projection onto the orthogonal complement of where PW the kernel of W. The solutions to Equations (2.5) and (2.8) are of the form ⊥ 0 , t = eit W PW

0 ∈ H .

(2.9)

The energy density at time t of a solution ≡ t (x) = (ψt (x), φt (x)) of the wave equation (2.1) is given by E (t, x) = 12 ψ(x), K(x)−1 ψt (x)Cn + φt (x), R(x)−1 φt (x)Cm . (2.10) The wave energy, a conserved quantity, is thus given by E = 12 t 2H

for any t.

(2.11)

Note that (2.9) gives the finite energy solutions to the wave equation (2.1). 2.3. WAVE LOCALIZATION Let = t (x) be a finite energy solution of the wave equation (2.1). There are many criteria for wave localization, e.g.: Simple localization: Almost all of the wave’s energy remains in a fixed bounded region at all times, more precisely: 1 E (t, x) dx = 1. (2.12) lim inf R→∞ t E |x|R Moment localization: For some (or we may require for all) q > 0, we have q |x|q E (t, x) dx = 12 sup |x| 2 t 2H < ∞. (2.13) sup t

Rd

t

102


Exponential localization (in the L2 -sense): For some C < ∞ and m > 0, we have sup χx E (t, ·) = √1 sup χx t H Ce−m|x| (2.14) t

2

2

t

for all x ∈ Rd , where χx denotes the characteristic function of a cube of side 1 centered at x. It is easy to see that exponential localization implies moment localization for all q > 0, and moment localization for some q > 0 implies simple localization. The fact that finite energy solutions are given by (2.9) suggests a method to obtain localized waves: if 0 ∈ H is an eigenfunction for the classical wave operator W with nonzero eigenvalue ω, i.e., W0 = ω0 with ω q= 0, then the wave t = eit ω 0 exhibits simple localization. If in addition |x| 2 0 2H < ∞, we have moment localization. If 0 is exponentially decaying (in the L2 -sense), we have exponential localization. In a homogenous medium, a classical wave operator cannot have nonzero eigenvalues. (This can be shown using the Fourier transform.) Thus an appropriate inhomogenous medium is required to produce nonzero eigenvalues, and hence localized waves. In Subsection 2.5 we will see that we can produce eigenvalues in spectral gaps of classical wave operators by introducing defects, i.e., by making local changes in the medium. Moreover, the corresponding waves will exhibit exponential localization. In the sequel [17] we show that random changes in the media can produce Anderson localization in spectral gaps of periodic classical wave operators. 2.4. CLASSICAL WAVE OPERATORS We now introduce the mathematical machinery needed to make the preceding discussion mathematically rigorous. It is convenient to work on L2 (Rd , dx; Ck ) instead of the weighted space 2 L (Rd , S(x)−1 dx; Ck ). To do so, note that the operator VS , given by multiplication by the matrix S(x)−1/2 , is a unitary map from the Hilbert space L2 (Rd , S(x)−1 dx;

= (VK ⊕ VR )W(V ∗ ⊕ V ∗ ), we have Ck ) to L2 (Rd , dx; Ck ), and if we set W K R √ √ ∗ 0√ −i K(x)D R(x)

= √ , (2.15) W 0 i R(x)D K(x) a formally self-adjoint operator on L2 (Rd , dx; Cn ) ⊕ L2 (Rd , dx; Cm ). In addition, if S− I S(x) S+ I with 0 < S− S+ < ∞, as it will be the case in this article, it turns out that if ϕ = VS ϕ, then the functions ϕ(x) and

ϕ (x) share the same decay and growth proporties (e.g., exponential or polynomial decay). Thus it will suffice for us to work on L2 (Rd , dx; Ck ), and we will do so in the remainder of this article. We set H (k) = L2 (Rd , dx; Ck ).

(2.16)


103

Given a closed densely defined operator T on a Hilbert space H , we will denote its kernel by ker T and its range by ran T ; note ker T ∗ T = ker T . If T is selfadjoint, it leaves invariant the orthogonal complement of its kernel; the restriction of T to (ker T )⊥ will be denoted by T⊥ . Note that T⊥ is a self-adjoint operator on the Hilbert space (ker T )⊥ = PT⊥ H , where PT⊥ denotes the orthogonal projection onto (ker T )⊥ . DEFINITION 2.1. A constant coefficient, first-order, partial differential operator D from H (n) to H (m) (CPDO(1) n,m ) is of the form D = D(−i∇), where, for a d-component vector k, D(k) is the m × n matrix D(k)r,s = ar,s · k, ar,s ∈ Cd . (2.17) D(k) = D(k)r,s r=1,...,m ; s=1,...,n

We set D+ = sup{D(k); k ∈ Cd , |k| = 1},

(2.18)

so D(k) D+ |k| for all k ∈ Cd . Note that D+ is bounded by the norm of the matrix [|ar,s |] r=1,...,m . s=1,...,n Defined on D(D) = {ψ ∈ H (n) : Dψ ∈ H (m) in distributional sense},

(2.19)

∞ d n a CPDO(1) n,m D is a closed, densely defined operator, and C0 (R ; C ) (the space of infinitely differentiable functions with compact support) is an operator core for D. We will denote by D∗ the CPDO(1) m,n given by the formal adjoint of the matrix in (2.17).

DEFINITION 2.2. A CPDO(1) n,m D is said to be partially elliptic if there exists a (1) ⊥ CPDOn,q D (for some q), satisfying the following two properties: D⊥ D∗ = 0, D∗ D + (D⊥ )∗ D⊥ ,[(−-) ⊗ In ],

(2.20) (2.21)

with , > 0 being a constant. (- = ∇ · ∇ is the Laplacian on L2 (Rd , dx); In denotes the n × n identity matrix.) If D is partially elliptic, we have H (n) = ker D⊥ ⊕ ker D,

(2.22)

D∗ D + (D⊥ )∗ D⊥ = (D∗ D)⊥ ⊕ ((D⊥ )∗ D⊥ )⊥ .

(2.23)

and

Note that D is elliptic if and only it is partially elliptic with D⊥ = 0. Note also ∗ that a CPDO(1) n,m D may be partially elliptic with D not being partially elliptic [18, Remark 1.1].

104


DEFINITION 2.3. A coefficient operator S on H (n) (COn ) is a bounded, invertible operator given by multiplication by a coefficient matrix: an n×n matrix -valued measurable function S(x) on Rd , satisfying S− In S(x) S+ In ,

with 0 < S− S+ < ∞.

(2.24)

DEFINITION 2.4. A multiplicative coefficient, first-order, partial differential operator from H (n) to H (m) (MPDO(1) n,m ) is of the form √ √ 1 (2.25) A = RD K on D(A) = K − 2 D(D), where D is a CPDO(1) n,m , K is a COn , and R is a COm . (We will write AK,R for A whenever it is necessary to make explicit the dependence on the medium, i.e., on the coefficient operators. D does not depend on the medium, so it will be omitted in the notation.) √ √ ∗ KD∗ R An MPDO(1) n,m A is a closed, densely defined operator with A = − 12 ∞ C0 (Rd ; Cn ) is an operator core for A. an MPDO(1) m,n . Note that K The following quantity will appear often in estimates: (2.26) /A ≡ D+ R+ K+ . DEFINITION 2.5. A first-order classical operator (CWO(1) n,m ) is an operator of the form 0 −iA∗ on H (n+m) ∼ (2.27) WA = = H (n) ⊕ H (m) , iA 0 ∗ where A is an MPDO(1) n,m . If either D or D is partially elliptic, WA will also be called partially elliptic. √ √ (1) SWD S, where A CWO(1) n,m is a self-adjoint MPDOn+m,n+m : WA = S = K ⊕ R is a COn+m and WD is a self-adjoint CPDO(1) n+m,n+m . (Note that our definition of a first-order classical wave operator is more restrictive than the one used in [18]. The definition of partial ellipticity is also different; [18] requires both D and D∗ to be partially eliptic.) The Schrödinger-like Equation (2.5) for classical waves with the auxiliary condition (2.8) may be written in the form:

−i

∂ t = (WA )⊥ t , ∂t

t ∈ (ker WA )⊥ = (ker A)⊥ ⊕ (ker A∗ )⊥ ,

(2.28)

with WA a CWO(1) n+m as in (2.27). Its solutions are of the form t = eit (WA )⊥ 0 ,

0 ∈ (ker WA )⊥ ,

which is just another way of writing (2.9).

(2.29)


Since

(WA )2 =

0 A∗ A 0 AA∗

105

(2.30)

,

if t = (ψt , φt ) ∈ H (n) ⊕ H (m) is a solution of (2.28), then its components satisfy the second-order wave equations (2.2) and (2.3), plus the auxiliary conditions, which may be all written in the form ∂2 ψt = −(A∗ A)⊥ ψt , ∂t 2 ∂2 φt = −(AA∗ )⊥ φt , ∂t 2

with ψt ∈ (ker A)⊥ ,

(2.31)

with φt ∈ (ker A∗ )⊥ .

(2.32)

The solutions to (2.31) may be written as 1 1 ψt = cos t(A∗ A)⊥2 ψ0 + sin t(A∗ A)⊥2 η0 ,

ψ0 , η0 ∈ (ker A)⊥ ,

(2.33)

with a similar expression for the solutions of (2.32). The operators (A∗ A)⊥ and (AA∗ )⊥ are unitarily equivalent (see Lemma A.1): the operator U defined by −1

U ψ = A(A∗ A)⊥ 2 ψ

1

for ψ ∈ ran(A∗ A)⊥2 ,

(2.34)

extends to a unitary operator from (ker A)⊥ to (ker A∗ )⊥ , and (AA∗ )⊥ = U (A∗ A)⊥ U ∗ . In addition, if 1 U= √ 2

IA iU

IA −iU

(2.35)

,

with IA the identity on (ker A)⊥ ,

(2.36)

U is a unitary operator from (ker A)⊥ ⊕ (ker A)⊥ to (ker A)⊥ ⊕ (ker A∗ )⊥ , and we have the unitary equivalence: 1 1 (2.37) U∗ (WA )⊥ U = (A∗ A)⊥2 ⊕ −(A∗ A)⊥2 . Thus the operator (A∗ A)⊥ contains full information about the spectral theory of the operator (WA )⊥ (e.g., [8, 18]). In particular 1 1 (2.38) σ ((WA )⊥ ) = σ (A∗ A)⊥2 ∪ −σ (A∗ A)⊥2 , and to find all eigenvalues and eigenfunctions for (WA )⊥ , it is necessary and sufficient to find all eigenvalues and eigefunctions for (A∗ A)⊥ . For if (A∗ A)⊥ ψω2 = ω2 ψω2 , with ω = 0, ψω2 = 0, we have i i (2.39) (WA )⊥ ψω2 , ± Aψω2 = ±ω ψω2 , ± Aψω2 . ω ω

106


Conversely, if (WA )⊥ (ψ±ω , φ±ω ) = ±ω(ψ±ω , φ±ω ), with ω = 0, it follows that (see [18, Proposition 5.2]) (A∗ A)⊥ ψ±ω = ω2 ψ±ω

and

i φ±ω = ± Aψ±ω . ω

(2.40)

DEFINITION 2.6. A second-order classical wave operator on H (n) (CWO(2) n ) is for some m. (We write W an operator W = A∗ A, with A an MPDO(1) K,R = n,m (2) ∗ AK,R AK,R .) If D in (2.25) is partially elliptic, the CWOn will also be called partially elliptic. Note that a first-order classical wave operator WA is partially elliptic if and only if one of the two second-order classical wave operators A∗ A and AA∗ is partially elliptic. DEFINITION 2.7. A classical wave operator (CWO) is either a CWO(1) n or a . If the operator W is a CWO, we call W a proper CWO. CWO(2) ⊥ n Remark 2.8. A proper classical wave operator W has a trivial kernel by construction, so 0 is not an eigenvalue. However, using a dilation argument, one can show that 0 is in the spectrum of W⊥ [18, Theorem A.1], so W⊥ and W have the same spectrum and essential spectrum.

2.5. DEFECTS AND WAVE LOCALIZATION We now describe our results on defects and wave localization. We can produce eigenvalues in spectral gaps of classical wave operators by introducing defects. Moreover, the corresponding waves exhibit exponential localization. The proofs and details are given in Section 5. A defect is a modification of a given medium in a bounded domain. Two media, described by coefficient matrices K0 (x), R0 (x) and K(x), R(x), are said to differ by a defect, if they are the same outside some bounded set 5, i.e., K0 (x) = K(x) / 5. The defect is said to be supported by the bounded and R0 (x) = R(x) if x ∈ set 5. We recall that the essential spectrum σess(H ) of an operator H consists of all the points of its spectrum, σ (H ), which are not isolated eigenvalues with finite multiplicity. Figotin and Klein [9] showed that the essential spectrum of Acoustic and Maxwell operators are not changed by defects. We extend this result to the class of classical wave operators: the essential spectrum of a partially elliptic classical wave operator (first or second order) is not changed by defects. THEOREM 2.9. Let W0 and W be partially elliptic classical wave operators for two media which differ by a defect. Then σess (W ) = σess (W0 ).

(2.41)


107

If (a, b) is a gap in the spectrum of W0 , the spectrum of W in (a, b) consists of at most isolated eigenvalues with finite multiplicity, the corresponding eigenmodes decaying exponentially fast away from the defect, with a rate depending on the distance from the eigenvalue to the edges of the gap. In view of the unitary equivalence (2.37), Theorem 2.9 is an immediate corollary to Theorem 5.1 and Corollary 5.3. For second-order classical wave operators, the exponential decay of an eigenmode is given in (5.11). For first-order classical wave operators, the exponential decay of an eigenmode follows from (2.40), (5.11), and (3.19). We now turn to the existence of midgap eigenmodes and exponentially localized waves. The next theorem shows that one can design simple defects which generate eigenvalues in a specified subinterval of a spectral gap of W0 , extending [9, Theorem 2] to the class of classical wave operators. We insert a defect that changes the value of K0 (x) and R0 (x) inside a bounded set of ‘size’ 8 to given positive constants K and R. If (a, b) is a gap in the spectrum of W0 , we will show that we can deposit an eigenvalue of W √ inside any specified closed subinterval of (a, b), by inserting such a defect with 8/ KR large enough. We provide estimates on how large is ‘large enough’. Note that the corresponding eigenmode is exponentially decaying by Theorem 2.9, so we construct an exponentially localized wave. THEOREM 2.10 (Existence of exponentially localized waves). Let (a, b) be a gap in the spectrum of a partially elliptic classical wave operator W0 = WK0 ,R0 , select µ ∈ (a, b), and pick δ > 0 such that the interval [µ − δ, µ + δ] is contained in the gap, i.e., [µ − δ, µ + δ] ⊂ (a, b). Given an open bounded set 5, x0 ∈ 5, 0 < K, R, 8 < ∞, we introduce a defect that produces coefficient matrices K(x) and R(x) that are constant in the set 58 = x0 + 8(5 − x0 ), with K(x) = KIn

and

R(x) = RIm

for x ∈ 58 .

(2.42)

Then there is a finite constant C, satisfying an explicit lower bound depending only on the order ( first or second ) of the classical wave operator, and on D+ , µ, δ, and the geometry of 5, such that if √

8 KR

> C,

(2.43)

then the operator W = WK,R has at least one eigenvalue in the interval [µ − δ, µ + δ]. Theorem 2.10 follows from Theorem 5.4 and (2.37). For second-order operators the explicit lower bound is given in (5.16), for first-order operators it can be calculated from (5.16) and (2.37).

108


3. Properties of Classical Wave Operators In this section we discuss several important properties of classical wave operators, which provide the necessary technical tools for proving localization in inhomogeneous and random (see [17]) media. 3.1. A TRACE ESTIMATE Partially elliptic second-order classical wave operators satisfy a trace estimate that provides a crucial ingredient for many results. THEOREM 3.1 ([18, Theorem 1.1]). Let W be a partially elliptic second-order classical wave operator on H (n) , and let PW⊥ denote the orthogonal projection onto (ker W )⊥ . Then tr(V ∗ PW⊥ (W + I )−2r V ) Cd,n,K± ,R± ,D+ ,D+⊥ ,, V 2∞,2 < ∞,

(3.1)

for r ν, where ν is the smallest integer satisfying ν > d/4. V is the bounded operator on H (n) given by multiplication by an n × n matrix-valued measurable function V (x), with χy,1 (x)V ∗ (x)V (x)∞ < ∞. (3.2) V 2∞,2 = y∈Zd

(χy,L denotes the characteristic function of a cube of side length L centered at y.) The constant Cd,n,K± ,R± ,D+ ,D+⊥ ,, depends only on the fixed parameters d, n, K± , R± , D+ , D+⊥ , ,. 3.2. FINITE VOLUME CLASSICAL WAVE OPERATORS Throughout this paper we use two norms in Rd and Cd : |x| =

d

1/2 |xi |2

,

(3.3)

i=1

x = max{|xi |, i = 1, . . . , d}.

(3.4)

We set Br (x) to be the open ball in Rd , centered at x with radius r > 0: Br (x) = {y ∈ Rd ; |y − x| < r}.

(3.5)

By =L (x) we denote the open cube in Rd , centered at x with side L > 0 =L (x) = {y ∈ Rd ; y − x < L/2},

(3.6)


109

and by =L (x) the closed cube. By = we will always denote some open cube =L (x). We will identify a closed cube =L (x) with a torus in the usual way, and use the following distance in the torus: √ d L for y, y ∈ =L (x). (3.7) dL (y, y ) = min |y − y + m| d 2 m∈LZ We set H=(n) = L2 (=, dx; Cn ). A CPDO(1) n,m D defines a closed densely defined operator D= from H=(n) to H=(m) with periodic boundary condition; an operator core ∞ is given by Cper (=, Cn ), the infinitely differentiable, periodic Cn -valued functions on =. The restriction of a COn S to = gives the bounded, invertible operator S= on H=(n) . Given an MPDO(1) n,m A as in (2.25), we define its restriction A= to the cube = with periodic boundary condition by A= =

R= D= K=

−1

on D(A= ) = K= 2 D(D= ),

(3.8)

a closed, densely defined operator on H=(n) . The restriction W= of the second-order classical wave operator W = A∗ A to = with periodic boundary condition is now defined as W= = A∗= A= . If the CPDO(1) n,m D is partially elliptic, then the restriction D= is also partially elliptic, in the sense that Equations (2.20) and (2.21) hold for D= , (D⊥ )= , and -= . (-= is the Laplacian on L2 (=, dx) with periodic boundary condition.) This can be easily seen by using the Fourier transform; here the use of periodic boundary condition plays a crucial role. We also have (2.22) and (2.23) with H=(n) . (n) , Wx,L , and so on. If = = =L (x), we write Hx,L Given a second-order classical wave operator W on H (n) , we define its finite volume resolvent on a cube = by R= (z) = (W= − z)−1

for z ∈ / σ (W= ).

(3.9)

If W is partially elliptic, it turns out that (W= )⊥ has compact resolvent, i.e., / σ (W= ). Note that it suffices to prove the R= (z)PW⊥= is a compact operator for z ∈ statement for z = −1. We will prove a stronger statement. In what follows, we write F G if the positive self-adjoint operators F and G are unitarily equivalent, and we write F ! G if F J for some positive selfadjoint operator J G. Note that, if 0 F ! G, then tr f (G) tr f (F ), for any positive, decreasing function f on [0, ∞). PROPOSITION 3.2. Let W be a partially elliptic second-order classical wave operator. Then for any finite cube = and p > d/2 we have (3.10) tr (W= + 1)−p PW⊥= n tr (K− R− ,(−-= ) + 1)−p < ∞.

110


Proof. Using Lemma A.1 and (2.24), we get (W= )⊥ = K= D∗= R= D= K= ⊥ R− K= D∗= D= K= ⊥ R− (D= K= D∗= )⊥ K− R− (D= D∗= )⊥ K− R− (D∗= D= )⊥ .

(3.11)

It follows from (3.11), (2.23), and (2.21) that tr (W= + 1)−p PW⊥= −p = tr (W= )⊥ + 1 tr (K− R− (D∗= D= )⊥ + 1)−p −p tr K− R− (D∗= D= )⊥ ⊕ (D⊥ )∗= (D⊥ )= ⊥ + 1 tr (K− R− ,[(−-= ) ⊗ In ] + 1)−p = n tr (K− R− ,(−-= ) + 1)−p < ∞,

(3.12) ✷

if p > d/2. Since (W= )⊥ 0 has compact resolvent, we may define NW= (E) = tr χ(−∞,E) ((W= )⊥ ),

(3.13)

the number of eigenvalues of (W= )⊥ that are less than E. If E 0, we have NW= (E) = 0, and if E > 0, NW= (E) is the number of eigenvalues of W= (or (W= )⊥ ) in the interval (0, E). Notice that NW= (E) is the distribution function of the measure nW= (dE) given by (3.14) h(E)nW= (dE) = tr(h((W= )⊥ )), for positive continuous functions h of a real variable. We have the following ‘a priori’ estimate: LEMMA 3.3. Let W be a partially elliptic second-order classical wave operator. Then for any finite cube = and E > 0 we have d2 E E nCd |=|, (3.15) NW= (E) nN−-= K− R− , K− R− , where Cd is some finite constant depending only on the dimension d. Proof. We have NW= (E) NK− R− (D∗= D= ) (E) NK− R− (D∗= D= +(D⊥ )∗= (D⊥ )= ) (E) NK− R− ,[(−-= )⊗In ] (E) = nN−-=

E , K− R− ,

(3.16) (3.17) (3.18)

111


where (3.16) follows from (3.11) and the Min-max Principle, (3.17) follows from (2.23), (3.18) follows from (2.21), plus a simple computation for the equality. The second inequality in (3.15) is given by a standard estimate. ✷

3.3. AN INTERIOR ESTIMATE The following interior estimate is an adaptation of [18, Theorem 4.1] to both finite or infinite volume. LEMMA 3.4. Let W = A∗ A be a second-order classical wave operator, and let = denote either an open cube or Rd . Let ρ ∈ C01 (=) and τ ∈ L∞ loc (=, dx), with 0 ρ(x) τ (x) and |∇ρ(x)| cτ (x) a.e., where c is a finite constant. Then, for any ψ ∈ D(W= ) we have 1 2 2 2 2 ρA= ψ aτ W= ψ + + 4c /A τ ψ2 (3.19) a for all a > 0, where /A is given in (2.26). Proof. This is proved as [18, Theorem 4.1], keeping track of the constants.

✷

3.4. IMPROVED RESOLVENT DECAY ESTIMATES IN A GAP We adapt an argument of Barbaroux, Combes and Hislop [1] to second-order classical wave operators, obtaining an improvement on the rate of decay given by the usual Combes–Thomas argument (e.g., [7, Lemma 12], [8, Lemma 15]). Our proof, while based on [1, Lemma 3.1], is otherwise different from the proof for Schrödinger operators, as we use an argument based on quadratic forms avoiding the analytic continuation of the operators. This way we can accomodate the nonsmoothness of the coefficients of our classical wave operators. We will prove the decay estimate for both infinite and finite volumes (with periodic boundary condition). We start with infinite volume. Recall that Br (x) denotes the open ball of radius r centered at x. THEOREM 3.5. Let W = A∗ A be a second-order classical wave operator with a spectral gap (a, b). Then for any E ∈ (a, b) and 8, 8 > 0 we have

χB8 (x) R(E)χB8 (y) CE emE (8+8 ) e−mE |x−y| for all x, y ∈ Rd , with 1 1 (E − a)(b − E) mE = 4/A (a + b + 2)(b + 1) 4/A

(3.20)

(3.21)

112


and

CE = max

a + b + 2 4(b + 1) , . E−a b−E

(3.22)

In addition, χB8 (x) AR(E)χB8 (y) 1 CE 2E + 16/2A 2 emE (8+8 +1) e−mE |x−y|

(3.23)

for all x, y ∈ Rd with |x − y| 8 + 8 + 1. Proof. We start by defining the operators formally given by Wα = eα·x W e−α·x ,

α ∈ Rd .

To do so, let us consider the bounded operator √ √ Gα = RD(α) K, Gα |α|/A .

(3.24)

(3.25)

Then Aα = eα·x Ae−α·x = A + iGα on D(A), (A∗ )α = eα·x A∗ e−α·x = A∗ + iG∗α on D(A∗ ),

(3.26) (3.27)

are closed, densely defined operators. (Note (A∗ )α = (Aα )∗ .) We define Wα = (A∗ )α Aα as a quadratric form. More precisely, for each α ∈ Rd ,we define a quadratic form with domain D(A) by

(3.28) Wα [ψ] = (A∗ )∗α ψ, Aα ψ . Note that if α = 0, W = W0 is the closed, nonegative quadratic form associated to the classical wave operator W . It follows from (3.26) and (3.27) that Wα [ψ] − W [ψ] = 2i ReAψ, Gα ψ − Gα ψ, Gα ψ,

(3.29)

so 1 Wα [ψ] − W [ψ] Gα ψ 4Aψ2 + Gα ψ2 2 1 1 + s |α|2 /2A ψ2 2sW [ψ] + 2 s

(3.30)

for any s > 0. It follows [15, Theorem VI.1.33] that Wα is a closed sectorial form on the form domain of W . We define Wα as the unique m-sectorial operator associated with it [15, Theorem VI.2.1]. For each α ∈ Rd , we set W (α) = W − G∗α Gα , θ (α) = 1 + |α|2 /2A ,

(3.31) (3.32)


113

note W (α) + θ (α) 1.

(3.33)

It follows from (3.29) that

− 1 − 1 W (α) + θ (α) 2 (Wa − E) W (α) + θ (α) 2 − 1 − 1 = W (α) + θ (α) 2 W (α) − E W (α) + θ (α) 2 + iYα ,

(3.34)

as everywhere defined quadratic forms, where (α) 1 1 W + θ (α) − 2 W (α) − E W (α) + θ (α) − 2 1 + θ (α) + E < ∞, (3.35) and − 1 − 1 Yα = W (α) + θ (α) 2 A∗ Gα + G∗α A W (α) + θ (α) 2

(3.36)

extends to a bounded self-adjoint operator with Yα 2|α|/A ,

(3.37)

in view of (3.25), (3.31), (3.32), (3.33), and (α) 1 A W + θ (α) − 2 ψ 2 (3.38) 1 1

(α) − − = W + θ (α) 2 ψ, A∗ A W (α) + θ (α) 2 ψ − 1 − 1

= ψ2 + W (α) + θ (α) 2 ψ, G∗α Gα − θ (α) W (α) + θ (α) 2 ψ − 1 2 (3.39) ψ2 − W (α) + θ (α) 2 ψ ψ2 . Since (a, b) is a gap in the spectrum of W and E ∈ (a, b), we have that the interval b − E − |α|2 /2A a−E , a + θ (α) b + θ (α) − |α|2 /2A b − E − |α|2 /2A a−E (3.40) , = b+1 a + 1 + |α|2 /2A is a gap in the spectrum of the operator

W (α) + θ (α)

− 12

− 1 W (α) − E W (α) + θ (α) 2 ,

containing 0, as long as √ b−E . |α| < /A

(3.41)

(3.42)

114


We now use [1, Lemma 3.1] to conclude, if in addition to (3.42) we also require 1 (E − a)(b − E − |α|2 /2A ) |α| < , (3.43) 4/A (a + 1 + |α|2 /2A )(b + 1) that 0 is not in the spectrum of the operator in (3.34), and (α) 1 1 W + θ (α) − 2 (Wa − E) W (α) + θ (α) − 2 −1 b+1 a + 1 + |α|2 /2A , . 5α ≡ 2 max E−a b − E − |α|2 /2A

(3.44)

Since 1 D(Wα ) ⊂ D(Wα ) = D(W ) = D W (α) + θ (α) 2 ,

(3.45)

we may use (3.33) and (3.44) to obtain, for all φ ∈ D(Wα ), (Wa − E)φ − 1 − 1 1 W (α) + θ (α) 2 (Wa − E) W (α) + θ (α) 2 W (α) + θ (α) 2 φ (α) 1 W + θ (α) 2 φ 5−1 φ. 5−1 α α

(3.46)

Since (3.46) holds for all α ∈ Rd , and Wα∗ = W−α , W (α) = W (−α) , θ (α) = θ (−α) , and 5(α) = 5(−α) , we see that we also have (3.46) for Wα∗ . We can conclude that E∈ / σ (Wα ) and (Wα − E)−1 5α , (3.47) for all α ∈ Rd satisfying (3.42) and (3.43). We now take |α| mE , where mE is given √ in (3.21). Then both (3.42) and (3.43) are satisfied, and we also have |α|/A (b − E)/2, so 5α CE , with CE as in (3.22), and (3.47) gives Rα (E) CE ,

with Rα (E) = (Wα − E)−1 .

We may now prove (3.20). Let x0 , y0 ∈ Rd , 8 > 0, and take α = We have

(3.48) mE (x −y0 ). |x0 −y0 | 0

χB8 (x0 ) R(E)χB8 (y0 ) = χB8 (x0 ) e−α·x Rα (E)eα·x χB8 (y0 ) = e−m|x0 −y0 | χB8 (x0 ) e−α·(x−x0) Rα (E)eα·(x−y0) χB8 (y0 ) ,

(3.49)

so χB8 (x0 ) R(E)χB8 (y0 ) CE χB8 (x0 ) e−α·(x−x0 ) ∞ χB8 (y0 ) ea·(x−y0) ∞ e−m|x0 −y0 | .

(3.50)


115

Since χB8 (x0 ) e±α·(x−x0) ∞ e|α|8 = emE 8 ,

(3.51)

(3.20) follows from (3.50) and (3.51). To prove (3.23), we use Lemma 3.4. We let |x0 − y0 | 8 + 8 + 1, and pick ρ ∈ C01 (Rd ), with χB8 (x0 ) ρ χB8+1 (x0 ) and |∇ρ(x)| 2. We have χB8 (x0 ) AR(E)χB8 (y0 ) ρAR(E)χB8 (y0 ) 1 2E + 16/2A 2 χB8+1 (x0 ) R(E)χB8 (y0 ) ,

(3.52) ✷

so (3.23) now follows from (3.20)

We now turn to the torus, i.e., we prove a version of Theorem 3.5 for the restriction of a second-order classical wave operator to a cube = with periodic boundary condition. We use the distance (3.7) in the torus. THEOREM 3.6. W be a second-order classical wave operator whose restriction with periodic boundary condition to a cube =L (x0 ) has a spectral gap (a, b). Then for any E ∈ (a, b) and 8 > 0, with L > 28 + 8, we have χB8 (x) Rx0 ,L (E)χB8 (y)x0 ,L CE e2mE,L,8 8 e−mE,L,8 dL (x,y) for all x, y ∈ =L (x0 ), where mE,L,8

mE = , cL,8

with cL,8 =

√ 2 d 1−

2(8+3) L−2

+1 ,

(3.53)

(3.54)

and mE and CE are as in Theorem 3.5. Proof. Let us fix x1 , y1 ∈ =L (x0 ), by redefining the coefficient operators we may assume x0 = 0 = 12 (x1 + y1 ) and x1 , y1 ∈ = L (0). In particular, dL (x1 , y1 ) = 2 |x1 − y1 |. Let L > 28 + 8, we pick a real valued function ξ ∈ C01 (R) with 0 ξ(t) 1 for all t ∈ R, such that ξ(t) = 1 for |t|

L 8 + , 4 2

ξ(t) = 0 for |t|

L − 1, 2

and

−1 L 8 − −2 for all t ∈ R. |ξ (t)| 4 2 We set (x) = di=1 ξ(xi ) for x ∈ Rd . Notice supp (x) ⊂ =L (0).

116


We now proceed as in the proof of Theorem 3.5 with =L (0) substituted for Rd and definition (3.24) replaced by (W0,L )α = e(x)α·x W0,L e−(x)α·x ,

α ∈ Rd ,

and instead of (3.25), consider the bounded operators (G0,L )α = R0,L D ∇ (x)α · x 0,L K0,L . Since ∇ (x)α · x

√ ( L2 − 1) d L 4

−

8 2

−2

+ 1 |α| = cL,8 |α|

(3.55)

(3.56)

(3.57)

for all x ∈ =L (0), with cL,8 as in (3.54), we have (G0,L )α cL,8 |α|/A .

(3.58)

We now proceed as in the proof of Theorem 3.5, except that we must now substitute cL,8 |α| for |α| in the estimates. Thus, if |α| mE /cL,8 , we conclude that E∈ / σ ((W0,L)α ) and (R0,L)α (E) CE , with (R0,L )α (E) = (W0,L )α − E −1 . (3.59) To prove (3.53), we take α=

mE (x1 − y1 ), cL,8 |x1 − y1 |

and complete the proof of as before (with x1 , y1 substituted for x, y in (3.53)), as χB8 (x1 ) e±(x)α·x−α·x1 ∞ = χB8 (x1 ) e±α·(x−x1) ∞ emE,L,8 8 .

(3.60) ✷

3.5. A SIMON – LIEB - TYPE INEQUALITY The norm in H=(r) and also the corresponding operator norm will both be denoted by = , or x,L in case = = =L (x). (We omit r from the notation.) If =1 ⊂ =2 are open cubes (possibly the whole space), let J==12 : H=(r)1 → H=(r)2 be the canonical 2 injection. If =i = =Li (xi ), i = 1, 2, we write xx21 ,L ,L1 for the operator norm from =L (x2 )

,L2 = J=L 2(x1 ) . H=(r)L (x1 ) to H=(r)L (x2 ) , and Jxx12,L 1 1 2 1 Given a function φ ∈ L∞ (Rd ), with supp φ ⊂ =, we do not distinguish in the notation between φ as a multiplication operator on H=(r) and on H (r) . If D is a 1 d CPDO(1) n,m , and φ ∈ C0 (R ), real-valued, with supp φ ⊂ =1 ⊂ =2 , we can verify that, as operators,

D=2 J==12 φ = J==12 D=1 φ

on D(D=1 ).

(3.61)


117

It follows for the MPDO(1) n,m A that A=2 J==12 φ = J==12 A=1 φ

on D(A=1 ).

(3.62)

We set A[φ] =

√

√ RD[φ] K,

(3.63)

where D[φ], given by multiplication by the matrix valued function D(−i∇φ(x)), is a bounded operator from H (n) to H (m) , with norm bounded by D+ ∇φ∞ . Thus A[φ] is a bounded operator given by multiplication by a matrix-valued, measurable function, with (see (2.26)) A[φ] /A ∇φ∞ .

(3.64)

We denote by A= [φ] its restriction to the cube =; it also satisfies the bound (3.64). We will use the fact that A= R= (z) is a bounded operator with (3.65) A= R= (z)2= R= (z)= |z|R= (z)= + 1 . The basic tool to relate the finite volume resolvents in different scales is the smooth resolvent identity (SRI) (see [2, 7, 8, 18]). LEMMA 3.7 (SRI). Let W = A∗ A be a second-order classical wave operator, and let =1 ⊂ =2 be either open cubes or Rd , and let φ ∈ C01 (Rd ) with supp φ ⊂ =1 . Then, for any z ∈ / σ (W=1 ) ∪ σ (W=2 ) we have R=2 (z)φJ==12 = J==12 φR=1 (z) + R=2 (z)A∗=2 [φ]J==12 A=1 R=1 (z) − − R=2 (z)A∗=2 J==12 A=1 [φ]R=1 (z) as bounded operators from H=(nL to H=(n)L . 1 2 Proof. This lemma can be proved as [18, Lemma 7.2].

(3.66)

✷

We will now state and prove a Simon–Lieb-type inequality (SLI) for secondorder classical wave operators. This estimate is a crucial ingredient in the multiscale analysis proofs of localization for random operators, where it is used to obtain decay in a larger scale from decay in a given scale [13, 12, 5, 2, 7, 8, 14]. Let us fix q ∈ N. (In [17] we will work with a periodic background medium, and we will take q to be the period.) We will take cubes =L (x) centered at sites x ∈ qZd with side L ∈ 2qN (so in a periodic background medium with period q the background medium will be the same in all cubes in a given scale L). For such

118


cubes (with L 4q), we set L d ϒL (x) = y ∈ qZ ; y − x = − q , 2

L (x) = =L−q (x)\=L−3q (x) = =q (y), ϒ

(3.67) (3.68)

y∈ϒL (x)

L (x) = = 3q (x)\= 5q (x), ϒ L− 2 L− 2 χy,q a.e., Mx,L = χϒ L (x) =

(3.69) (3.70)

y∈ϒL (x)

Mx,L = χϒL (x).

(3.71)

Note that |ϒL (x)| d(L − 2q + 1)d−1 .

(3.72)

In addition each cube =L (x) will be equipped with a function φx,L constructed in the following way: we fix an even function ξ ∈ C01 (R) with 0 ξ(t) 1 for all t ∈ R, such that ξ(t) = 1 for |t| q/4 , ξ(t) = 0 for |t| 3q/4, and |ξ (t)| 3/q for all t ∈ R. (Such a function always exists.) We define 1, if |t| L2 − 5q , ξL (t) = (3.73) L 43q L 3q ξ |t| − 2 − 2 , if |t| 2 − 2 , and set φx,L (y) = φL (y − x)

for y ∈ Rd , with φL (y) =

d

ξL (yi ).

(3.74)

i=1

We have φx,L ∈ C01 (Rd ), with supp φx,L ⊂ =L (x) and 0 φx,L 1. By construction, we have χx, L − 5q φx,L = χx, L − 5q , 2

4

2

4

Mx,L (∇φx,L ) = ∇φx,L ,

χx, L − 3q φx,L = x,L , 2 4 √ 3 d . |∇φx,L | q

(3.75) (3.76)

Similarly, we also construct a function ρx,L ∈ C01 (Rd ), 0 ρx,L 1, such that Mx,L , Mx,L ρx,L = √ 5 d . |∇ρx,L | q

Mx,L ρx,L = ρx,L,

(3.77) (3.78)

LEMMA 3.8 (SLI). Let W be a second-order classical wave operator. Let x, y ∈ qZd , L, 8 ∈ 2qN, and 5 be a set, with 5 ⊂ =8−3q (y) ⊂ =L−3q (x). Then, if z∈ / σ (Wx,L) ∪ σ (Wy,8 ), we have Mx,L Rx,L(z)χ5 x,L γz My,8 Ry,8(z)χ5 y,8 Mx,L Rx,L (z)My,8x,L ,

(3.79)


with

√ 1 6 d 100d 2 2 /A 2|z| + 2 /A , γz = q q

119

(3.80)

where /A is given in (2.26). Proof. We proceed as in [7, Lemma 26]. Using (3.75), (3.66), and Mx,L φy,8 = 0, we obtain x,L χ5 Mx,L Rx,L (z)Jy,8 x,L x,L = Mx,L Rx,L (z)Jy,8 φy,8 χ5 = Mx,L Rx,L(z)A∗x,L [φy,8 ]Jy,8 Ay,8 Ry,8 (z)χ5 − x,L Ay,8 [φy,8 ]Ry,8(z)χ5 . −Mx,L Rx,L (z)A∗x,LJy,8

(3.81)

We now use (3.76) and (3.64) to get x,L Ay,8 Ry,8(z)χ5 x,L Mx,L Rx,L(z)A∗x,L [φy,8 ]Jy,8 y,8 x,L = Mx,L Rx,L(z)My,8 A∗x,L [φy,8 ]Jy,8 My,8 Ay,8 Ry,8 (z)χ5 x,L y,8 √ 3 d /A Mx,L Rx,L (z)My,8x,L My,8 Ay,8 Ry,8 (z)χ5 y,8 , q

(3.82)

and x,L Ay,8 [φy,8 ]Ry,8 (z)χ5 x,L Mx,L Rx,L(z)A∗x,L Jy,8 x,L = Mx,L Rx,L(z)A∗x,L Ay,8 [φy,8 ]My,8 Ry,8 (z)χ5 x,L My,8 Jy,8 √ 3 d /A Mx,L Rx,L(z)A∗x,L My,8 x,L My,8 Ry,8(z)χ5 y,8 q √ 3 d /A = My,8 Ax,L Rx,L (¯z)Mx,L x,L My,8 Ry,8 (z)χ5 y,8 . q

(3.83)

(n) and a > 0 We now appeal to Lemma 3.4 using (3.77) and (3.78). For ψ ∈ Hy,8 we get

My,8 Ay,8 Ry,8 (z)χ5 ψ2y,8 ρy,8 Ay,8 Ry,8 (z)χ5 ψ2y,8 aMy,8 Wy,8 Ry,8 (z)χ5 ψ2y,8 + 1 100d 2 + 2 /A My,8 Ry,8(z)χ5 ψ2y,8 + a q 100d 2 1 2 a|z| + + 2 /A My,8 Ry,8(z)χ5 ψ2y,8 . a q

(3.84)

Choosing a = |z|−1 , we get My,8 Ay,8 Ry,8 (z)χ5 y,8 1 100d 2 2 2|z| + 2 /A My,8 Ry,8 (z)χ5 y,8 . q

(3.85)

120


Similarly, we get My,8 Ax,L Rx,L (¯z)Mx,L x,L 1 100d 2 2 2|z| + 2 /A My,8 Rx,L (¯z)Mx,L x,L q 1 100d 2 2 = 2|z| + 2 /A Mx,L Rx,L (z)My,8x,L . q

(3.86)

Since x,L χ5 x,L Mx,L Rx,L(z)χ5 x,L = Mx,L Rx,L (z)Jy,8 y,8 ,

the lemma follows from (3.81)–(3.86).

(3.87) ✷

3.6. THE EIGENFUNCTION DECAY INEQUALITY The eigenfunction decay inequality (EDI) estimates decay for generalized eigenfunctions from decay of finite volume resolvents. We start by introducing generalized eigenfunctions for classical wave operators. (We refer to [18] for the details.) Given ν > d/4, we define the weighted spaces (we will omit ν from the notation) H±(r) as follows: H±(r) = L2 (Rd , (1 + |x|2 )±2ν dx; Cr ). H−(r) is the space of polynomially L2 -bounded functions. The sesquilinear form (3.88) φ1 , φ2 H (r) ,H (r) = φ1 (x) · φ2 (x) dx, +

−

where φ1 ∈ H+(r) and φ2 ∈ H−(r) , makes H+(r) and H−(r) conjugate duals to each other. By O † we will denote the adjoint of an operator O with respect to this duality. By construction, H+(r) ⊂ H (r) ⊂ H−(r) , the natural injections ı+ : H+(r) → H (r) and ı− : H (r) → H−(r) being continuous with dense range, with ı+† = ı− . Given a second-order classical wave operator W = A∗ A, where A is a MPDO(1) n,m , (n) we define operators W± on H as follows: A+ is the restriction of the operator A to H+(n) , i.e., A+ is the operator from H+(n) to H+(m) with domain D(A+ ) = {φ ∈ D(A) ∩ H+(n) ; Aφ ∈ H+(m) }, defined by A+ φ = Aφ for φ ∈ D(A+ ). A+ is a closed densely defined operator, and we set A− = (A∗+ )† , a closed densely defined operator from H−(n) to H−(m) . We define W+ = A∗+ A+ = A†− A+ , which is a closed densely defined operator on H+(n) with domain D(W+ ) = {φ ∈ D(W ) ∩ H+(n) ; W φ ∈ H+(n) }, and W+ φ = W φ for φ ∈ D(W+ ) [18, Theorem 4.2]. We define W− = W+† , a closed densely defined operator on H−(n) . Note that W is the restriction of W− to H (n) .


121

A measurable function ψ: Rd → Cn is said to be a generalized eigenfunction of W with generalized eigenvalue λ, if ψ ∈ H−(n) (for some ν > d/4) and is an eigenfunction for W− with eigenvalue λ, i.e., ψ ∈ D(W− ) and W− ψ = λψ. In other words, ψ ∈ H−(n) and W+ φ, ψH (n) ,H (n) = λφ, ψH (n) ,H (n) +

−

+

−

for all φ ∈ D(W+ ).

(3.89)

Eigenfunctions of W are always generalized eigenfunctions. Conversely, if a generalized eigenfunction is in H (n) , then it is a bona fide eigenfunction. LEMMA 3.9 (EDI). Let W be a second-order classical wave operator, and let ψ be a generalized eigenfunction of W with generalized eigenvalue E. Let x ∈ qZd and L ∈ 2qN be such that E ∈ / σ (Wx,L ). Then for any set 5 ⊂ =L−3q (x) we have χ5 ψ γE Mx,L Rx,L(E)χ5 x,L Mx,L ψ,

(3.90)

with γE as in (3.80). Proof. We fix ν > d/4 such that ψ ∈ D(W− ) and W− ψ = Eψ. We write Rd . Jx,L = Jx,L (n) , Using [18, Lemma 4.1], we can show that, weakly in Hx,L ∗ ∗ ∗ χ5 ψ = χ5 Jx,L φx,L ψ = χ5 Rx,L (E)(Wx,L − E)Jx,L φx,L ψ Jx,L ∗ ∗ = χ5 Rx,L (E)Ax,L Jx,L A[φx,L ]ψ + ∗ A∗ [φx,L ]A− ψ. + χ5 Rx,L (E)Jx,L

Proceeding as in the proof of Lemma 3.8, we have χ5 Rx,L (E)A∗ J ∗ A[φx,L ]ψ x,L x,L x,L ∗ ∗ = χ5 Rx,L (E)Ax,L Mx,L Jx,L A[φx,L ]Mx,L ψ x,L √ 3 d ∗ /A χ5 Rx,L(E)A∗x,L Mx,L ψx,L Mx,L x,L Jx,L q √ 3 d ∗ /A = Mx,L Ax,L Rx,L(E)χ5 x,L Jx,L Mx,L ψx,L q √ 1 3 d 100d 2 2 /A 2|E| + 2 /A × q q ×Mx,L Rx,L(E)χ5 x,L Mx,L ψ. Similarly, χ5 Rx,L (E)J ∗ A∗ [φx,L ]A− ψ x,L x,L ∗ ∗ = χ5 Rx,L (E)Mx,L Jx,L A [φx,L ] Mx,L A− ψ x,L √ 3 d /A Mx,L Rx,L (E)χ5 x,L Mx,L A− ψ, q

(3.91)

(3.92)

(3.93)

122


and, using Lemma 3.4, which is also valid for the operator A− (see [18, Theorem 4.1], we have Mx,L A− ψ2 ρx,L A− ψ2

1 100d 2 aMx,L W− ψ2 + + 2 /A Mx,L ψ2 a q 1 100d = a|E|2 + + 2 /2A Mx,L ψ2 , a q

(3.94)

for any a > 0. Choosing a = |E|−1 in (3.94), (3.90) follows from (3.91)–(3.94).

✷

4. Periodic Classical Wave Operators In this section we study classical wave operators in periodic media. The main theorem gives the spectrum of a periodic classical wave operator in terms of the spectra of its restriction to finite cubes with periodic boundary condition. DEFINITION 4.1. A coefficient operator S on H (n) is periodic with period q > 0 if S(x) = S(x + qj ) for all x ∈ Rd and j ∈ Zd . DEFINITION 4.2. A medium is called periodic if the coefficient operators K and R that describe the medium are periodic with the same period q. (We will always take the period q ∈ N without loss of generality.) The corresponding classical wave operators will be said to be periodic with period q (q-periodic). If k, n ∈ N, we say that k ! n if n ∈ kN and that k ≺ n if k ! n and k = n. THEOREM 4.3. Let W be a q-periodic second-order classical wave operator. Let {8n ; n = 0, 1, 2, . . .} be a sequence in N such that 80 = q and 8n ≺ 8n+1 for each n = 0, 1, 2, . . .. Then σ (W0,8n ) ⊂ σ (W0,8n+1 ) ⊂ σ (W )

for all n = 0, 1, 2, . . . ,

(4.1)

and σ (W ) =

∞

σ (W0,8n ).

(4.2)

n=0

Proof. The analogous result for periodic Schrödinger operators is well known [6]. Periodic acoustic and Maxwell operators are treated in [7, Theorem 14] and [8, Theorem 25], respectively. We will sketch a proof, using Floquet theory. We refer to [19, Section XIII.6] for the definitions and notations of direct integrals of Hilbert spaces.


123

˘ 2π (0) the dual basic cell, We let Q = =q (0) be the basic period cell, Q˜ = = q L L d ˘ L (x) = y ∈ R ; xi − yi < xi + , i = 1, . . . , d . = 2 2 ˘ q (0), but we will not since it will make no difference (We should also take Q = = in what follows.) For any r ∈ N we define the Floquet transform ⊕ (r) ˜ dk; HQ(r) HQ(r) dk ≡ L2 Q, (4.3) F :H → Q˜

by (F ψ)(k, x) =

q 2π

d2

eik·(x−m) ψ(x − m),

˜ x ∈ Q, k ∈ Q,

(4.4)

m∈qZd

if ψ has compact support; it extends by continuity to a unitary operator. The q-periodic operator W is decomposable in this direct integral representation, more precisely, ⊕ ∗ WQ (k) dk, (4.5) F WF = Q˜

where for each k ∈ Rd we set DQ (k) to be the restriction to Q with periodic boundary condition of the operator given by the matrix D(−i∇ + k) (see (2.17) ), a closed, densely defined operator, and let WQ (k) = A∗Q (k)AQ (k) with AQ (k) = RQ DQ (k) KQ . (If for p ∈ 2π /qZd , Up denotes the unitary operator on HQ(r) given by multiplication by the function e−ip·x , then for all k ∈ Rd we have WQ (k + p) = Up∗ WQ (k)Up .) Since AQ (k + h) − AQ (k) |h|/A ,

(4.6)

follows from the resolvent identity that the map k ∈ Rd (→ (WQ (k) + I )−1 ∈ B HQ(n)

(4.7)

is operator norm continuous, so we conclude from (4.5) that σ (WQ (k)). σ (W ) =

(4.8)

k∈Q˜

If 8 ∈ qZd , similar considerations apply to the operator W0,8 , which is q-periodic on the torus =8 (0). The Floquet transform (r) → HQ(r) (4.9) F8 : H0,8 d ˜ k∈ 2π 8 Z ∩Q

124


is a unitary operator now defined by d2 q (F8 ψ)(k, x) = 8 d

eik·(x−m) ψ(x − m),

(4.10)

˘ 8 (0) m∈qZ ∩=

where x ∈ Q,

k∈

2π d ˜ Z ∩ Q, 8

(r) ψ ∈ H0,8 ,

ψ(x − m) being properly interpreted in the torus =8 (0). We also have WQ (k), F8 W0,8 F8∗ =

(4.11)

d ˜ k∈ 2π 8 Z ∩Q

and σ (W0,8 ) =

σ (WQ (k)).

(4.12)

d ˜ k∈ 2π 8 Z ∩Q

Theorem 4.3 follows from (4.8) and (4.12).

✷

5. Defects and Midgap Eigenmodes We now prove the results in Subsection 2.5. THEOREM 5.1 (Stability of essential spectrum). Let W0 and W be second-order partially elliptic classical wave operators for two media which differ by a defect. Then σess (W ) = σess (W0 ).

(5.1)

Proof. We will first prove the theorem when the defect only changes R, i.e., we will show σess (WK0 ,R ) = σess (WK0 ,R0 ). The general case will follow, using Remark 2.8 and Lemma A.1, as then σess (WK0 ,R0 ) = σess (WK0 ,R ) = σess (WK0 ,R )⊥ = σess (WR,K0 )⊥ = σess (WR,K0 ) = σess (WR,K ) = σess (WR,K )⊥ = σess (WK,R )⊥ = σess(WK,R ).

(5.2)

(5.3)

To prove (5.2), we proceed as in [9, Theorem 1]. Let T (x) = R(x) − R0 (x), by our hypotheses it is a bounded, measurable, self-adjoint matrix-valued function with compact support. We write T (x) = T+ (x) − T− (x),


125

with T± (x) the positive/negative part of the self-adjoint matrix T (x). We let T± denote the bounded operators given by the matrices T± (x), they would be coefficients operators except for the fact that the functions T± (x) have compact support, so they are not bounded away from zero. We may still define operators define nonnegative self-adjoint operators WK0 ,T± . We have WK0 ,R = (WK0 ,R0 + WK0 ,T+ ) − WK0 ,T− ,

(5.4)

as quadratic forms. (Note that Q(WK0 ,R ) = Q(WK0 ,R0 ) ⊂ Q(WK0 ,T± ), where Q(W ) denotes the form domain of the operator W .) Thus (5.2) follows from [19, Corollary 4 to Theorem XIII.14] and the following lemma. LEMMA 5.2. Let WK,R be a second-order partially elliptic classical wave operator, and let T be like a coefficient operator, except for the fact that the function T (x) has compact support, so it is not bounded away from zero (i.e., T− = 0). Then (5.5) tr (WK,R + I )−r WK,T (WK,R + I )−r < ∞ if r ν + 1, where ν is the smallest integer satisfying ν > d/4. Proof. Let 5 denote the support of T (x), we pick a function ρ ∈ C01 (Rd ) with

= supp ρ is a compact set. We have χ5 ρ(x) χ5

, where 5 T T+ ρ 2 T+ R−−1 ρ 2 R.

(5.6)

Thus, using HS to denote the Hilbert–Schmidt norm, and setting c = ∇ρ∞ , we have (5.7) tr (WK,R + I )−r WK,T (WK,R + I )−r T+ R−−1 tr (WK,R + I )−r A∗K,R ρ 2 AK,R (WK,R + I )−r 2 (5.8) = T+ R−−1 ρAK,R (WK,R + I )−r HS −1 −r 2 + (5.9) T+ R− χ5

WK,R (WK,R + I ) HS −r 2 + 1 + 4c2 /2AK,R χ5

(WK,R + I ) HS 2 −r+1 −r χ5 + (W + I ) + T+ R−−1 χ5

(WK,R + I )

K,R HS HS 2 2 2 −r (5.10) + 1 + 4c /AK,R χ5

(WK,R + I ) HS < ∞, where the final bound in (5.10) follows from Theorem 3.1 if r − 1 ν. To go from (5.8) to (5.9) we used Lemma 3.4. ✷ This finishes the proof of Theorem 5.1.

✷

COROLLARY 5.3 (Behavior of midgap eigenmodes). Let W0 and W be secondorder partially elliptic classical wave operators for two media which differ by

126


a defect. If (a, b) is a gap in the spectrum of W0 , the spectrum of W in (a, b) consists of at most isolated eigenvalues with finite multiplicity, the corresponding eigenmodes decaying exponentially fast from the defect, with a rate depending on the distance from the eigenvalue to the edges of the gap. If the defect is supported by some ball Br (x0 ), and E ∈ (a, b) is an eigenvalue for W with a corresponding eigenmode ψ, ψ = 1, then χx ψ2

√ 1 1 d 2CE /A0 E 2 + emE 2E + 16 /2A0 2 emE ( 2 +r+2) e−|x−x0 |

(5.11)

for all x ∈ Rd such that √ d + r + 3, |x − x0 | 2 where mE and CE are as in Theorem 3.5. Proof. By Theorem 5.1 W has no essential spectrum in (a, b). Thus, if E ∈ σ (W ) ∩ (a, b), it must be an isolated eigenvalue with finite multiplicity; let ψ be a corresponding eigenvector. To estimate the decay of ψ we have to deal with the fact that the form domains of W and W0 may be different, and ψ may not be in the form domain of W0 . Thus we pick ρ ∈ C 1 (Rd ) such that 1 − χBr+2 (x0 ) (x) ρ(x) 1 − χBr+1 (x0 ) (x),

|∇ρ(x)| 2.

(5.12)

Since W and W0 differ by a defect supported by Br (x0 ), it follows from (2.25) that Dρ ≡ ρD(A) = ρD(A0 ), and Aϕ = A0 ϕ for ϕ ∈ Dρ . Thus, if φ ∈ D(A0 ), we have A0 φ, A0 ρψ = = = = = = =

A0 φ, Aρψ = A0 φ, ρAψ + A0 φ, A[ρ]ψ ρA0 φ, Aψ + A0 φ, A[ρ]ψ ρA0 φ, Aψ + A0 φ, A[ρ]ψ A0 ρφ, Aψ − A0 [ρ]φ, Aψ + A0 φ, A[ρ]ψ Aρφ, Aψ − A0 [ρ]φ, Aψ + A0 φ, A[ρ]ψ ρφ, W ψ − A0 [ρ]φ, Aψ + A0 φ, A0 [ρ]ψ Eρφ, ψ − A0 [ρ]φ, Aψ + A0 φ, A0 [ρ]ψ.

(5.13)

−1

Taking φ = (W0 − E) χx ψ, we get

χx ψ2 = − A0 [ρ](W0 − E)−1 χx ψ, Aψ +

+ A0 (W0 − E)−1 χx ψ, A0 [ρ]ψ

= − A0 [ρ]χBr+2 (x0 ) (W0 − E)−1 χx ψ, Aψ +

+ χBr+2 (x0 ) A0 (W0 − E)−1 χx ψ, A0 [ρ]ψ √ 2/A0 EχBr+2 (x0 ) (W0 − E)−1 χx + + χBr+2 (x0 ) A0 (W0 − E)−1 χx ψ2 , where we used (3.64) and Aψ = ψ, W ψ = Eψ . 2

2

(5.14)


127

The estimate (5.11) now follows from (5.14), using (3.20) and (3.23) in Theorem 3.5. ✷ The next theorem shows that one can design simple defects which generate eigenvalues in a specified subinterval of a spectral gap of W0 , extending [9, Theorem 2] to the class of classical wave operators. Let 5 be a an open bounded subset of Rd , x0 ∈ 5. Typically, we take 5 to be the cube =1 (x0 ), or the ball B1 (x0 ). We set 58 = x0 + 8(5 − x0 ) for 8 > 0. We insert a defect that changes the value of K0 (x) and R0 (x) inside 58 to given positive constants K and R. If (a, b) is a gap in the spectrum of W0 , we will show that we can deposit an eigenvalue of W inside √ any specified closed subinterval of (a, b), by inserting such a defect with 8/ KR large enough, how large depending only on D+ , the geometry of 5, and the specified closed subinterval. THEOREM 5.4 (Creation of midgap eigenvalues). Let (a, b) be a gap in the spectrum of a second-order partially elliptic classical wave operator W0 = WK0 ,R0 , select µ ∈ (a, b), and pick δ > 0 such that the interval [µ − δ, µ + δ] is contained in the gap, i.e., [µ − δ, µ + δ] ⊂ (a, b). Given an open bounded set 5, x0 ∈ 5, 0 < K, R, 8 < ∞, we introduce a defect that produces coefficient matrices K(x) and R(x) that are constant in the set 58 = x0 + 8(5 − x0 ), with K(x) = KIn

and

R(x) = RIm

for x ∈ 58 .

(5.15)

If 12 µ δ 2 2 > D+ inf ∇η2 + ∇η2 + -η2 , √ δ µ KR 8

√

(5.16)

where the infimum is taken over all real valued C 2 -functions η on Rd with support in 5 and η2 = 1, the operator W = WK,R has at least one eigenvalue in the interval [µ − δ, µ + δ]. Proof. We proceed as in [9, Theorem 2]. In view of Corollary 5.3, it suffices to show that σ (W ) ∩ [µ − δ, µ + δ] = ∅

(5.17)

if (5.16) is satisfied. To prove (5.17), it suffices to find ϕ ∈ D(W ) such that (W − µ)ϕ δϕ. (5.18) To do so, we will construct a function ϕ ∈ D(W ), with ϕ = 1 and support in 58 , such that (5.18) holds. In this case the inequality (5.18) takes the following simple form: (KRD∗ D − µ)ϕ δ, (5.19)

128


which is the same as ∗ (D D − µ )ϕ δ ,

(5.20)

with µ = µ/KR and δ = δ/KR. We start by constructing generalized eigenfunctions for the nonnegative operator D∗ D corresponding to µ . In order to do this, we consider κ ∈ Sd , pick an eigenvalue λ = λκ > 0 and a corresponding eigenvector ξ = ξκ,λ ∈ Cn , |ξ | = 1, of the n × n matrix D(κ)∗ D(κ) (see (2.17)). We set µ λ κ·x

f (x) = fκ,λ,ξ (x) = ei

ξ ∈ C ∞ (Rn ; Cn ).

(5.21)

Note that, pointwise, we have |f (x)| = 1, and (D∗ Df )(x) = µ f (x).

(5.22)

To produce the desired ϕ satisfying (5.18), we will restrict f to 58 in suitable manner, and prove (5.20). To do so, let η8 be a real valued C 2 function on Rd with support in 58 and η8 2 = 1. We set ϕ(x) = η8 (x)f (x),

note

ϕ = η8 2 = 1.

(5.23)

We have ϕ ∈ D(D∗ D) with support in 58 , and (D∗ D − µ )ϕ ! ∗ µ ∗ D (−i∇η8 )D(κ)f + = D (−i∇)D(−i∇η8 ) f + λ ! µ ∗ D (κ)D(−i∇η8 )f. + λ Thus

! ∗ (D D − µ )ϕ D 2 -η8 2 + 2 µ D 2 ∇η8 2 . + λ +

(5.24)

(5.25)

We now use a scaling argument (i.e., write η8 (x) = η(8−1 (x − x0 ) + x0 )) to conclude that to obtain (5.20), it suffices to find η ∈ C 2 (Rd , R) with support in 5, η2 = 1, and a unit vector κ ∈ Rd , such that ! −2 2 −1 µ D 2 ∇η2 δ , (5.26) 8 D+ -η2 + 28 λ + which will be satisfied if

√ √ 8−2 KRD+2 -η2 + 28−1 KRD+ µ∇η2 δ,

where we used the fact that λ D+2 . Thus (5.20) holds if (5.16) is satisfied.

(5.27) ✷


129

Appendix: A Useful Lemma The following well known lemma (e.g., [4, Lemma 2]) is used throughout this paper. We recall that, given a closed densely defined operator T on a Hilbert space H, we denote its kernel by ker T and its range by ran T . If T is self-adjoint, it leaves invariant the orthogonal complement of its kernel; the restriction of T to (ker T )⊥ is denoted by T⊥ , a self-adjoint operator on the Hilbert space (ker T )⊥ . LEMMA A.1. Let B be a closed, densely defined operator from the Hilbert space H1 to the Hilbert space H2 . Then the operators (B ∗ B)⊥ and (BB ∗ )⊥ are unitarily equivalent. More precisely, the operator U defined by −1

U ψ = B(B ∗ B)⊥ 2 ψ

1

for ψ ∈ ran(B ∗ B)⊥2 ,

(A.1)

extends to a unitary operator from (ker B)⊥ to (ker B ∗ )⊥ , and (BB ∗ )⊥ = U (B ∗ B)⊥ U ∗ .

(A.2)

Acknowledgements The authors thanks Maximilian Seifert for many discussions and suggestions. A. Klein also thanks Alex Figotin, François Germinet, and Svetlana Jitomirskaya for enjoyable discussions. References 1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11.

Barbaroux, J. M., Combes, J. M. and Hislop, P. D.: Localization near band edges for random Schrödinger operators, Helv. Phys. Acta 70 (1997), 16–43 . Combes, J. M. and Hislop, P. D.: Localization for some continuous, random Hamiltonian in d-dimension, J. Funct. Anal. 124 (1994), 149–180. Combes, J. M., Hislop, P. D. and Tip, A.: Band edge localization and the density of states for acoustic and electromagnetic waves in random media, Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), 381–428. Deift, P. A.: Applications of a commutation formula, Duke Math. J. 45 (1978), 267–310. von Dreifus, H. and Klein, A.: A new proof of localization in the Anderson tight binding model, Comm. Math. Phys. 124 (1989), 285–299. Eastham, M.: The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, 1973. Figotin, A. and Klein, A.: Localization of classical waves I: Acoustic waves, Comm. Math. Phys. 180 (1996), 439–482. Figotin, A. and Klein, A.: Localization of classical waves II: Electromagnetic waves, Comm. Math. Phys. 184 (1997), 411–441. Figotin, A. and Klein, A.: Localized classical waves created by defects, J. Statist. Phys. 86 (1997), 165–177. Figotin, A. and Klein, A.: Midgap defect modes in dielectric and acoustic media, SIAM J. Appl. Math. 58 (1998), 1748–1773. Figotin, A. and Klein, A.: Localization of light in lossless inhomogeneous dielectrics, J. Opt. Soc. Amer. A 15 (1998), 1423–1435.

130 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.


Fröhlich, J., Martinelli, F., Scoppola, E. and Spencer, T.: Constructive proof of localization in the Anderson tight binding model, Comm. Math. Phys. 101 (1985), 21–46. Fröhlich, J. and Spencer, T.: Absence of diffusion with Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), 151–184. Germinet, F. and Klein, A.: Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys., to appear. Kato, T.: Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1976. Klein, A.: Localization of light in randomized periodic media, In: J.-P. Fouque (ed.), Diffuse Waves in Complex Media, Kluwer, Dordrecht, 1999, pp. 73–92. Klein, A. and Koines, A.: A general framework for localization of classical waves: II. Random media, in preparation. Klein, A., Koines, A. and Seifert, M.: Generalized eigenfunctions for waves in inhomogeneous media, J. Funct. Anal., to appear. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, Vol. IV, Analysis of Operators, Academic Press, New York, 1978. Schulenberger, J. and Wilcox, C.: Coerciveness inequalities for nonelliptic systems of partial differential equations, Arch. Rational Mech. Anal. 88 (1971), 229–305. Wilcox, C.: Wave operators and asymptotic solutions of wave propagation problems of classical physics, Arch. Rational Mech. Anal. 22 (1966), 37–78.


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Toda Equations, bi-Hamiltonian Systems, and Compatible Lie Algebroids ATTILIO MEUCCI Bain & Co, via Crocefisso 10, I-20122 Milan, Italy. e-mail: [email protected] (Received: 23 March 2001) Abstract. We present the bi-Hamiltonian structure of Toda3 , a dynamical system studied by Kupershmidt as a restriction of the discrete KP hierarchy. We derive this structure by a suitable reduction of the set of maps from Zd to GL(3, R), in the framework of Lie algebroids. Mathematics Subject Classifications (2000): 37K10, 70H06. Key words: Toda lattice, Lie algebroids, bi-Hamiltonian manifolds, Marsden–Ratiu reduction.

1. Introduction It is well known (see [12] and references therein) that the periodic Toda lattice is a bi-Hamiltonian system and that its integrability properties can be easily derived by its bi-Hamiltonian structure. In [11], this structure is investigated by means of a new approach: it stems from a reduction process of a special kind of Lie algebroids [6]. This approach parallels the work of [2], where the ‘continuous counterpart’ of the Toda lattice is studied, namely, the KdV equation. Indeed, the KdV equation is a bi-Hamiltonian system obtained by reducing the space Map(S 1 , gl(2, R)) of C ∞ maps from S 1 to gl(2, R). If instead of the space Map(S 1 , gl(2, R)) one considers the space Map(S 1 , gl(3, R)), one obtains the Boussineq hierarchy, which also displays a bi-Hamiltonian structure. The discrete version of the KdV equation, the Toda lattice, is obtained in [11] by replacing the circle S 1 with the cyclic group Zd and the algebra gl(2, R) with the group GL(2, R): therefore, the space to reduce is Map(Zd , GL(2, R)). In this paper we analyze the discrete version of the Boussineq equation. We consider therefore the reduction of the space Map(Zd , GL(3, R)). The equations we obtain also display a bi-Hamiltonian structure, which, to the best of our knowledge, is not known in the literature. We call Toda3 the integrable dynamical system that arises naturally from this structure. This dynamical system is studied under a different perspective by Kupershmidt in [5]. The plan of the paper is the following. In Section 2 we introduce the geometrical structure of the phase space to reduce: the set of maps Map(Zd , GL(3, R)). Following the analysis in [11], this space can be endowed with the structure of

132

ATTILIO MEUCCI

a Poisson bi-anchored manifold. In Section 3 we present the reduction of the Poisson bi-anchored manifold Map(Zd , GL(3, R)). This reduction is an adaptation of the Marsden–Ratiu reduction scheme for Poisson manifolds [9] and gives rise to a bi-Hamiltonian manifold. In Section 4 we apply the theory of Gelfand and Zakharevich [4] to study the integrability properties of the bi-Hamiltonian flows obtained previously. The last section contains an example. 2. Poisson Bi-anchored Manifolds In this section we introduce the geometric objects we need for our approach to the Toda lattice. The discussion closely follows [11]. We will endow the manifold M of the maps from the cyclic group Zd to GL(3, R) with several structures, namely a Poisson tensor and two compatible Lie algebroids (see [6]) suitably soldered together: this will make M into a Poisson bi-anchored manifold. First we introduce the manifold M. A point q of M is simply a d-tuple of invertible 3 × 3 matrices (1) q = q1, . . . , qd , where 

 q1k q2k q3k q k =  q4k q5k q6k  . q7k q8k q9k

(2)

We will always be dealing with d-tuples of matrices and the following condition is supposed to hold throughout the discussion: (·)k+d = (·)k .

(3)

For convenience, we will say that a matrix as in (2) represents a d-tuple as in (1). Vector fields on M will be represented by d-tuples of 3 × 3 matrices q˙ k whose entries are functions of the point q ∈ M. The same way, one-forms on M are represented as d-tuples of 3 × 3 matrices α k where each entry is a function of the point. The value of the one-form α on the vector field q˙ is given by the scalar function: α, q ˙ =

d

Tr α k q˙ k .

(4)

k=1

Next, we endow M with a Poisson manifold structure. A quick computation shows that the map P : T ∗ M → T M defined as q˙ k = P (α)k = q k α k b − bα k q k ,

(5)

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where b is any fixed matrix, is indeed a Poisson tensor, i.e., it defines a Poisson bracket {f, g} = df, P dg . In order to recover the Toda lattice we choose   1 0 0 b =  0 0 0 . (6) 0 0 0 To obtain the Hamiltonian vector field XH associated with a (Hamiltonian) function H : M → R, we simply have to plug its differential α = dH into Equation (5). At this point we endow M with an additional structure: a pencil of Lie algebroids. We recall (see [6]) that (M, E, A, {·, ·}) is a Lie algebroid if (i) E is a vector bundle on M (ii) {·, ·} is a bilinear composition law on the sections of E that makes them into a Lie algebra (iii) the map A: E → T M, called an anchor, is a Lie algebra morphism: A({s, t}) = [A(s), A(t)],

(7)

where [·, ·] is the usual commutator of vector fields.

If two different Lie algebroid structures (M, E, A, {·, ·}) and (M, E, A , {·, ·} ) coexist on the same manifold M and vector bundle E we can consider the pencil of brackets {s, t}λ = {s, t} + λ{s, t}

(8)

and the pencil of maps

Aλ (s) = A(s) + λA (s),

(9)

where λ is a complex parameter. The two Lie algebroid structures are said to be compatible if (M, E, Aλ , {·, ·}λ ) is a Lie algebroid for every value of λ. In our case, we consider the trivial vector bundle E = M × [Mat(3, R)]d , where d (·) denotes the d-times Cartesian product (·)×· · ·×(·). The sections of this bundle are represented by d-tuples of 3 × 3 matrices s k whose entries are functions of the point q ∈ M. Then we define the pencil of anchors Aλ : E → T M as (10) q˙ k = Aλ (s)k = s k+1 q k + λb − q k + λb s k , and a composition law {·, ·}λ as

{s, t}kλ = ∂A(s)t k − ∂A(t )s k + t k , s k + λ ∂A (s) t k − ∂A (t )s k ,

(11)

˙ It is where by ∂q˙ t we mean the derivative of the section t along the vector field q. easy to prove that the manifold (M, E, Aλ , {·, ·}λ ) is a pencil of Lie algebroids. The pencil of Lie algebroids structure defines a useful relation among oneforms. We define a one-form α to be related with a one-form β (and we denote this by α ∼ β) if A∗ α = A ∗ β. In order to explicitly calculate this relation we

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need to derive the expression of the dual pencil of anchors A∗λ = A∗ + λA ∗ . An element ξ of the dual vector bundle E ∗ can be naturally identified with a point of E by means of the pairing ξ, s =

d

Tr ξ k s k .

k=1

Therefore the dual pencil A∗λ : T ∗ M → E ∗ can be viewed as a map that with a oneform α associates a d-tuple ξ k of matrices whose entries are functions of the point q ∈ M. A quick calculation shows that A∗λ reads: (12) ξ k = A∗λ (α)k = q k−1 + λb α k−1 − α k q k + λb . Thus far we have defined a Poisson structure and a pencil of Lie algebroids on the manifold M. We arrived at the last step: soldering these structures by means of two maps J, J : T ∗ M → E that verify the following conditions: if α ∼ β, then P (α) = A (J α + J β), P (β) = A(J α + J ).

(13) (14)

It is easy to see that the intertwining maps defined as J : sk = αk q k ,

J : s k = −α k b,

satisfy the above relation. This ends the definition of the geometrical structures of the manifold M = Map(Zd , GL(3, R)): it is a Poisson manifold endowed with two compatible Lie algebroid structures that define a relation on one-forms and two intertwining maps that solder everything. We call such a structure a Poisson bi-anchored manifold. 3. The Reduction In this section we perform the reduction of the manifold M = Map(Zd , GL(3, R)) introduced in the previous section. Combining a restriction and a projection we obtain a new manifold N of lower dimension endowed with the same geometrical structure as the original manifold M, but with the important additional property of being bi-Hamiltonian. As a first step, we consider the distribution Im P + Im A . It is easy to verify by formulas ( 5) and (10) that this distribution is integrable and therefore it foliates Map(Zd , GL(3, R)) in maximal integral leaves, which are the 5d-dimensional hyperplanes of the form  k k k q1 q2 q3 k  (15) q = q4k ν5k ν6k  , q7k ν8k ν9k


135

where the νik are constants. The restriction process we mentioned above consists in selecting one of these leaves. To obtain the Toda3 system, we pick the leaf L defined by points of the form  k k k q1 q2 q3 (16) q k =  q4k 0 0  . q7k 1 0 The pencil of anchors allows us to define another distribution, namely D = A(ker A ), which is integrable. As opposed to the GL(2, R) case, this distribution is not tangent to L. We are interested only in the restriction of this distribution to the leaf L, which we denote by D|L . The distribution E = D|L ∩ T L of L is also integrable and an explicit computation shows that E is spanned by the vector fields of the form q˙1k = 0, q˙5k = 0,

q˙2k = µk q2k − q3k s8k , q˙3k = µk−1 q3k , q˙4k = −µk+1 q4k , q˙6k = 0, q˙7k = τ k q4k − µk q7k , q˙8k = 0, q˙9k = 0

(17)

for arbitrary µk , τ k . From this expression we see that along the vector fields in E the following equations are satisfied: k+1 k k k+2 • • q2 q4 + q3k+1 q7k = 0, q4 q3 = 0. q˙1k = 0, This means that the distribution E admits the three invariants a1k = q1k , a2k = q2k+1 q4k + q3k+1 q7k ,

(18) (19)

a3k = q4k q3k+2 .

(20)

At this point we can operate the projection we mentioned above: we define the reduced manifold N to be the quotient of the leaf L with respect to the foliation induced by the distribution E. By (18), (19) and (20) we argue that N is a 3d-dimensional manifold that can be regarded as R3d , endowed with the set of coordinates (a1k , a2k , a3k )k=1,...,d . The above formulas also yield the expression of the canonical projection π : L → N . After obtaining the reduced manifold N we endow it with a Poisson structure. This can be done observing that (M, P , D, L) is Poisson reducible, in the terminology of [9]. In this context, to find the expression of the reduction of P we have to extend a generic one-form ϕ on N to a one-form α on M (possibly defined only at the points of the leaf L) which annihilates the distribution D. This means: α, D = 0,

˙ α, q ˙ = ϕ, π∗ q .

Let us denote by α = ext(ϕ) any such extension. Then the expression p (ϕ) = π∗ ◦ P (ext(ϕ))

(21)

This is true in general, provided that for any two sections s, t such that A (s) = 0 and A (t) = 0 we have {s, t} = 0. It is evident from (11) that this condition holds in our case.

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does not depend on the of ext(ϕ) and determines a Poisson structure on N .

dchoice k If we denote by ϕ = k=1 ϕ1 da1k + ϕ2k da2k + ϕ3k da3k the generic one-form on N , an easy calculation shows that an extension α = ext(ϕ) has the form   ϕ3k q3k+2 + q2k+1 ϕ2k ϕ2k q3k+1 ϕ1k (22) αk =  ϕ2k−1 q4k−1 α5k q3k+1 ϕ3k−1 q4k−1  , k−2 k−2 k−1 k−1 k ϕ3 q4 + q7 ϕ2 α8 α9k where α9k = q7k−1 ϕ3k−1 q3k+1 − q2k ϕ3k−2 q4k−2 + α5k−1 . As we said, this matrix is not completely determined: the components α5k , α8k are free, since the extension gives an equivalence class of one-forms. Now we can apply formula (21) to obtain the expression of the reduced Poisson tensor p : a˙ 1k = ϕ2k−1 a2k−1 − ϕ2k a2k + ϕ3k−2 a3k−2 − ϕ3k a3k , a˙ 2k = ϕ1k − ϕ1k+1 a2k − ϕ2k+1 a3k + ϕ2k−1 a3k−1 , a˙ 3k = ϕ1k − ϕ1k+2 a3k .

(23)

Recalling the periodic Toda case [11], to find another Poisson structure we have to determine the reduced relation on the one-forms of N . Therefore we need the expression of the reduced dual pencil of anchors a∗λ . To have this, in turn, we have to define a proper reduced vector bundle U based on N on which the reduced anchors act: aλ : U → T N . It is convenient to define first the dual vector bundle U∗ and the dual pencil a∗λ . The definition of the vector bundle U and the pencil aλ will then follow by duality. A lenghty computation [10] allows us to find all these characters. We are only interested in the reduced relation on one-forms, which turns out to be the following: ϕ ∼ ψ (i.e., a∗ (ϕ) = a ∗ (ψ)) if and only if they satisfy ψ1k − ψ1k+1 = ϕ1k a1k − ϕ1k+1 a1k+1 + ϕ2k−1 a2k−1 − ϕ2k+1 a2k+1 + + ϕ3k−2 a3k−2 − ϕ3k+1 a3k+1 ,

(24)

−ψ2k−2 a3k−2 + ψ2k a3k−1 = ϕ1k−2 a3k−2 − ϕ1k+1 a3k−1 − ϕ2k−2 a3k−2 a1k−1 + + ϕ2k a3k−1 a1k − ϕ3k−2 a3k−2 a2k−1 + ϕ3k−1 a3k−1 a2k−1 , ψ2k a2k − ψ2k−1 a2k−1 + ψ3k a3k − ψ3k−2 a3k−2 =

ϕ1k−1 a2k−1 − ϕ1k+1 a2k + + ϕ2k−2 a3k−2 − ϕ2k+1 a3k

(25) (26)

+ a1k ϕ2k a2k − ϕ2k−1 a2k−1 + a1k ϕ3k a3k − ϕ3k−2 a3k−2 .

Now we focus on the special feature of the reduced Poisson bi-anchored manifold N . By Equations (24), (25) and (26) for a fixed one-form ϕ there is a whole class


137

of one-forms [ψ] = ψ + ker a ∗ that is related with it. In the reduced structure, though, ker a ∗ ⊂ ker p . Therefore the following tensor, p(ϕ) := p (ψ),

(27)

is well defined. A lengthy calculation shows that the bracket induced by this tensor verifies the Jacobi identity. Furthermore, p is compatible with p , i.e., the pencil pλ = p + λp is a Poisson tensor for all values of the complex parameter λ. Explicitly, a˙ = p(ϕ) reads a˙ 1k = a1k ϕ2k−1 a2k−1 − ϕ2k a2k + ϕ3k−2 a3k−2 − ϕ3k a3k + + a2k ϕ1k+1 − a2k−1 ϕ1k−1 + a3k ϕ2k+1 − a3k−2 ϕ2k−2 , a˙ 2k = a1k ϕ1k a2k + ϕ2k−1 a3k−1 − (28) k+1 k k+1 k+1 k k+2 k k−1 k−1 − a1 ϕ1 a2 + ϕ2 a3 + ϕ1 a3 − ϕ1 a3 + + a2k ϕ2k−1 a2k−1 − ϕ2k+1 a2k+1 + ϕ3k−2 a3k−2 + ϕ3k−1 a3k−1 − − ϕ3k a3k − ϕ3k+1 a3k+1 , a˙ 3k = a3k ϕ1k a1k − ϕ1k+2 a1k+2 + ϕ2k−1 a2k−1 + ϕ2k a2k − ϕ2k+1 a2k+1 − ϕ2k+2 a2k+2 + + ϕ3k−2 a3k−2 + ϕ3k−1 a3k−1 − ϕ3k+1 a3k+1 − ϕ3k+2 a3k+2 . Our goal has been achieved: we arrived at a bi-Hamiltonian manifold by means of a systematic procedure of reduction of the original Poisson bi-anchored manifold. Now we have to investigate the information provided by the bi-Hamiltonian structure. 4. The Toda3 System In this section we show how the bi-Hamiltonian structure obtained in the previous section defines specific vector fields and, moreover, accounts for their integrability. To obtain an integrable system we will focus on the Casimirs of the Poisson pencil, i.e., the functions C such that their differentials are in the kernel of Pλ . Indeed, if we expand a Casimir in powers of λ, H i λi , (29) C= i

it is immediate to check that the coefficients Hi satisfy the Lenard relations P (dHi ) = −P (dHi+1 ).

(30)

It is easily shown (see, e.g., [7]) that this in turn implies that the Hi ’s are in involution with respect to both Poisson bracket. If these coefficients are enough, the We recall that a manifold endowed with two compatible Poisson tensors is said to be

bi-Hamiltonian.

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system is integrable in the classical sense of Liouville and Arnold [1]. Nevertheless, in general finding the Casimirs of a Poisson pencil is not easy. In the present case we can make use of the following PROPOSITION 1. If hk solves hk hk+1 hk+2 = a1k+2 + λ hk hk+1 + a2k+1 hk + a3k ,

(31)

then C(λ) = h1 · · · hd is a Casimir of the Poisson pencil (23)–(28). The solutions hk and thus the Casimirs C can be calculated explicitly as Laurent series in the parameter λ. Proof. See the appendix. ✷ We call (31) the characteristic equation. Proposition 1 allows us in principle to calculate the Casimirs of the Poisson pencil pλ , but the computation is lengthy. Fortunately there is a shortcut: if in (31) we set hk =

ψk+1 µ, ψk

(32)

the characteristic equation becomes the linear system 0 = ψk+3 µ3 − a1k+2 + λ ψk+2 µ2 − a2k+1 ψk+1 µ − a3k ψk .

(33)

We can express (33) in matrix form as Lψ = 0, where L is the matrix  −µ3 µ2 a11 + λ   µa21 µ2 a12 + λ   L= a31 µa22   ..  . 0 −µ3 0

(34)

0

a3d−1 .. . .. .

µa2d



  −µ3 a3d   ..  . 0   .. d−1 2 3 . µ a1 + λ −µ  a3d−2 µa2d−1 µ2 a1d + λ

and ψ is the vector of the ‘homogeneous coordinates’   ψ1   ψ =  ...  . ψd For (34) to admit nontrivial solutions we must have det L = 0. It can be proved that the cyclicity of the matrix L implies that its determinant is a polynomial of degree 3 in µd . Therefore we must have 0 = det L = ±µ3d + K1 µ2d + K2 µd + K3 ,

(35)


139

where K1 , K2 , K3 are polynomials in λ (in particular, C3 does not depend on λ). By Proposition 1 and Equation (32), for all µ that satisfy (35) we have that µd = h1 . . . hd is a Casimir, thus K1 , K2 and K3 are Casimirs as well. Their coefficients provide all information about the geometry of the system at hand. The family of dynamical systems associated with the Casimirs K1 , K2 and K3 is what we call Toda3 . It can be proved that they coincide with Kupershmidt’s reduction of the discrete KP hierarchy [5]. We will show in an example how the integrability of these systems stems from their bi-Hamiltonian structure. 5. An Example of the Toda3 System To illustrate how the scheme described above works we consider the specific case where d = 4. This example is easy to handle, but at the same time general enough. In order to make the equations easier to read we will change notation: we set a1k = bk , a2k = ak , a3k = ck , ϕ1k = βk , ϕ2k = αk , ϕ3k = γk . Thus our bi-Hamiltonian manifold N becomes R12

with coordinates (a1 , . . . , c4 ). The Poisson pencil pλ associates with a one-form 4k=1 (αk dak + βk dbk + γk dck ) the vector field b˙k = (bk + λ)(αk−1 ak−1 − αk ak + γk+2 ck+2 − γk ck ) + + ak βk+1 − ak−1 βk−1 + ck αk+1 − ck+2 αk+2 , a˙ k = (ck−1 αk−1 + ak βk )(bk + λ) − − (ck αk+1 + ak βk+1 )(bk+1 + λ) + ck βk+2 − βk−1 ck−1 + + ak (αk−1 ak−1 − αk+1 ak+1 + γk−1 ck−1 − γk ck − − γk+1 ck+1 + γk+2 ck+2 ), c˙k = ck (βk (bk + λ) − βk+2 (bk+2 + λ)) + + ck (−αk+2 ak+2 + αk ak − αk+1 ak+1 + αk−1 ak−1 + + γk−1 ck−1 − γk+1 ck+1 ),

(36)

where k = 1, 2, 3, 4. Equation (35) provides the Casimirs of the Poisson pencil (36). Indeed, from that equation one expects to find only three Casimirs of the whole pencil. The interesting feature of this approach is that in fact we obtain four Casimirs which prove to be enough to guarantee integrability. The three coefficients are K1 (λ) = λ4 + λ3 C1 + λ2 H1 + λH2 + H3 , K2 (λ) = λ2 C2 + λC3 + H4 , K3 (λ) = C4 , where C1 = b1 + b2 + b3 + b4 , H1 = a1 + a2 + a3 + a4 + b1 b2 + b2 b3 + b3 b4 + b4 b1 + b1 b3 + b2 b4 ,

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H2 = c1 + c2 + c3 + c4 + b1 b2 b3 + b2 b3 b4 + b3 b4 b1 + b4 b1 b2 + + b1 (a2 + a3 ) + b2 (a3 + a4 ) + b3 (a4 + a1 ) + b4 (a1 + a2 ), H3 = b1 b2 b3 b4 + b1 c2 + b2 c3 + b3 c4 + b4 c1 + a1 a3 + a2 a4 + + b1 b2 a3 + b2 b3 a4 + b3 b4 a1 + b4 b1 a2 , C2 = −c2 c4 − c1 c3 , C3 = −b1 c2 c4 − b2 c3 c1 − b3 c4 c2 − b4 c1 c3 + + c1 a3 a4 + c2 a4 a1 + c3 a1 a2 + c4 a2 a3 , H4 = −a1 a2 a3 a4 + b1 a2 a3 c4 + b2 a3 a4 c1 + b3 a4 a1 c2 + b4 a1 a2 c3 + + a1 c2 c3 + a2 c3 c4 + a3 c4 c1 + a4 c1 c2 − b2 b4 c1 c3 − b1 b3 c2 c4 , C4 = c1 c2 c3 c4 . Nonetheless, the second coefficient K2 (λ) is composed itself of two independent Casimirs of the pencil: λ2 C2 and λC3 + H4 . Therefore we obtain the four Casimirs: K1 (λ) = λ4 + λ3 C1 + λ2 H1 + λH2 + H3 , K2

(λ) = λC3 + H4 , K3 (λ) = C4 .

K2 (λ) = C2 ,

In the theory of Gelfand–Zakharevich the important objects are these eight functions C1 , . . . , C4 , H1 , . . . , H4 . Since the K’s are Casimir functions of the Poisson pencil, we must have: p (dH1 ) = −p(dC1 ), p (dH3 ) = −p(dH2 ),

p (dH2 ) = −p(dH1 ), p (dH4 ) = −p(dC3 )

(37)

and 0 = p (dC1 ) = p (dC2 ) = p (dC3 ) = p (dC4 ) = p(dC2 ) = p(dH3 ) = p(dC4 ) = p(dH4 ).

(38)

The 4-particle periodic Toda3 system are the vector fields X1 , . . . , X4 defined as follows: X1 = p (dH1 ),

X2 = p (dH2 ),

X3 = p (dH3 ),

X4 = p (dH4 ).

The explicit expression of these four vector fields is X1 :

X2 :

X3 :

b˙1 = a4 − a1 , a˙ 1 = a1 (b2 − b1 ) − c1 + c4 , c˙1 = c1 (b3 − b1 ), b˙1 = c3 − c1 − (b3 + b4 )a1 + (b2 + b3 )a4 , a˙ 1 = a1 (−b4 b1 − b1 b3 + b2 b3 + b4 b2 − a4 + a2 )− −c1 (b4 + b1 ) + c4 (b2 + b3 ), c˙1 = c1 (−b4 b1 − b1 b2 + b2 b3 + b3 b4 − a4 − a1 + a2 + a3 ), b˙1 = −b4 c1 + b2 c3 − (a3 + b3 b4 )a1 + (a2 + b2 b3 )a4 , a˙ 1 = a1 (−b3 b4 b1 + b2 b3 b4 − b3 a4 − b1 a3 + b2 a3 + b4 a2 −


X4 :

141

−c3 + c2 ) − c1 (a4 + b4 b1 ) + c4 (a2 + b2 b3 ), c˙1 = c1 (−b4 b1 b2 + b2 b3 b4 − b4 a1 − b2 a4 + b2 a3 + b4 a2 − c4 + c2 ), b˙1 = −c1 c4 a3 + c3 c4 a2 , a˙ 1 = −c4 b1 c1 a3 − c4 c1 c3 + c1 b2 c4 a3 + c1 c4 c2 , c˙1 = c1 (−a4 a1 c2 + b1 c4 c2 + a2 a3 c4 − b3 c2 c4 ).

Of course, in the above formulas the cyclic condition holds and yields the other components. Due to (37) these vector fields are bi-Hamiltonian. In order to show that they are integrable we choose a symplectic leaf of the Poisson tensor p . It can be shown that this leaf is given by C1 = constant, C3 = constant,

C2 = constant, C4 = constant.

Therefore, the leaf is an eight-dimensional symplectic submanifold of the twelvedimensional original phase space. As we said in Section 4, the functions H1 , . . . , H4 commute with respect to both Poisson brackets. Therefore their restrictions to the symplectic leaf also commute, and, since they can be checked to be functionally independent, constitute an integrable system. 6. Conclusions In this paper we showed that reducing a special Poisson bi-anchored manifold, namely the set of maps from Zd to GL(3, R), we obtain a new bi-Hamiltonian structure. This bi-Hamiltonian structure gives rise to an integrable system, which we called Toda3 and already appeared in the work of Kupershmidt [5]. This system represents a generalization of the periodic Toda lattice (which corresponds to GL(2, R)). There are several further developments of this approach. First of all, it is easy to endow the set of maps from Zd to GL(n, R) for a generic n ∈ N with the structure of bi-anchored Poisson manifold. It is possible to show that the reduction of these manifolds gives rise to other Toda systems, which are the discrete analog of the Gelfand–Dickey hierarchies [3]. Secondly (see [8, 10]) the study of the conservation laws of the periodic Toda lattice allows to define the discrete analog [5] of the KP equations on the Sato Grassmannian [13]. These represent flows on an infinite-dimensional phase space that admit invariant submanifolds. These submanifolds are the different phase spaces of the Toda system, and the restriction of the KP equation to these phase spaces are the Toda equations. This way it is possible to extend to the discrete case the description given for the continuous case in [2], where the KdV hierarchy and the (usual) KP equations are considered.

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Appendix. Proof of Proposition 1 We will split the proof in three parts. (1) We are looking for Casimir functions, i.e. exact one-forms in the kernel of the Poisson pencil pλ , which we rewrite like this a˙ 1k = a1k + λ ϕ2k−1 a2k−1 − ϕ2k a2k + ϕ3k−2 a3k−2 − ϕ3k a3k − − ϕ1k−1 a2k−1 + ϕ1k+1 a2k − ϕ2k−2 a3k−2 + ϕ2k+1 a3k , a˙ 2k = ϕ2k−1 a3k−1 a1k + λ − ϕ2k+1 a3k a1k+1 + λ −

(39)

− ϕ1k−1 a3k−1 + ϕ1k+2 a3k + ϕ3k−1 a3k−1 a2k − ϕ3k a3k a2k + + a2k ϕ1k a1k + λ − ϕ1k+1 a1k+1 + λ + ϕ2k−1 a2k−1 − − ϕ2k+1 a2k+1 + ϕ3k−2 a3k−2 − ϕ3k+1 a3k+1 , a˙ 3k = a3k ϕ1k a1k + λ − ϕ1k+1 a1k+1 + λ + ϕ2k−1 a2k−1 − − ϕ2k+1 a2k+1 + ϕ3k−2 a3k−2 − ϕ3k+1 a3k+1 + + a3k ϕ1k+1 a1k+1 + λ − ϕ1k+2 a1k+2 + λ + ϕ2k a2k − − ϕ2k+2 a2k+2 + ϕ3k−1 a3k−1 − ϕ3k+2 a3k+2 . From this expression, it is straightforward to see that in order for the one-form ϕ to be in the kernel of the Poisson pencil it is enough that it satisfies the following equations: 0 = a1k + λ ϕ2k−1 a2k−1 − ϕ2k a2k + ϕ3k−2 a3k−2 − ϕ3k a3k − − ϕ1k−1 a2k−1 + ϕ1k+1 a2k − ϕ2k−2 a3k−2 + ϕ2k+1 a3k , 0 = ϕ2k−1 a3k−1 a1k + λ − ϕ2k+1 a3k a1k+1 + λ − 0 =

− ϕ1k−1 a3k−1 + ϕ1k+2 a3k + ϕ3k−1 a3k−1 a2k − ϕ3k a3k a2k , ϕ1k a1k + λ − ϕ1k+1 a1k+1 + λ + ϕ2k−1 a2k−1 − − ϕ2k+1 a2k+1 + ϕ3k−2 a3k−2 − ϕ3k+1 a3k+1 .

(40)

(41)

A change of variables reduces this system to a single equation of Riccati type. Indeed, let hk be any solution of what we call the characteristic equation, which is of Riccati type: (42) hk hk+1 hk+2 = a1k+2 + λ hk hk+1 + a2k+1 hk + a3k . Let us set ϕ1k+2 = hk hk+1 ρk ,

ϕ2k+1 = hk ρk ,

ϕ3k = ρk .

Let us choose the multiplier ρ is such a way that the following equation holds: ϕ1k a1k + λ + ϕ2k−1 a2k−1 + ϕ2k a2k + ϕ3k−2 a3k−2 + ϕ3k−1 a3k−1 + ϕ3k a3k = L, where L is any constant different from zero.


143

(2) Such a one-form ϕ solves (40) and therefore it is an element of the kernel of the Poisson pencil pλ . Indeed, let us consider a one-form ϕ that satisfies the system: hk hk+1 hk+2 = a1k+2 + λ hk hk+1 + a2k+1 hk + a3k , ϕ1k = hk−2 hk−1 ϕ3k−2 ,

ϕ2k = hk−1 ϕ3k−1 ,

(43)

L = ϕ1k (a1k + λ) + ϕ2k−1 a2k−1 + ϕ2k a2k + + ϕ3k−2 a3k−2 + ϕ3k−1 a3k−1 + ϕ3k a3k . We will show that ϕ satisfies the following equations: 0 = a1k + λ ϕ2k−1 a2k−1 − ϕ2k a2k + ϕ3k−2 a3k−2 − ϕ3k a3k − − ϕ1k−1 a2k−1 + ϕ1k+1 a2k − ϕ2k−2 a3k−2 + ϕ2k+1 a3k , 0 = ϕ2k−1 a3k−1 a1k + λ − ϕ2k+1 a3k a1k+1 + λ −

(44)

− ϕ1k−1 a3k−1 + ϕ1k+2 a3k + ϕ3k−1 a3k−1 a2k − ϕ3k a3k a2k , 0 = ϕ1k a1k + λ − ϕ1k+1 a1k+1 + λ + ϕ2k−1 a2k−1 − ϕ2k+1 a2k+1 + + ϕ3k−2 a3k−2 − ϕ3k+1 a3k+1 . We immediately see that ϕ fulfills the third equation of (44): just compare with the last equation of (43). Before we verify that the two other equations are satisfied as well, we need a formula. Namely, we have hk−3 hk−2 hk−1 ϕ3k−3 − hk−2 hk−1 a1k + λ ϕ3k−2 = hk−1 ϕ3k−1 a2k + ϕ3k a3k . (45) We will show now that ϕ satisfies the first equation of (44): 0 = a1k + λ ϕ2k−1 a2k−1 − ϕ2k a2k + ϕ3k−2 a3k−2 − ϕ3k a3k − − ϕ1k−1 a2k−1

+

ϕ1k+1 a2k

−

ϕ2k−2 a3k−2

+

(46)

ϕ2k+1 a3k .

From the last equation of (43) we obtain

L = ϕ3k−2 hk−2 hk−1 (a1k + λ) + hk−2 ϕ3k−2 a2k−1 + hk−1 ϕ3k−1 a2k + ϕ3k−2 a3k−2 + + ϕ3k−1 a3k−1 + ϕ3k a3k = ϕ3k−2 (hk−2 hk−1 (a1k + λ) + hk−2 a2k−1 + a3k−2 ) + hk−1 ϕ3k−1 a2k + ϕ3k−1 a3k−1 + ϕ3k a3k = ϕ3k−2 (hk−2 hk−1 hk ) + ϕ3k−1 (hk−1 a2k + a3k−1 ) + ϕ3k a3k . This implies ϕ3k−2 (hk−2 hk−1 hk ) = ϕ3k−1 (hk−1 hk hk+1 − hk−1 a2k − a3k−1 ) + ϕ3k (hk a2k+1 ) + ϕ3k+1 a3k+1 and the result follows using the characteristic equation hk−1 hk hk+1 − a2k hk−1 − a3k−1 = (a1k+1 + λ)hk−1 hk .

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ATTILIO MEUCCI

We replace in this expression the values of ϕ1k , ϕ2k obtained from (43), and we multiply both sides by hk−2 hk−1 . We obtain that (46) holds if and only if 0 = hk−2 hk−1 a1k + λ hk−2 ϕ3k−2 a2k−1 − hk−1 ϕ3k−1 a2k + ϕ3k−2 a3k−2 − ϕ3k a3k − − hk−2 hk−1 hk−3 hk−2 ϕ3k−3 a2k−1 + hk−2 hk−1 hk−1 hk ϕ3k−1 a2k − − hk−2 hk−1 hk−3 ϕ3k−3 a3k−2 + hk−2 hk−1 hk ϕ3k a3k . Rearranging the terms in the last expression, we end up having to prove that

k−1 k k k ϕ a + ϕ a hk−3 h k−2 hk−1 ϕ3k−3 − hk−2 hk−1 a1k + λ ϕ3k−2 hk−1 2 3 3 3 0 = det hk−2 a2k−1 a3k−2 hk−2 hk−1 hk − hk−2 hk−1 a1k + λ but this is true, since the characteristic equation and (45) hold. We will show now that ϕ satisfies the second equation of (44): 0 = ϕ2k−1 a3k−1 a1k + λ − ϕ2k+1 a3k a1k+1 + λ − − ϕ1k−1 a3k−1 + ϕ1k+2 a3k + ϕ3k−1 a3k−1 a2k − ϕ3k a3k a2k .

(47)

We replace in this expression the values of ϕ1k , ϕ2k obtained from (43), and we multiply both sides by hk−1 . We obtain that (47) holds if and only if: 0 = hk−1 hk−2 ϕ3k−2 a3k−1 a1k + λ − hk−1 hk ϕ3k a3k a1k+1 + λ − − hk−1 hk−3 hk−2 ϕ3k−3 a3k−1 + hk−1 hk hk+1 ϕ3k a3k + + hk−1 ϕ3k−1 a3k−1 a2k − hk−1 ϕ3k a3k a2k . Rearranging the terms in the last expression, we end up having to prove that

k−1 hk−1 hk hk+1 − hk−1 hk a1k+1 + λ −a2k a3 0 = det ϕ3k a3k hk−1 hk−3 hk−2 ϕ3k−3 − hk−1 hk−2 ϕ3k−2 a1k + λ − ϕ3k−1 a2k but this is true, since the characteristic equation and (45) hold. (3) Furthermore, ϕ is exact. The system hk hk+1 hk+2 = a1k+2 + λ hk hk+1 + a2k+1 hk + a3k , ϕ2k = hk−1 ϕ3k−1 , ϕ1k = hk−2 hk−1 ϕ3k−2 , L = ϕ1k a1k + λ + ϕ2k−1 a2k−1 + ϕ2k a2k + ϕ3k−2 a3k−2 + ϕ3k−1 a3k−1 + ϕ3k a3k is equivalent to

hk hk+1 hk+2 = a1k+2 + λ hk hk+1 + a2k+1 hk + a3k ,

ϕ2k = hk−1 ϕ3k−1 , ϕ1k = hk−1 hk−2 ϕ3k−2 , 0 = L − ϕ3k a3k + ϕ2k a1k+1 + λ hk − ϕ1k hk − ϕ2k hk hk+1 . Combining the first and the fourth equations we obtain ϕ3k a1k+2 + λ hk+1 + ϕ3k a2k+1 L = − + ϕ3k hk+1 hk+2 − ϕ2k (a1k+1 + λ) + ϕ1k + ϕ2k hk+1 . hk

145


We will use this in the next calculation. We move on to evaluating d k+2 k+2 ϕ1 a˙ 1 + ϕ2k+1 a˙ 2k+1 + ϕ3k a˙ 3k k=1

=

d

hk hk+1 ϕ3k a˙ 1k+2 + hk ϕ3k a˙ 2k+1 + ϕ3k a˙ 3k

k=1

=

d

ϕ3k hk hk+1 a˙ 1k+2 + hk a˙ 2k+1 + a˙ 3k

k=1

=

d

ϕ3k

k=1

= =

d

˙ k hk+1 hk+2 + hk h˙ k+1 hk+2 + hk hk+1 h˙ k+2 − hk+2 − a1 + λ h˙ k hk+1 − a1k+2 + λ hk h˙ k+1 − a2k+1 h˙ k

k=1

ϕ3k h˙ k hk+1 hk+2 + ϕ3k hk h˙ k+1 hk+2 + ϕ3k hk hk+1 h˙ k+2 − k+2 k k+2 k −ϕ3 a1 + λ h˙ k hk+1 − ϕ3 a1 + λ hk h˙ k+1 − ϕ3k a2k+1 h˙ k

k=1

k˙ hk+1 hk+2 + ϕ2k+1 h˙ k+1 hk+2 + ϕ1k+2 h˙ k+2 − ϕk3 hkk+2 k+2 k+1 k+1 k − ϕ2 a1 + λ h˙ k+1 −h˙ k ϕ3 (a1 + λ)hk+1 + ϕ3 a2

d

 k+1 ˙ k+2 ˙ k˙ h h + ϕ h + ϕ − h h h ϕ k k+1 k+2 k+1 k+2 k+2 3 2 1 d    −h˙ k ϕ k + ϕ k hk+1 + ϕ k hk+1 hk+2 − L − ϕ k a k+1 + λ −  = 2 1 1 2 3   hk k=1 k+1 k+2 k+2 ˙ ˙ −ϕ2 a1 + λ hk+1 ϕ1 hk+2 .   k L d k k+1 k k ˙  −hk ϕ1 + ϕ2 hk+1 + ϕ3 hk+1 hk+2 − hk − ϕ2 a1 + λ −  −ϕ3k hk+1 hk+2 + ϕ2k+1 hk+2 − ϕ2k+1 a1k+2 + λ h˙ k+1 k=1 =



d

• L = L log(h1 · · · hd ) . h˙ k hk k=1

The last equation follows from L being a constant (independent of both site k and variables aik ). Therefore d k k ϕ1 da1 + ϕ2k da2k + ϕ3k da3k = d(L log(h1 · · · hd )). k=1

Thus, ϕ is exact, and C = h1 . . . hd is a Casimir of the Poisson pencil. This ends the proof. Acknowledgements I wish to thank Marco Pedroni for his suggestions and continuous support, as well as my former advisor, Franco Magri, for useful discussions on the topic.

146

ATTILIO MEUCCI

References 1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12. 13.

Arnold, V. I.: Mathematical Methods of Classical Mechanics, Grad. Texts in Math., Springer, New York, 1989. Falqui, G., Magri, F. and Pedroni, M.: Bihamiltonian geometry, Darboux coverings, and linearization of the KP hierarchy, Comm. Math. Phys. 197 (1998), 303–324. Gelfand, I. M. and Dickey, L. A.: Fractional powers of operators and Hamiltonian systems, Funct. Anal. Appl. 10 (1976), 259–273. Gelfand, I. M. and Zakharevich, I.: On the local geometry of a bi-Hamiltonian structure, In: L. Corvin et al. (eds), The Gelfand Mathematical Seminars 1990–1992, Birkhäuser, Boston, 1993, pp. 51–112. Kupershmidt, B. A.: Discrete Lax equations and differential-difference calculus, Asterisque 123 (1985), 212–245. Mackenzie, K.: Lie Groupoids and Lie Algebroids in Differential Geometry 124, Cambridge Univ. Press, Cambridge, 1987. Magri, F.: Eight lectures on integrable systems, In: Integrability of Nonlinear Systems, Lecture Notes in Phys. 495, Springer, New York, 1997, pp. 256–296. Magri, F.: The bihamiltonian route to Sato Grassmannian, Proc. CRM Conf. on Bispectral Problems, Montreal, CRM Proc. Lecture Notes 14, Amer. Math. Soc., Providence, 1998, pp. 203–209. Marsden, J. E. and Ratiu, T.: Reduction of Poisson manifolds, Lett. Math. Phys. 11 (1986), 161–169. Meucci, A.: The bi-Hamiltonian route to the discrete Sato Grassmannian, PhD thesis, Universitá Statale di Milano, 1999. Meucci, A.: Compatible lie algebroids and the periodic Toda lattice, To appear in J. Geom. Phys. Morosi, C. and Pizzocchero, L.: R-matrix theory, formal Casimirs and the periodic Toda lattice, J. Math. Phys. 37 (1996), 4484–4513. Sato, M. and Sato, Y.: Soliton equations as dynamical systems on infinite-dimensional Grassmannian manifolds, In: P. Lax and H. Fujita (eds), Nonlinear PDE’s in Applied Sciences (US/Japan Seminar, Tokio), North-Holland, Amsterdam, 1982, pp. 259–271.


147

Ergodicity for the Randomly Forced 2D Navier–Stokes Equations SERGEI KUKSIN1 and ARMEN SHIRIKYAN2

1 Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K. e-mail: [email protected] and Steklov Institute of Mathematics, 8 Gubkina St., 117966 Moscow, Russia. 2 Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K. e-mail: [email protected] and Institute of Mechanics of MSU, 1 Michurinskii Av., 119899 Moscow, Russia.

(Received: 18 April 2001) Abstract. We study space-periodic 2D Navier–Stokes equations perturbed by an unbounded random kick-force. It is assumed that Fourier coefficients of the kicks are independent random variables all of whose moments are bounded and that the distributions of the first N0 coefficients (where N0 is a sufficiently large integer) have positive densities against the Lebesgue measure. We treat the equation as a random dynamical system in the space of square integrable divergence-free vector fields. We prove that this dynamical system has a unique stationary measure and study its ergodic properties. Mathematics Subject Classifications (2000): 37H99, 35Q30. Key words: Navier–Stokes equations, kick-force, stationary measure, random dynamical system, Ruelle–Perron–Frobenius theorem.

0. Introduction We continue our study of the randomly forced 2D space-periodic Navier–Stokes system (NS), started in [KS1, KS2]. That is, we consider the equations u˙ − νu + (u, ∇)u + ∇p = ηω (t, x),

div u = 0,

(0.1)

where x ∈ T = R /Z , 0 < ν 1 is the viscosity, u = u(t, x) is the velocity field, and p = p(t, x) is the pressure. Equations (0.1) are supplemented by the conditions 2

2

u ≡ η ≡ 0,

2

div η = 0.

The brackets · signify the space averaging. The right-hand side ηω is a random process with range in the functional space H = {u ∈ L2 (T2 , R2 ) : div u = 0, u = 0}, and Equations (0.1) defines a random dynamical system in H . We provide H with the usual orthonormal basis {e1 , e2 , . . .} formed by the trigonometric vector fields

148

SERGEI KUKSIN AND ARMEN SHIRIKYAN

Figure 1. Evolution defined by (0.1), (0.2).

2 2 Cs −s sin(s · x) and Cs −s cos(s · x), s ∈ Z2 \ {0}. The ej ’s are eigenvectors of s1 s1 the Laplacian, −ej = αj ej . We assume that the eigenvalues αj are indexed in non-decreasing order. In [KS1], we consider the NS equations forced by a bounded random kick-force ηω =

δ(t − kT )ηk (x),

ηk =

∞

bj ξj k ej (x),

(0.2)

j =1

k∈Z

where bj 0 are some constants such that b2 := b12 + b22 + · · · < ∞, and {ξj k } are independent random variables. It is assumed in [KS1] that the distribution D(ξj k ) of the random variable ξj k is k-independent and has the form D(ξj k ) = pj (r) dr

for j 1, k ∈ Z,

(0.3)

where the pj ’s are Lipschitz continuous functions such that pj (0) > 0, supp pj ⊂ [−1, 1]. Let {St , t 0} be flow-maps of the free NS equation (0.1) with η ≡ 0. If u(t, x) is a solution for (0.1) with a kick-force (0.2) normalised to be a continuous from the right curve in H , then for any integer k and for t ∈ [T k, T (k + 1)] we have (see Figure 1) t < T (k + 1), St −T k (u(T k)), (0.4) u(t) = t = T (k + 1). ST (u(T k)) + ηk , Accordingly, long-time behaviour of solutions for (0.1), (0.2) is described by longtime behaviour of solutions for the following random dynamical system with discrete time: uk = S(uk−1 ) + ηk ,

(0.5)

where S = ST and uk = u(T k, ·) ∈ H . In [KS1], we show that if relations (0.3) hold with densities pj as above and bj = 0

for 1 j N0

(0.6)

ERGODICITY FOR THE RANDOMLY FORCED 2D NAVIER–STOKES EQUATIONS

149

for some finite N0 = N0 (ν, b) 1, then the random dynamical system (0.5) has in H a unique stationary measure λ. Moreover, if (uk , k 0) satisfies (0.5) for k > 0 and u0 = u, then λ as k → ∞

D(uk )

(0.7)

for any choice of the initial vector u ∈ H .! We note that if bj = 0 for all j 1 and bj2 < ∞, then these results apply to Equation (0.1) with any ν > 0, any T > 0 and with arbitrarily large kick-force η as above. See the introduction to [KS1] for discussion of this result and see [G, KS1] for its relations with statistical hydrodynamics. Next, E, Mattingly, and Sinai [EMS] and Bricmont, Kupiainen, and Lefevere [BKL] considered the 2D NS equations perturbed by a white noise force η=

∞

bj w˙ j (t)ej (x),

j =1

where w1 , w2 , . . . are independent standard Brownian motions. Under the assumption that bj = 0 for 1 j N0 (ν) and bj = 0 for j > N with some ∞ > N N0 (ν), they obtained results similar to those reviewed above. We do not discuss these results here, but we mention that, as it is shown in [BKL], for almost all initial functions u(0, x) the distribution of a solution converges to the stationary measure exponentially fast. In this work we study the NS equation with unbounded random kick-forces. That is, we consider Equations (0.1), (0.2), where the independent random variables ξj k have k-independent distributions as in (0.3), the densities pj are absolutely continuous and everywhere positive, ∞ ∂pj (r) ∂r dr < ∞ for all j 1; −∞ (0.8) pj (r) > 0 for all j 1, r ∈ R, and decay at infinity faster than any negative degree of r. We consider in fact the following two extreme cases which are allowed by our techniques: (A) (finite moments) the densities pj satisfy (0.8) and ∞ |r|m pj (r) dr Cm for all m 1, j 1,

(0.9)

−∞

with some fixed constants Cm , m 1. ! In [KS1], we study in fact the system (0.3) restricted to the domain of attainability from zero A,

which is a compact subset of H , invariant for (0.3), and prove that the restricted system has a unique stationary measure and satisfies (0.7) for u ∈ A. In the short paper [KS2] we show that this measure is a unique stationary measure for the system in the whole space H and prove that (0.7) holds for any u ∈ H.

150


(B) (finite second exponential moments) the densities pj satisfy (0.8) and ∞ 2 e%0 r pj (r) dr C0 for any j 1, −∞

with some fixed positive constants %0 and C0 ;

We stress that in (A) and (B) it is not assumed that rpj (r) dr = 0. For any s > 0, we denote H s = H ∩ H s (T2 ; R2 ), where H s (T2 ; R2 ) is the Sobolev space of order s with the corresponding norm · s . MAIN THEOREM. Let us assume that condition (A) is satisfied and ∞

bj2 αjs < ∞

for some s > 0.

j =1

Then there is an integer N0 < ∞ with the following property: if (0.6) holds, then the random dynamical system (0.5) has a unique stationary measure λ such that |u|m λ(du) < ∞ for all m 1. H

Moreover, the following assertions hold: (a) λ(H s ) = 1; (b) if (uk , k 0) is a solution of Equation (0.5) with a deterministic initial function u0 = u, then, for λ-almost all u, convergence (0.7) holds. Moreover, Ef (uk ) → (λ, f )

as k → ∞,

(0.10) p

where f is any continuous function on H s such that |f (u)| C1 + C2 us for some finite constants C1 , C2 , and p. (c) if bj = 0 for all j , then supp λ = H , and convergence (0.10) holds uniformly in u ∈ H s , us R, for any R > 0. 2 Finally, if condition (B) is also satisfied, then H eβ|u| λ(du) < ∞ for some β > 0, and convergence (0.10) holds for λ-almost all u ∈ H and any function f ∈ C(H s ) such that |f (u)| C exp(σ uκs ), where the positive constants σ and κ are sufficiently small. If s > 1, then the delta-function is a continuous functional on the space H s . Accordingly, if s > 1 and u(t, x) is a solution for (0.1) such that u(0, x) = u0 ∈ H , then, for λ-almost all u0 ∈ H , the correlation tensor of the solution j Eui (k, x)u i(k, y)j converges as k → ∞ to the correlation tensor of the measure λ, equal to u (x)u (y)λ(du). If u0 is an arbitrary vector in H , then in this statement the convergence should be replaced by the Cesàro convergence. The proof of the Main Theorem remains true if condition (A) is replaced by the following weaker assumption with M 20:


151

(AM ) the densities pj satisfy (0.8), and (0.9) holds for m M and all j 1. In this case, the stationary measure λ has M M finite moments, where M goes to infinity with M, and (0.10) holds for any continuous functional f : H → R satisfying the inequality |f (u)| C1 + C2 uM s . The proof of the Main Theorem, which occupies Sections 1–5, follows the scheme developed in [KS1] to work with bounded kick-forces. It is based on a Foia¸s–Prodi type reduction of (0.5) to a finite-dimensional abstract Gibbs system which has a unique stationary solution due to a version of the Ruelle–Perron– Frobenius theorem. In fact, the Main Theorem can be strengthened as follows: AMPLIFICATION. Under the assumption of the above theorem, convergence (0.10) holds for any u ∈ H , uniformly on bounded subsets of H . This result can be derived from the Main Theorem (and some intermediate assertions), using the methods of [KS2]. Since the corresponding arguments differ from those used in this work, we shall present them in another publication. NOTATION

We denote by Z be the set of all integers and by Z0 be the set of non-positive integers. Let X be a topological space. We shall use the following notation. [B]X is the closure in the space X of its subset B. BX (x, r) is a closed ball in X of radius r centred at x ∈ X. B(X) is the σ -algebra of Borel subsets of X. P (X) is the set of probability measures on (X, B(X)). C(X) is the space of real-valued continuous functions on X. Cb (X) is the space of bounded functions f ∈ C(X). It is endowed with the supremum-norm f ∞ . L1 (X, µ) is the space of Borel functions on X with finite norm f µ := f (x) dµ(x). X

The integral of a function f (x) over the space X with respect to a measure µ will sometimes be denoted by (µ, f ): f (x) dµ(x) = f dµ. (µ, f ) = X

X

D(ξ ) is the distribution of a random variable ξ . a ∨ b (a ∧ b) is the maximum (minimum) of real numbers a and b. We denote by Ci , i = 1, 2, . . . , unessential positive constants.

152


1. Preliminaries: Equations, Estimates and the Markov Chain 1.1. DESCRIPTION OF THE CLASS OF PROBLEMS IN QUESTION Let us consider the Navier–Stokes (NS) system (0.1). Applying the L2 -orthogonal projection 1 onto the space H of divergence-free vector fields with zero mean value (see the Introduction), we can write this system as u˙ + νLu + B(u, u) = η(t, x),

x ∈ T2 , 0 < ν 1

(1.1)

(for instance, see [CF]). Here u(t) is a two-dimensional vector field with values in the functional space H . The operators L and B have the form Lu = −u,

B(u, v) = 1(u, ∇)v.

It is assumed that the right-hand side of (1.1) is a kick-force as in the Introduction. To simplify notations, we assume that T = 1. Then η takes the form η(t, x) =

+∞

δ(t − k)ηk (x),

(1.2)

k=−∞

where δ(·) is the Dirac measure and ηk , k ∈ Z, is a sequence of i.i.d. random variables with range in H . We note that if g(t): R → H is a continuous function with compact support, then +∞ +∞ η(t), g(t) dt = ηk , g(k), −∞

k=−∞

where ·, · denotes the scalar product in H . We now turn to a description of the sequence {ηk }. Let α1 α2 · · · be eigenvalues of the positive self-adjoint operator L acting in H and let ej (x), j 1, be the corresponding eigenfunctions as in the Introduction. We shall assume that the random vector ηk has the form ηk (x) =

∞

bj ξj k ej (x),

(1.3)

j =1

where {ξj k } is a family of independent scalar random variables satisfying condition (A) (see the Introduction), and {bj } is a sequence of real numbers such that ∞

bj2 αjs < ∞,

s 0.

(1.4)

j =1

In what follows, we always assume that inequality (1.4) and condition (A) are satisfied. In particular, it follows that Eηk m s 0 such that abj2 αjs %0

for all j 1.

To simplify notation, we shall write |u| and u instead of u0 and u1 , respectively. In what follows, the constants bj are assumed to be fixed, and we shall not specify dependence of different parameters on them. We now define the notion of a solution for Equation (1.1). For any s 0 we s s 2 2 with the norm · s . We note introduce the space √ H = H ∩ H (T , R ) endowed that the operator L defines an isomorphism H s → H s−1, s 1. Let I ⊂ R be an open interval (which can be of infinite length). DEFINITION 1.1. A mapping u(t): I → H is called a regular curve if it belongs to L1loc (I, H 1 ) and is continuous at non-integer points of I while at integer points it is continuous from the right and has a limit from the left. For a Banach space X, let C01 (I, X) be the set of continuously differentiable functions f (t): I → X with compact support. DEFINITION 1.2. A regular curve u(t): I → H is called a solution of Equation (1.1) with a deterministic force of the form (1.2) if the left- and right-hand sides of (1.1) coincide as linear functionals on the space C01 (I, H 1 ). That is, √ √ −u, v ˙ + ν Lu, Lv + B(u, u), v dt I ηk , v(k) (1.6) = η, v dt = I

k∈Z∩I

for any v ∈ C01 (I, H 1 ). A random process u = uω (t), t ∈ I , with range in H is called a solution of Equation (1.1) with a random force of the form (1.2), (1.3) if for almost all ω the mapping uω (t): I → H is a regular curve satisfying (1.1). We note that if u(t, x) is a solution of Equation (1.1), then, due to (1.6), we have u(k, x) − u(k − 0, x) = ηk (x)

for any integer k ∈ I,

(1.7)

while on any interval not containing integer points the function u(t, x) satisfies the free Navier–Stokes equations u˙ + νLu + B(u, u) = 0,

u(t) ∈ H.

(1.8)

154


In particular, (0.4) holds with T = 1. 1.2. CAUCHY PROBLEM AND A PRIORI ESTIMATES We now consider the Cauchy problem for Equation (1.1): u(0, x) = u0 (x),

(1.9)

where u0 (x) is a random variable in H . We shall assume that it is independent of η1 , η2 , . . . and that all of its moments are finite: E|u0 |m < ∞ for any m 1.

(1.10)

We have the following theorem on the correctness of the Cauchy problem: THEOREM 1.3. Assume that (1.10) is satisfied. Then the problem (1.1), (1.9) has a unique solution defined for t 0. Moreover, for any m 1 we have the estimate E|u(k)|m q k E|u0 |m + C(m)ν −(m−1) dν (k)E|ηk |m ,

k 1,

(1.11)

where 0 < ν 1, q = e−να1 , C(m) > 0 is a constant not depending on u0 , k, and ν, and dν (k) = 1 + q + · · · + q k−1 α1−1 eα1 ν −1 . Finally, if (1.4) holds for some s > 0 and l = l(s) 1 is the smallest integer no less than s, then there is a constant C(l, m) > 0 such that −m/2 ν E|uk−1 |m + E ηk m l = 1, m s , (1.12) Eu(k)s C(l, m) −5lm/2 ml m E |uk−1 | + E ηk s , l 2, 1+ν where k 1 and ml = m(2l + 1). In case the random variables u0 and ηk have finite second exponential moments, stronger estimates for the solutions hold: THEOREM 1.4. Suppose that the random variables ξj k satisfy condition (B) and there is ρ > 0 such that (1.13) E exp ρν|u0 |2 < ∞. Then the solution of the problem (1.1), (1.9) constructed in Theorem 1.3 satisfies the inequality q k , k 1, (1.14) E exp σ0 ν|u(k)|2 d(k) E exp σ0 ν|u0 |2 where 0 < ν 1, q = e−α1 ν , σ0 = ρ ∧ (aα1 e−α1 ), and 1+q+···+q k−1 1 E exp(a|ηk |2 ) 1−q . d(k) = E exp(a|ηk |2 )


155

Moreover, if (1.4) holds for some s > 0 and l is the smallest integer no less than s, then there are positive constants Cl and σl , depending only on σ0 and l, such that l E exp σl ν pl u(k)2κ s (1.15) Cl E exp aηk 2s E exp σ0 ν|u(k − 1)|2 , k 1. Here κ1 = 1, p1 = 2, and κl =

1 , 2l + 1

pl =

7l + 1 2l + 1

for l 2.

The proof of Theorems 1.3 and 1.4 is carried out by standard methods and is given in the Appendix (see Section 6). We shall also need some estimates for the rate of growth and for the mean value of solutions (and of the right-hand side of the equation). For any sequence of non-negative numbers ak and arbitrary integers m n, we set ak nm

n 1 = ak . n − m + 1 k=m

In the case m > n, we set ak nm = ak m n. PROPOSITION 1.5. Let k− k0 k+ be some integers, where k+ (k− ) can take the value +∞ (−∞), and let u(t, x) be a solution of (1.1) that is defined for k− t k+ and satisfies the inequality sup k− kk+

E |u(k)|m Nm ν −m

for 0 < ν 1, m 1,

(1.16)

where the constants Nm > 0 do not depend on ν. Then there is a constant M > 1, not depending on Nm and ν, and a non-negative random variable Tν (ω) ∈ Z such that T |u(k)|2 + ηk 2s k Mν −2 for k− T k+ , |T − k0 | Tν (ω). (1.17) 0

Moreover, for any m > 1 there is a constant Cm > 0 such that E Tνm Cm N2m + E|η1 |4(m+2) ν −m for 0 < ν 1.

(1.18)

In this proposition and everywhere below, we assume that k < k+ if k+ = +∞ and k > k− if k− = −∞. Remark 1.6. If in Proposition 1.5 we assume that condition (B) is also satisfied and replace inequality (1.16) by the stronger estimate sup

k− kk+

2

E eσ0 ν|u(k)| N0

for 0 < ν 1,

(1.19)

156


then (1.17) holds with a constant M > 1 (depending on σ0 solely) and an integervalued non-negative random variable Tν (ω) ∈ Z such that E eσ Tν C0

for 0 < ν 1,

where the positive constants C0 and σ depend only on N0 and σ0 , respectively. Proof of this assertion follows the same scheme as that of Proposition 1.5, and we shall not dwell on it. We also note that, due to Theorems 1.3 and 1.4, Proposition 1.5 and its modification above apply to any solution of the problem (1.1), (1.9), where the random variable u0 satisfies condition (1.10) or (1.13). Proof of Proposition 1.5. To simplify notation, we confine ourselves to the case when k0 = k− = 0 and k+ = +∞. Moreover, we shall only show that T |u(k)|2 0 Mν −2 for T Tν (ω). It will be clear from the proof that the same arguments apply in the general case. (1) We first note that |u(T )|2 + 2ν

T

u(t)2 dt

k−1

k=1

= |u(0)|2 +

k

T

|ηk |2 + 2ηk , u(k − 0) ,

(1.20)

k=1

where T 1 is an arbitrary integer. Indeed, since on any open interval (k − 1, k) the solution u(t, x) satisfies the free NS equations (1.8), we have (see (6.3) with l = 0) k 2 2 u(t)2 dt = 0. (1.21) |u(k − 0)| − |u(k − 1)| + 2ν k−1

Besides, relation (1.7) implies that |u(k)|2 = |u(k − 0)|2 + |ηk |2 + 2ηk , u(k − 0).

(1.22)

Combining (1.21) and (1.22), we derive k u(t)2 dt = |u(k − 1)|2 − |u(k)|2 + |ηk |2 + 2ηk , u(k − 0). 2ν k−1

Taking the sum over k = 1, . . . , T , we obtain (1.20). (2) We now recall that (see [CF]) |St (v)| e−να1 t |v|,

t 0,

(1.23)


157

where α1 > 0 is the first eigenvalue of L. It follows from (1.23) that 2|ηk , u(k − 0)| (να1 )−1 |ηk |2 + να1 |u(k − 0)|2 k −1 2 (να1 ) |ηk | + ν u(t)2 dt. k−1

Substitution of this inequality into (1.20) results in |u(T )| + ν 2

T k=1

k k−1

u(t)2 dt |u(0)|2 + (1 + α1−1 ν −1 )

T

|ηk |2 . (1.24)

k=1

Now note that, by (1.21) and (1.23), we have k 1 − e−2α1 ν |u(k − 1)|2 . u(t)2 dt ν 2 k−1 Combining this with (1.24), we derive T

|u(k)|2 cν −2 ν|u(0)|2 + T E |η1 |2 + 0 is a constant and },

u0 ∈ H, > ∈ B(H ).

Denote by Pk : Cb (H ) → Cb (H ),

Pk∗ : P (H ) → P (H )

the Markov operators corresponding to P (k, u0 , >).! It follows from Theorem 1.3 that if condition (1.4) is satisfied for some s 0, then P1∗ µ(H s ) = 1 for any µ ∈ P (H ). In particular, when µ is the delta-measure concentrated at u0 , we obtain P (k, u0 , H s ) = 1 for any k 1.

(1.29)

In what follows, we shall need some properties of the operators Pk and Pk∗ . The following two lemmas show that Pk can be extended to a broader class of functionals. LEMMA 1.7. Suppose that condition (1.4) holds for some s > 0. Then Pk can be extended to a continuous operator from Cb (H s ) to Cb (H ) whose norm is equal to 1. Proof. It suffices to consider the case k = 1. Let f ∈ Cb (H s ). In view of Theorem 1.3, for any initial function u0 ∈ H the solution u1 = u(1, x) belongs to H s with probability 1, so that the random variable f (u1 ) is well-defined. Moreover, since the operator S: H → H s is continuous (see Lemma 6.1), we conclude from (1.28) that u1 continuously depends (in H s -norm) on u0 ∈ H for all ω. Therefore the function f (u1 ) is also continuous. The continuity of the function ! Since the map S: H → H is continuous, for any f ∈ C (H ) the function P f (u) = b k P (k, u, dv)f (v) is continuous in u. Hence, P maps the space C (H ) into itself. k b H


159

P1 f (u0 ) = Ef (u1 ) follows now from the Lebesgue theorem on dominated convergence. It remains to note that if |f (u)| 1 for all u ∈ H , then |Ef (u1 )| 1, that is, the norm of the operator P1 : Cb (H s ) → Cb (H ) does not exceed 1. We now show that the operators Pk can be continued to a class of functionals growing at infinity. For any increasing positive function β(r), r 0, we denote by C(H s ; β) the space of continuous functions f (u): H s → R such that |f (u)| const β us , u ∈ H s . It is clear that C(H s ; β) is a Banach space with respect to the norm

f s,β := sup |f (u)|/β us . u∈H s

We recall that the integer l = l(s) 1 and the constants ml (l 2), κl , σl , and pl are defined in Theorems 1.3 and 1.4, and set ml = m for l = 0, 1. LEMMA 1.8. Under the conditions of Theorem 1.3, for any m > 1 and m , 1 m < m, the operator Pk can be extended to a continuous map from C(H s , βm ) to C(H ; βml ), where βd (r) = 1 + r d . Moreover, for any ν, 0 < ν 1, the norms of the extended operators are bounded uniformly in k 1. Remark 1.9. Under the assumptions of Theorem 1.4, the operator Pk extends to a bounded map from C(H s ; γ ) to C(H ; γ ). Here γ (r) = exp(cr 2κl ) and γ (r) = exp(c r 2 ), where κl is defined in Theorem 1.4 and c and c are some positive constants that can be easily recovered from Theorem 1.4. Proof. The proofs of all assertions are similar, and to simplify notation, we confine ourselves to the case s = 0. Let f ∈ C(H ; βm ) and let hR (r) be a continuous function equal to 1 and 0 for r R and r R + 1, respectively. Obviously, the function fR (u) = hR (|u|)f (u) belongs to Cb (H ). It follows from Lemma 1.7 and inequality (1.11) that for any R2 > R1 1 we have Pk fR (u) − Pk fR (u) (|v|) − h (|v|) f (v)µ (dv) h R R k 1 2 2 1 H m (1.30) 1 + |v| µk (dv) const R1 |v|R2 +1

const (1 + R1 )m −m ,

(1.31)

where µk = P (k, u, ·). We note that inequality (1.31) holds uniformly in u from bounded subsets of H . Letting R1 to go to infinity, we conclude that there is a limit lim Pk fR (u) =: Pk f (u),

R→∞

160


and the limiting function Pk f is continuous in u ∈ H . Moreover, it follows from (1.11) that m Pk f (u) f (v)µk (dv) f 0,β 1 + |v| µk (dv) m H H m const ν −m f 0,βm 1 + |u| . This completes the proof in the case s = 0.

✷

We now turn to the problem of existence of a stationary measure. DEFINITION 1.10. A probability measure λ ∈ P (H ) is said to be stationary for Equation (1.1) if P1∗ λ = λ. We recall that the support supp µ of a measure µ is defined as the minimal closed set of full measure and that D(ξ ) denotes the distribution of a random variable ξ . PROPOSITION 1.11. Suppose that condition (1.4) is satisfied for some s > 0. Then there is a stationary measure λ ∈ P (H ) such that λ(H s ) = 1 and  −m for r = 0, ν −3m/2 for 0 < r 1, ν (1.32) um λ(du) C(l, m) r  −(5l+2)m/2 H for 1 < r s, ν where m 1, 0 < ν 1, l = l(r) is the smallest integer no less than r, and C(l, m) is a constant not depending on ν. Moreover, there is a stationary Markov chain (uk , k ∈ Z) satisfying (1.28) for all k ∈ Z such that D(uk ) = λ. Finally, if all the constants bj in (1.3) are non-zero and λ0 ∈ P (H ) is an arbitrary stationary measure for Pk∗ , then supp λ0 = H . Proof. The existence of a stationary measure and inequality (1.32) can easily be proved by the Bogolyubov–Krylov argument using Theorem 1.3 and the Prokhorov theorem on the weak compactness of a tight family of measures (cf. [DZ]). The fact that λ(H s ) = 1 follows immediately from (1.29) and the Chapman–Kolmogorov relation P (1, u, >)λ(du), u ∈ H, > ∈ B(H ). (1.33) λ(>) = H

The existence of a stationary solution of (1.28) with distribution λ follows from the Prokhorov and Skorokhod theorems. (For the proof of this assertion in the case when the support of the distribution of ηk is a bounded subset in H , see [KS1, Section 1.2].) To prove the last assertion of the theorem, we note that if γ is the distribution of the random variable ηk defined by the formula (1.3) in which all bj are non-zero, then γ (U ) > 0 for any open set U ⊂ H (see Lemma 6.2 in the Appendix). It


161

follows that P (1, u, U ) > 0. Setting > = U and λ = λ0 in (1.33), we conclude that λ(U ) > 0 for any open set U . ✷ Combining Propositions 1.5 and 1.11, we obtain the following assertion. PROPOSITION 1.12. Suppose that (1.4) holds for some s > 0. Let λ0 ∈ P (H ) be a stationary measure for Pk∗ that satisfies the condition |u|m λ0 (du) Nm ν −m for m 1, 0 < ν 1, (1.34) H

where Nm > 0 do not depend on ν, and let (uk , k ∈ Z) be a stationary solution of (1.28) such that D(uk ) = λ0 . Then there is a constant M 1 and for any k0 ∈ Z there exists an integer-valued nonnegative random variable Tν (ω) satisfying (1.18) such that (1.17) holds for |T − k0 | Tν . Remark 1.13. Analogues of Propositions 1.11 and 1.12 are true in the case when condition (B) holds. In this situation, the stationary measure λ satisfies the inequalities l exp σ ν pl u2κ λ(du) Cr , 0 < ν 1, s Hr

where 0 r s, l = l(r) is the smallest integer no less than r, κ0 = p0 = 1, the constants pl and κl with l 1 are defined in Theorem 1.4, and Cr > 0 is a constant not depending on ν. Moreover, the random variable Tν (ω) has a finite exponential moment.

2. Lyapunov–Schmidt Reduction In this section we prove a result of the Foia¸s–Prodi type and show that if a Markov chain {uk } is a stationary solution of Equation (1.28), then sufficiently high Fourier modes of uk are uniquely defined by low modes of the sequence (ul , l k). This will enable us to reduce the problem of uniqueness of a stationary solution for (1.28) to a similar question for a Gibbs system with a finite-dimensional phase space. 2.1. FORMULATION OF THE RESULT To simplify notations, from now on we assume that ν = 1. Let us define H s as the closure in H s of the linear manifold spanned by those vectors ej whose coefficients bj in expansion (1.3) are nonzero. It is clear that if all the coefficients bj are nonzero, then H s = H s . For any integer N 2, let HN be the subspace

162


in H spanned by the vectors ej , j = 1, . . . , N − 1, and let HN⊥ be its orthogonal complement. We set HNs = H s ∩ HN ,

HNs⊥ = H s ∩ HN⊥

and note that −1/2

|w| αN

w for any w ∈ HN⊥ ,

(2.1)

where αj , j 1, are the eigenvalues of L indexed in increasing order. We denote by PN and QN the orthogonal projections onto HN and HN⊥ , respectively. Finally, we set Hs = H × H s ,

HsN = HN × HNs⊥ ,

and for any s 0 and any integer N 1 we define the projections u PN u s s . !→ 1N : H → HN , QN η η We shall also use the corresponding projections in the spaces of sequences: s Z0 s Z0 u PN u N : H , → HN , !→ QN η η where PN u = (PN ul , l 0) and QN η = (QN ηl , l 0). In the case N = ∞, we set s Z0 u u s Z0 !→ u, ∞ : H →H , !→ u. 1∞ : H → H, η η Applying QN to (1.28), we obtain wk = QN S(vk−1 + wk−1 ) + ψk ,

(2.2)

where vk = PN uk ,

wk = QN uk ,

ψk = QN ηk .

We wish to show that, for a sufficiently large class of sequences (vl , l 0) and (ψl , l 0), Equation (2.2) with k 0 has a unique solution (wl , l 0), and the dependence of the zeroth component w0 on vl and ψl decays exponentially as a function of l. To formulate the corresponding results, we have to introduce some notations. For a sequence u = (ul , l k) with ul ∈ H and integers m n k we set

|u|2

n m

n ≡ |ul |2 m =

1 |ul |2 . n − m + 1 l=m n


163

In what follows, we shall need the following two-sided estimate for |u|2 nm : 2unm |u|2 nm cunm

(2.3)

where c = 2(1 − e−α1 )−1 and 1 = n − m + 1 l=m n

unm

St (ul )2 dt.

1 0

To prove (2.3), we note that if u(t) is a solution of the homogeneous NS system (1.8), then t 2 u(θ)2 dθ = |u(0)|2 , t 0. (2.4) |u(t)| + 2 0

This estimate immediately implies the left-hand inequality in (2.3). Combining (1.23) with ν = 1 and (2.4), we derive t 1 u(θ)2 dθ (1 − e−2α1 t )|u(0)|2 , 2 0 whence follows the right-hand estimate in (2.3). For any K > 0 and any integer R 0, we denote by Fs (K, R) the set of sequences! u uk (2.5) = , k 0 , uk ∈ H, ηk ∈ H s , ηk η such that Equation (1.28) is satisfied for k 0, and the following inequality holds: 0 (2.6) |uk |2 + ηk 2s T K, T ∈ Z, T −R. It is clear that (2.6) is equivalent to the inequality 0 (|T | + 1) |uk |2 + ηk 2s T K(|T | ∨ R + 1),

T 0,

(2.7)

which implies, in particular, that |uk |2 + ηk 2 K(R ∨ |k| + 1),

k 0.

(2.8)

We also introduce the space Fs (K) of sequences (2.5) that satisfy the inequality 0 lim sup |uk |2 + ηk 2s T K. T →−∞

It is clear that Fs (K, R) ⊂ Fs (K) for any integer R 0. The sets Fs (K) and Fs (K, R) are subsets of the linear space H = (H0 )Z0 . We endow H with the ! The choice of the space Fs (K, R) is implied by the fact that if {u } is a stationary solution k for (1.28) all of whose moments are finite, then with probability 1 the sequence (ul , ηl , l 0) belongs to Fs (M, R) for an integer R 0, where M > 0 is the constant in Proposition 1.12.

164


uk

Tikhonov topology. That is, a sequence ηk converges to uη if ukl → vl in H and ηlk → ηl in H for each l 0. This topology metrisable; for instance, one can use the distance 1 2 0 1 u u |ul − u2l | + |ηl1 − ηl2 | ∧ 2l . , = dist 1 2 η η l=−∞ The sets Fs (K) and Fs (K, R) are provided with the topology of H. We stress that the topology in the spaces Fs (K) and Fs (K, R) is defined in terms of the L2 -norm | · |, rather than the H s -norm · s . In the theorem below, we have compiled some properties of the spaces Fs (K) and Fs (K, R). We abbreviate F0 (K) = F(K) and F0 (K, R) = F(K, R). THEOREM 2.1. (i) Let s > 0 and let M > 0 be the constant defined in Proposition 1.5. Then for any K M the space Fs (K) is nonempty. Moreover, for any integer R 0, F(K, R) is closed in H and Fs (K, R) is compact in H. (ii) There is a constant C∗ > 0 such that if N ∈ [N0 , ∞], where N0 = N0 (K) 1 is the smallest integer satisfying the condition log αN0 > C∗ K, then the restriction of the projection N to F(K) is injective. Moreover, for any integer l 0 the operator Wl : FN (K) ≡ N F(K) → HN⊥ taking each ϒ = ψv = N uη to wl = QN ul satisfies the inequality Wl (ϒ 1 ) − Wl (ϒ 2 )

|ψl1

−

ψl2 |

+

l−1

−1/2 l−k

CαN

×

k=−∞

l−1 l−1 1

|ϒk − ϒk2 |. × exp C(l − k) |u1 |2 k + |u2 |2 k Here ϒ i =

vki ψki

(2.9)

∈ FN (K), i = 1, 2, C > 0 is a constant not depending on K, N,

and ϒ i , and |ϒk1 − ϒk2 | = |vk1 − vk2 | + |ψk1 − ψk2 |. Theorem 2.1 will be proved in Subsection 2.3. In the next subsection, we use this result to establish equivalence of two families of Markov chains related to a stationary measure for the original equation.


165

2.2. THEOREM ON ISOMORPHISM In what follows, we assume that condition (1.4) is satisfied for some s > 0. According to assertion (i) of Theorem 2.1, in this case Fs (K, R) is a compact subset of H for any R 0 and K M. We denote by B s (K) the Borel σ -algebra on the topological space Fs (K) and by P s (K) the set of all probability measures on (Fs (K), B s (K)). In the case s = 0 we shall simply write B(K) and P (K). We recall that to Equation (1.28) there corresponds an RDS and a family of Markov chains {θ k } in H given by the formulas θ 0 = u, θ k = S(θ k−1 ) + ηk ,

(2.10) (2.11)

where k 1. Let us fix arbitrary stationary measure λ0 ∈ P (H ) for (2.10), (2.11) with finite moments (see (1.34)) and denote by M > 0 the constant in Proposition 1.12. Along with {θ k }, let us consider another family of Markov chains in Fs (K), K M, defined by the rule u 0 , (2.12) = η ηk , (2.13) k = k−1 , S(k−1 ) + ηk where k 1 and Z S: Hs ≡ Hs 0 → Hs ,

S(U ) =

S(u0 ) for U ∈ Hs . 0

It is easy to see that (2.13) defines an RDS in Fs (K) in the sense that if k ∈ Fs (K), K M, then k+1 ∈ Fs (K) for all ω ∈ F. Accordingly, Equations (2.12) and (2.13) define a family of Markov chains in Fs (K). Moreover, if (uk , k ∈ Z) is a stationary solution for (2.11) uk such that D(uk ) = λ0s(see Propositions 1.5 and 1.11), then the random vector ( ηk , k 0) belongs to F (K) with probability 1, and its distribution 0 is a stationary measure for (2.12), (2.13). We now consider the image of {k } under the projection N . Here and everywhere below, we assume that N0 N ∞,

log αN0 > C∗ K,

(2.14)

where C∗ > 0 is the constant in Theorem 2.1. We shall see that all these projections are equivalent to the original chain {k }. For any integers K M and N N0 , we set FsN (K, R) = N Fs (K, R),

FsN (K) = N Fs (K).

Thus, for N < ∞ the set FsN (K, R) consists of those sequences ϒ = ψv for u which there is η ∈ Fs (K, R) such that v = PN u and ψ = QN η. By assertion (ii)

166


u

of Theorem 2.1, the pair η is uniquely determined. Similarly, Fs∞ (K, R) consists u = (uk , k 0) that are the first component of an element ofuthe sequences s (K, R), which is also unique since ηk = uk − S(uk−1 ). The spaces FsN (K) ∈ F η s and F∞ (K) can be described in a similar way. In what follows, we assume that FsN (K) is endowed with the Tikhonov topology of the space HN = (HN )Z0 . We confine ourselves to the case K = 2M (although the arguments below remain valid for all K M). To simplify notations, we shall write Fs and FsN instead of Fs (2M) and FsN (2M), respectively. Since N : Fs → FsN is a one-to-one continuous mapping, we can define its inverse −1 N . We claim that for any integer N N0 = N0 (2M) the mapping N : (Fs , B(Fs )) → (FsN , B(FsN )) is an isomorphism of measurable spaces. Indeed, the fact that N is measurable s s (that is, −1 N (>) ∈ B(F ) for any > ∈ B(FN )) follows from the continuity of N . Therefore, it suffices to show that N (>) ∈ B(FsN ) for any > ∈ B(Fs ). To this end, we first note that Fs =

∞

Fs (K, R).

K>2M R=1

It follows that Fs is a Borel subset of H and, hence, the Borel σ -algebra B(Fs ) coincides with the collection of subsets > ⊂ Fs for which there is a Borel set > ∈ B(H) such that > = > ∩ Fs . We now fix an arbitrary > ∈ B(Fs ). Since the restriction of N to the compact set Fs (K, R) is continuous together with its inverse, the set N (>) = N (>) ∩ FsN belongs to B(FsN ). What has been proved implies, in particular, that the composition mapping s s G = ∞ ◦ −1 N : FN → F∞

defines an isomorphism of measurable spaces with the inverse s s H = N ◦ −1 ∞ : F∞ → FN .

We also note that G(ϒ) = (ul , l 0), ul = vl + W0 (ϒ), vl , l 0 , vl = PN ul , ψl = QN ul − S(ul−1 ) , H(u) = ψl

(2.15) (2.16)

where the operator W0 is defined in Theorem 2.1. We now describe the families of Markov chains resulting from application of N to {k }. It is a matter of direct verification to show that for N = ∞ we obtain θ 0 = u, θ k = θ k−1 , S(θ0k−1 ) + ηk ,

(2.17) (2.18)


167

where k 1 and θ k = (θlk , l 0), and for N0 N < ∞ we have v 0 ϒ = , ψ ϕk k k−1 k−1 , ϒ = ϒ , T (ϒ ) + ψk

(2.19)

where ϕk = PN ηk , ψk = QN ηk , and PN S(v0 + W0 (v, ψ)) v . T = 0 ψ

(2.20)

(2.21)

We shall treat (2.18) and (2.20) as either random dynamical systems or Markov chains in the corresponding phase spaces. Note that the mapping G conjugates the two dynamical systems: if θ k = G(ϒ k ), then θ k+1 = G(ϒ k+1 ). Let us denote by P(k, u, >) and P(k, ϒ, >) the transition probabilities for the families {θ k } and {ϒ k }, respectively, and by Pk and Pk the Markov semigroups associated with them. The above construction implies that P(k, G(u), G(>)) = P(k, ϒ, >),

ϒ ∈ FsN ,

> ∈ B(FsN ),

and, hence, (Pk f ) ◦ G = Pk (f ◦ G), We now set k u k , = ηk

f ∈ Cb (FsN ).

uk = (ul , l k), ηk = (ηl , l k),

where (uk , k ∈ Z) is a stationary solution such that D(uk ) = λ0 . It is clear that {k } is a stationary Markov chain in Fs satisfying (2.13) for all k ∈ Z. Let us consider its image under the projections N and ∞ : vl k k , l k , uk = ∞ k = (ul , l k), ϒ = N = ψl where vl = PN ul and ψl = QN ηl . What has been said implies that if N satisfies (2.14), then ϒ k and uk are stationary Markov chains in FsN and Fs∞ that satisfy (2.20) and (2.18), respectively. Moreover, the distribution of each of the sequences ϒ k and uk uniquely determines the distribution of k . Thus, we obtain a one-to-one correspondence between some classes of stationary measures for (2.11), (2.18), and (2.20). More exactly, we have the following theorem. THEOREM 2.2. Let λ0 ∈ P (H ) be a stationary measure for (2.11) satisfying (1.34) and let (uk , k ∈ Z) be a stationary solution of (2.11) with distribution λ0 . Then the distribution µ of the corresponding stationary Markov chain ϒ k in FsN is

168


uniquely defined. Moreover, the measure µ uniquely determines λ0 . In particular, if Equation (2.20) has at most one stationary measure concentrated on the set ∞

FsN (2M, R) ⊂ FsN (2M) ≡ FsN ,

R=1

then Equation (2.11) has a unique stationary measure that satisfies (1.34). Remark 2.3. If {ϒ k = (ϒlk , l 0), k ∈ Z} is a stationary solution for (2.20), then {ϒ0k , k ∈ Z} is a stationary process. Its distribution in the space of sequences {ϒl , l ∈ Z} is an (abstract) Gibbs measure in the sense of Ruelle, Sinai and Bowen, see discussion in [KS1]. Therefore, uniqueness of a stationary solution for (2.20) which we prove in Section 4 below implies (is in fact equivalent to) uniqueness of the corresponding 1D Gibbs system.

2.3. PROOF OF THEOREM 2.1 (i) Since s > 0, Proposition 1.11 implies that there is a stationary solution (uk , k ∈ Z) of (1.28) whose distribution satisfies inequality (1.19) with ν = 1 and k± = ±∞. By Proposition 1.5, almost every realisation of the random variable uηkk , k 0 belongs to Fs (M), and therefore Fs (K) = ∅ for K M. The proofs of the assertions on compactness and closedness are similar, and we confine ourselves to proving that Fs (K, R) is compact in the space H with i Tikhonov topology. Let uηi ∈ Fs (K, R) be an arbitrary sequence. The definition of Fs (K, R) implies that for any l 0 the sequence ηli is bounded in H s . Furthermore, it follows from Equation (1.28) and the continuity of the map S from H to H s that the sequence uil is contained in a bounded subset of H s . Therefore, there i i are subsequences of u{ul } and {ηl } that converge in H . Its is clear that the limiting pair of sequences η satisfies (1.28) and belongs to F (K, R). This implies the required assertion. (ii) The case N = ∞ is trivial, and therefore we shall assume that N < ∞. We shall need the following lemma whose proof is given in the Appendix (see Section 6.4). LEMMA 2.4. There is a constant C > 0 such that the resolving semigroup of the free NS system (1.8) satisfies the inequalities t Sθ (u0 )2 dθ , St (u0 ) − St (u0 ) u0 − u0 exp C (2.22) 1 2 1 2 1 0 St (u0 ) − St (u0 ) C (t −3/2 ∨ 1)u0 − u0 × 1 2 1 2 t 0 2 0 2 Sθ (u1 ) + Sθ (u2 ) dθ , (2.23) × exp C 0

where t 0 and u01 , u02 ∈ H .


Let

vi ϒ = ψi i

i u = N i ∈ FN (K), η

169

i = 1, 2.

We set wli = QN uil and wli− = QN S(uil−1 ). By (2.1), (2.3), and (2.23), for any l 0 we have −1/2

|wl1 − wl2 | |wl1− − wl2− | + |ψl1 − ψl2 | αN wl1− − wl2− + |ψl1 − ψl2 | 1 −1/2 2 1 2 − vl−1 | + |wl−1 − wl−1 | +|ψl1 − ψl2 | CαN D(l − 1, l) |vl−1 where for any integers p < q 0 we set q−1 q−1

. D(p, q) = exp C(q − p) |u1 |2 p + |u2 |2 p Arguing by induction, for any m < l − 1 we derive |wl1

−

wl2 |

l−1

−1/2 l−k

CαN

D(k, l) vk1 − vk2 + ψk1 − ψk2 +

k=m+1

−1/2 l−m D(m, l)u1m − u2m . + ψl1 − ψl2 + CαN

(2.24)

It follows from (2.7) and (2.8) that D(k, l) const e2KC|k|,

|uik | K 1/2 |k|1/2 + const,

i = 1, 2.

Therefore, we can pass to the limit in (2.24) as m → −∞ on condition that log αN > 4KC + 2 log C. This results in 1 w − w 2 l l l−1 −1/2 l−k D(k, l) vk1 − vk2 + ψk1 − ψk2 . CαN ψl1 − ψl2 +

(2.25)

k=−∞

In particular, if ϒ 1 = ϒ 2 , then u1 = u2 and, in view of (1.28), η1 = η2 . It remains to note that (2.25) coincides with (2.9). 3. A Version of the Ruelle–Perron–Frobenius (RPF) Theorem In this section, we prove a version of the RPF theorem which is a generalisation of the corresponding result from [KS1] to systems with unbounded phase space. Its application to the Markov semi-group corresponding to the family (2.19), (2.20) will give us the required uniqueness of a stationary measure.

170


3.1. STATEMENT OF THE RESULT Let X0 ⊂ X1 ⊂ · · · be an increasing family of compact metric spaces which are subsets of a topological space X. We assume that the embeddings XR ⊂ XR+1 ⊂ X are isometries for any integer R 0. Let B(X) be the Borel σ -algebra on X and let P (X) be the set of all probability measures on (X, B(X)). Let P(k, υ, >) be a family of Feller transition probabilities on (X, B(X)) and let Pk : Cb (X) → Cb (X),

P∗k : P (X) → P (X),

k 0,

be the corresponding Markov semi-groups. Recall that a subset R ⊂ Cb (X) is called a determining family for P (X) if for arbitrary measures µ1 , µ2 ∈ P (X) the condition f (υ) dµ1 (υ) = f (υ) dµ2 (υ) for any f ∈ R X

X

implies that µ1 = µ2 . For any function f (υ), denote by f + and f − its positive and negative parts, respectively: f + = 12 (|f | + f ),

f − = 12 (|f | − f ).

For a function f ∈ Cb (X), we shall write fk+ = (Pk f )+ ,

fk− = (Pk f )− .

We shall assume that the condition below is satisfied (cf. hypothesis (H) in [KS1, Section 4.1]): (H) There is a determining family R for P (X) such that f − c belongs to R for all f ∈ R and c ∈ R, and for any f ∈ R and α > 0 and arbitrary integers R 0 and ρ 0 there are k0 = k0 (α, f, ρ, R) ∈ N and A = Af (α, ρ, R) > 1 such that the following property holds: if sup fk+ (υ) α

for all k 0,

(3.1)

sup fk− (υ) α

for all k 0,

(3.2)

υ∈Xρ υ∈Xρ

then for any k k0 there is l = l(k, α, f, ρ, R) > 0 such that sup Pl fk+ (υ) Af (α, ρ, R) inf Pl fk+ (υ),

(3.3)

sup Pl fk− (υ) Af (α, ρ, R) inf Pl fk− (υ).

(3.4)

υ∈XR

υ∈XR

υ∈XR

υ∈XR

Sufficient conditions guaranteeing the validity of (H) are given in Section 3.3. The following result is a generalisation of Theorem 4.1 in [KS1].


171

THEOREM 3.1. Suppose that condition (H) is satisfied. Then the assertions below hold. (i) Let µ ∈ P (X) be a stationary measure of P∗k such that Af (α, ρ, R)µ(X \ XR ) → 0

as R → ∞

(3.5)

for all f ∈ R, α > 0, and ρ 0. Then, for any f ∈ R, Pk f → (µ, f )

as k → ∞ in L1 (X, µ).

(3.6)

(ii) The operator P∗k has at most one stationary measure µ ∈ P (X) satisfying (3.5).

3.2. PROOF OF THEOREM 3.1 (1) As in the case of a single metric space (see [KS1]), (i) implies (ii). Indeed, if µ1 , µ2 ∈ P (X) are two different stationary measures, then there is f ∈ R such that (µ1 , f ) = (µ2 , f ). By (i), Pk f → (µi , f )

as k → ∞ in L1 (X, µi ), i = 1, 2.

Therefore, there is a sequence of integers ks such that Pks f → (µi , f )

as s → ∞ µi -almost everywhere.

(3.7)

Let Ci ⊂ X be set of convergence in (3.7). We have µ1 (C1 ) = µ2 (C2 ) = 1 and C1 ∩ C2 = ∅ and, hence, µ1 and µ2 are singular. We now compare the measures µ1 and µ = (µ1 +µ2 )/2. Applying the above argument to them, we see that µ1 and µ are singular, which contradicts the definition of µ. (2) Thus, it suffices to establish (i). We can assume without loss of generality that (µ, f ) = 0. Since Pk f µ is a nonincreasing sequence, the required assertion will be established if we show that for any ε > 0 there is an integer kε 1 such that Pkε f µ ε.

(3.8)

Let us assume that for any integer ρ 0 there is a sequence ks (ρ) such that sup fk+s (ρ) (υ) → 0 as s → ∞.

υ∈Xρ

In this case, we have + (Pks (ρ) f ) dµ(υ) = fk+s (ρ) dµ(υ) X

X

f ∞ µ(X \ Xρ ) + sup fk+s (ρ) (υ). υ∈Xρ

(3.9)

172


It is clear that the right-hand side of (3.9) can be made arbitrarily small by an appropriate choice of ρ and s. Moreover, it follows from the relation (µ, f ) = 0 that (µ, fk+ ) = (µ, fk− ), and therefore a subsequence of (µ, fk+ ) + (µ, fk− ) = fk µ goes to zero. What has been said obviously implies (3.8). Similar arguments apply in the case when, for any integer ρ 0, sup fk−s (ρ) (υ) → 0 as l → ∞,

υ∈Xρ

where ks (ρ) is a sequence going to +∞ with s. (3) Thus, we can assume that inequalities (3.1) and (3.2) hold for some positive constants α and ρ. In this case, by condition (H), for any integers R 0 and k k0 (α, f, ρ, R) there is l = l(k, α, f, ρ, R) 0 such that (3.3) and (3.4) are satisfied. We now fix arbitrary integer R 0 and, repeating the scheme applied in [KS1], construct a sequence of integers ks = ks (R) such that Pks f µ εf (R) 1 + af (R) + · · · + af (R)s−1 × (3.10) ×f ∞ + af (R)s f µ , where s 0 and εf (R) = 1 + 4Af (R)−1 µ(X \ XR ), af (R) = 1 − µ(XR )Af (R)−1 < 1.

(3.11)

Here and henceforth, the dependence on α and ρ is not indicated explicitly. The proof of (3.10) is by induction on s. For s = 0, in view of the relation Pk∗0 µ = µ, we have Pk0 f µ = f µ , which coincides with (3.10) for s = 0. Assuming that (3.10) is established for s r, we now prove it for s = r + 1. We set kr+1 = kr + lr , where lr = l(kr , α, f, ρ, R) 0 is the integer entering condition (H). In view of (3.3) and (3.4), we have! ± ± fkr dµ = Plr fkr dµ = + X X XR X\XR sup Plr fk±r (υ) µ(XR ) + f ∞ µ(X \ XR ) υ∈XR

Af (R) inf Plr fk±r (υ) µ(XR ) + f ∞ µ(X \ XR ). υ∈XR

! Here and henceforth a formula involving the symbol ± is a brief notation for the two formulas

corresponding to the upper and lower signs.


173

It follows that Plr fk±r (υ) − Af (R)−1 fk±r µ + Af (R)−1 f ∞ µ(X \ XR ) 0 for υ ∈ XR . Let us estimate the expression Pkr+1 f µ = Plr fkr µ . We have Plr fkr dµ = + X

XR

where Dr (fk±r )

= XR

Now note that Dr (fk±r )

X\XR + Dr (fkr ) + Dr (fk−r )

+ f ∞ µ(X \ XR ),

(3.12)

Pl f ± − Af (R)−1 f ± µ dµ. r kr kr

Plr fk±r (υ) − Af (R)−1 fk±r µ + XR + Af (R)−1 f ∞ µ(X \ XR ) dµ + + Af (R)−1 f ∞ µ(X \ XR ).

This implies that Dr (fk+r ) + Dr (fk−r ) 4Af (R)−1 f ∞ µ(X \ XR )+

Plr (fk+r + fk−r ) − Af (R)−1 (fk+r µ + fk−r µ ) dµ + XR 1 − µ(XR )Af (R)−1 fkr µ + 4Af (R)−1 f ∞ µ(X \ XR ). Substituting this expression into (3.12) and using the induction hypothesis, we obtain Pk f dµ r+1 X

r+1 f µ + 1 − µ(XR )Af (R)−1 r j 1 − µ(XR )Af (R)−1 , + 1 + 4Af (R)−1 f ∞ µ(X \ XR ) j =0

which completes the proof of (3.10). It follows from (3.10) and (3.11) that εf (R) f ∞ + af (R)s f µ 1 − af (R)

µ(X \ XR )Af (R) µ(XR )(1 + 4Af (R)−1 ) f ∞ + (3.13) + af (R)s f µ .

Pks (R) f µ

174


The expression in the brackets on the right-hand side of (3.13) is no greater than 5. Hence, in view of (3.5), the right-hand side of (3.13) can be made arbitrarily small by a suitable choice of R and s. This completes the proof of (3.8). 3.3. SUFFICIENT CONDITIONS FOR APPLICATION OF THEOREM 3.1 Let P(k, υ, >), υ ∈ X, > ∈ B(X), be a Feller transition function. Suppose that there is a determining family R for P (X) such that R is invariant with respect to addition of a constant, and the following two conditions hold: (H1 ) For any f ∈ R, R 0, and β > 0 and an arbitrary υ ∈ XR there is an integer k0 = k0 (f, R, β) 1, not depending on υ, and a Borel subset O(f, υ, R, β) ⊂ X such that Pk f (υ ) − Pk f (υ) β for k k0 , υ ∈ O(f, υ, R, β). (H2 ) There is an integer ρ0 0 such that for any ρ ρ0 , R 0, β > 0, and f ∈ R there is a constant ε = ε(f, ρ, β) > 0, not depending on R, and an integer l = l(f, ρ, β, R) 1 for which (3.14) P l, υ 0 , O(f, υ, ρ, β) ε for any υ 0 ∈ XR , υ ∈ Xρ , where the set O(f, υ, ρ, β) is defined in condition (H1 ). THEOREM 3.2. Suppose that conditions (H1 ) and (H2 ) are satisfied. Then (H) holds for R with Af (α, ρ, R) = Af (α, ρ) =

4 f ∞ , α ε(f, ρ, α/2)

(3.15)

where ε(f, ρ, α/2) is the constant in condition (H2 ). In particular, there is at most one stationary measure µ ∈ P (X) concentrated on the union of XR , R 0. Proof. Let f ∈ R be arbitrary function satisfying (3.1) and (3.2) for an integer ρ 1. We must prove that (3.3) and (3.4) hold. To simplify notation, we confine ourselves to the case of the index +. Without loss of generality, it can be assumed that ρ ρ0 , where ρ0 is the integer in condition (H2 ). Let υ k ∈ Xρ be such that fk+ (υ k )

α , 2

k 0.

By condition (H1 ), there is an integer k0 = k0 (f, ρ, α/2) 1 and a sequence of Borel sets Ok = O(f, υ k , ρ, α/2) such that fk+ (υ )

α 4

for υ ∈ Ok , k k0 .

(3.16)


175

Let ε = ε(f, ρ, α/2) > 0 and l = l(f, ρ, α/2, R) 1 be the constants entering condition (H2 ). In view of (3.14) and (3.16), we have (3.17) sup Pl fk+ (υ) f ∞ , υ∈XR P(l, υ, dυ )fk+ (υ ) inf Pl fk+ (υ) = inf υ∈XR υ∈XR X P(l, υ, dυ )fk+ (υ ) inf υ∈XR

Ok

α ε(f, ρ, α/2) α . P(l, υ, Ok ) 4 4

(3.18)

Combining (3.17) and (3.18), we arrive at the required inequality. We now prove the assertion on the uniqueness of a stationary measure. Since the constant Af (α, ρ, R) is in fact independent of R (see (3.15)), there is at most one stationary measure such that µ(X \ XR ) → 0 as R → ∞.

(3.19)

It remains to note that (3.19) is equivalent to the condition that the measure µ is ✷ concentrated on the union of XR , R 0.

4. Uniqueness of a Stationary Measure for the Reduced Chain 4.1. MAIN RESULT We denote by P(k, ϒ, >) the transition probabilities for the family of Markov chains {ϒ k } defined in the space measurable space (FsN , B(FsN )) (see (2.19), (2.20)) and by Pk and P∗k the corresponding Markov semi-groups. We shall also need the following metric generating the Tikhonov topology on HN : 0 1 2 |ϒl1 − ϒl2 | ∧ 2l . dist ϒ , ϒ = l=−∞

THEOREM 4.1. Suppose that condition (1.4) is satisfied for some s > 0. There is a constant K∗ 2M such that if a finite integer N satisfies (2.14) with K = K∗ and bj = 0

for j = 1, . . . , N,

(4.1)

then P∗k has a unique stationary measure µ that is concentrated on the union of the sets FsN (2M, R), R 0. Moreover, for any f ∈ Cb (FsN ) and an arbitrary integer R 0, we have Pk f (ϒ) → (µ, f )

uniformly in ϒ ∈ FsN (2M, R) as k → ∞.

(4.2)

176


Proof. The existence of a stationary measure follows from Proposition 1.11 and Theorem 2.2. To prove the uniqueness and convergence (4.2), we apply the RPF type theorem established in Section 3. (1) We set XR = FsN (2M, R),

X = FsN .

Let R ⊂ Cb (FsN ) be the set of continuous cylindrical functions on FsN , i. e., the set of functions f : FsN → R for which there is an integer m 0 and a bounded continuous function F : (HN )m+1 → R such that v f (ϒ) = F (v−m , ψ−m , . . . , v0 , ψ0 ), ϒ = ∈ FsN . (4.3) ψ Clearly, R is a determining family for P (FsN ). It will be proved in Sections 4.2 and 4.3 that if an integer N satisfies (2.14) with sufficiently large K 2M, then the transition function P(k, ϒ, >) obeys conditions (H1 ) and (H2 ) in which

(4.4) O(f, ϒ, R, β) = ϒ ∈ FsN ∩ FN (K, R) : dist(ϒ , ϒ) r , where K is a fixed constant not depending on f , ϒ, R, β, and N, while r depends only on f , R, and β. By Theorems 3.1 and 3.2, this will imply the uniqueness of a stationary measure concentrated on the union of FsN (2M, R), R 0, and also convergence (4.2) in L1 (X, µ)-norm for any f ∈ R. Moreover, as is shown in Proposition 4.4, the sequence formed of the restrictions of the functions Pk f to XR is uniformly equicontinuous for any integer R 0. Therefore, by Arzelà–Ascoli theorem, a subsequence Pkl f converges uniformly on any XR . In view of the L1 -convergence, the limit is uniquely determined, and hence the whole sequence uniformly converges to (µ, f ). (2) We now show that (4.2) holds for any function f ∈ Cb (FsN ). Since Xρ is a compact subset of X, the restriction of f to Xρ is uniformly continuous for any integer ρ 0. Let us denote by fρ an arbitrary uniformly continuous extension of f Xρ to HN such that fρ ∞ 3f ∞ . For instance, we can take

) + ω ) , f (ϒ d(ϒ, ϒ fρ (ϒ) = inf ρ ϒ ∈Xρ

where ωρ (r), r 0, is the modulus of continuity of f Xρ :

ωρ (r) = sup |f (ϒ 1 ) − f (ϒ 2 )| : ϒ 1 , ϒ 2 ∈ Xρ , d(ϒ 1 , ϒ 2 ) r . Let us denote by JL : HN → HN the operator taking each ϒ = (ϒl , l 0) to (. . . , 0, ϒ−L , . . . , ϒ0 ). We define the function fρL (ϒ) = fρ (JL ϒ),

ϒ ∈ HN .


177

Clearly, we have fρL ∈ R. Thus, convergence (4.2) holds for f = fρL . Let us fix arbitrary R 0 and write Pk f (ϒ) − (µ, f ) Pk fρL (ϒ) − (µ, fρL ) + (µ, f − fρL )+ +Pk f (ϒ) − Pk fρL (ϒ).

(4.5)

As it was mentioned above, sup Pk fρL (ϒ) − (µ, fρL ) := ε1 (k, L, ρ),

(4.6)

ϒ∈XR

where ε1 (k, L, ρ) → 0 as k → ∞ for any fixed L 1 and ρ 0. Furthermore, it is clear that sup d(ϒ, JL ϒ) → 0 as L → ∞ for any ρ 0. ϒ∈Xρ

Therefore, in view of the uniform continuity of fρ , we have sup fρL (ϒ) − f (ϒ) = sup fρ (JL ϒ) − fρ (ϒ) ε2 = ε2 (L, ρ), ϒ∈Xρ

ϒ∈Xρ

where ε2 (L, ρ) → 0 as L → ∞ for any ρ 1. It follows that (µ, f − fρL ) |f − fρL | dµ + X Xρ X\Xρ |f − fρL | dµ + 4f ∞ µ(X \ Xρ ) Xρ

ε2 (L, ρ) + 4f ∞ µ(X \ Xρ ).

(4.7)

Finally, to estimate the third term on the right-hand side of (4.5), we note that Pk f (ϒ) − Pk fρL (ϒ) P(k, ϒ, dϒ )f (ϒ ) − fρL (ϒ ) X + ε2 (L, ρ) + Xρ

X\Xρ

+ 4f ∞ P(k, ϒ, X \ Xρ ). Combining (4.5), (4.6), (4.7), and (4.8), we derive Pk f (ϒ) − (µ, f ) ε1 (k, L, ρ) + ε2 (L, ρ) + + 4f ∞ P(k, ϒ, X \ Xρ ) + µ(X \ Xρ ) .

(4.8)

(4.9)

To conclude that the right-hand side of (4.9) goes to zero, we need the lemma below. We formulate two estimates the first of which is used here and the other will be needed in the next subsection.

178


LEMMA 4.2. For any integer R 0 and any m 1 there is a constant CRm > 0 such that P(k, ϒ, FsN \ FsN (2M, ρ)) CRmρ −m P(k, ϒ, FsN \ FsN (3M, ρ)) CRmρ −m

for k ρ 1, for k, ρ 1,

(4.10) (4.11)

where ϒ ∈ FsN (2M, R). Let us fix an arbitrary ε > 0. In view of (4.10) and the fact that µ is concentrated on ρ0 Xρ , there is an integer ρ 0 such that the third term on the right-hand side of (4.9) is less than ε for k ρ. We then choose integers L 1 and k0 ρ so large that ε1 (k, L, ρ) ε for k k0 and ε2 (L, ρ) ε. Combining all these estimates, we see that (4.9) does not exceed 4ε for k k0 . Thus, to complete the proof of (4.2), it remains to establish Lemma 4.2. ✷ Proof of Lemma 4.2. Let us fix arbitrary m 1. It is clear that it suffices to estabs lish (4.10) and (4.11) for usufficiently large ρ.s We fix an arbitrary ϒ ∈ FN (2M, R) and denote by U = η the element of F (2M, R) such that N U = ϒ. Let (ul , l 0) be the solution of the problem (1.27), (1.28) with the initial function u0 = v0 + W0 (ϒ) (note that |u0 | (2MR)1/2 ) and let al := |ul |2 + ηk 2s . Application of Proposition 1.5 with k− = 0, k+ = k0 = k and Remark 1.6 to the solution ul , 0 l k, shows that, with probability no less than ερm := 1−Cm ρ −m , we have (4.12) (k − T + 1)al kT 2M (k − T ) ∨ ρ + 1 , 0 T k. Since U ∈ Fs (2M, R), we conclude that if ρ R, then al 0T 2M

for T −R.

(4.13)

Combining (4.12) and (4.13), we see that for T 0, with probability ερm , al kT = (|T | + k + 1)−1 (|T | + 1)al 0T + kal k1 M(|T | + k + 1)−1 2(R ∨ |T | + 1) + k ∨ ρ 2M if k ρ 2R, 3M if |T | + k ρ R. We have thus proved that P(k, ϒ, FsN (2M, ρ)) ερm = 1 − Cm ρ −m , P(k, ϒ, FsN (3M, ρ)) ερm = 1 − Cm ρ −m ,

k ρ 2R, k 1, ρ R.

This implies the required inequalities (4.10) and (4.11).

✷

In Section 5, we shall need a corollary of Theorem 4.1. Let us recall that {ϒ k } is isomorphic to the family of Markov chains {θ k } defined by (2.17), (2.18). We denote by Pk and P∗k the Markov semigroups for {θ k }.


179

COROLLARY 4.3. Under the conditions of Theorem 4.1, the Markov semigroup P∗k has a unique stationary measure λ ∈ P (Fs∞) that is concentrated on the union of Fs∞ (2M, R), R 0. Moreover, for any f ∈ Cb (Fs∞ ) we have Pk f (u) → (λ, f ) uniformly in u ∈ Fs∞ (2M, R) as k → ∞.

4.2. CHECKING CONDITION ( H1 ) For any integer m 0, let Rm be the set of those f ∈ R for which the corresponding function F in (4.3) is defined on (HN )m+1 . We recall that the set O(f, ϒ, R, β) is defined in (4.4). PROPOSITION 4.4. Let the conditions of Theorem 4.1 be fulfilled and let K 2M be arbitrary constant. Then for any integer R 0 and any β > 0 there is r = r(R, β, K) > 0 satisfying the following property: if f ∈ Rm for an integer m 1, then Pk f (ϒ 1 ) − Pk f (ϒ 2 )| βf ∞ for k m + 1, 2 1 2 s where ϒ 1 ∈ FsN (2M, R), ϒ ∈ FN ∩ FN (K, R), and dist(ϒ , ϒ ) r. In particular, the sequence Pk f XR , k m + 1, is uniformly equicontinuous for any R 0, and condition (H1 ) holds with any domain O(f, ϒ, R, β) of the form (4.4). Proof. (1) Let dv be the Lebesgue measure on the finite-dimensional space HN and let dα(ψ) be the distribution of the random variables ψk on HNs⊥ . We denote to dv. (It by D(v), v ∈ HN , the density of the random variables ϕk with respect p follows from (1.3) and the conditions imposed on ξj k that D(v) = N j =1 j (bj xj ), where v = (x1 , . . . , xN ) ∈ HN .) Direct verification shows that for f ∈ Rm and k m + 1 we have (cf. [KS1, Section 1.3]) F (ϒk−m , . . . , ϒk )Dk (ϒ; ϒ k ) Pk (dϒ k ), (4.14) Pk f (ϒ) = (HsN )k

where ϒ k = (ϒ1 , . . . , ϒk ) and Pk (dϒ k ) = dv1 · · · dvk dα(ψ1 ) · · · dα(ψk ), Dk (ϒ; ϒ1 , . . . , ϒk ) =

k

D vl − T0 (ϒ, ϒ1 , . . . , ϒl−1 ) ,

(4.15)

l=1

and T0 is the first component of the operator T defined in (2.21), that is, T0 (ϒ) = PN S(v0 + W0 (v, ψ)). (2) Now let ϒ 1 ∈ FsN (2M, R) and ϒ 2 ∈ FsN ∩ FN (K, R). For any k 1, we denote by Vk = Vk (ϒ 1 , ϒ 2 ) the doubled variational distance between the two measure on (HsN )k defined by the densities Dk (ϒ i , ϒ k ), i = 1, 2. In other words, Dk (ϒ 1 , ϒ k ) − Dk (ϒ 2 , ϒ k ) Pk (dϒ k ). Vk = (HsN )k

180


Since F ∞ = f ∞ , it follows from (4.14) and (4.15) that Pk f (ϒ 1 ) − Pk f (ϒ 2 ) f ∞ Vk . Thus, it is sufficient to estimate Vk . To this end, we note that Dk−1 (ϒ 1 , ϒ k−1 )k (ϒ 1 , ϒ 2 ; ϒ k ) Pk (dϒ k ) Vk Vk−1 + (HsN )k

=: Vk−1 + Ik ,

(4.16)

where

k (ϒ 1 , ϒ 2 ; ϒ k ) = D vk − T0 (ϒ 2 , ϒ k−1 ) − D vk − T0 (ϒ 1 , ϒ k−1 ) .

We now derive an estimate for Ik = Ik (ϒ 1 , ϒ 2 ). (3) Let us fix arbitrary K 2M and B 1. To estimate Ik , we represent the domain of integration (HsN )k as the union of a sequence of nonintersecting subsets on each of which the expression k (ϒ 1 , ϒ 2 ; ϒ k ) admits a uniform estimate. Namely, for any integer ρ R we set Ak (ρ) = A˜ k (ρ) \ A˜ k (ρ − 1), where A˜ k (R − 1) = ∅ and A˜ k (ρ) is the set of those (ϒ1 , . . . , ϒk−1 ) ∈ (HsN )k−1 for which (ϒ 1 , ϒ1 , . . . , ϒk−1 ) ∈ FsN (3M, ρ). It is easy to see that the union of Ak (ρ), ρ R, coincides with (HsN )k−1 for any k 1. Let us write the integral Ik as Ik =

∞

(4.17)

Ikρ ,

ρ=R

where Ikρ = Ikρ (ϒ 1 , ϒ 2 ) Dk−1 (ϒ 1 , ϒ k−1 )k (ϒ 1 , ϒ 2 ; ϒ k ) Pk (dϒ k ). =

(4.18)

By the mean value theorem, we have k (ϒ 1 , ϒ 2 ; ϒ k ) Qk (vk ) T0 (ϒ 1 , ϒ k−1 ) − T0 (ϒ 2 , ϒ k−1 ),

(4.19)

Ak (ρ)×HsN

where

Qk (vk ) =

1

∇D vk − θT0 (ϒ 1 , ϒ k−1 ) − (1 − θ)T0 (ϒ 2 , ϒ k−1 ) dθ.

0

It is clear that Qk (vk ) P1 (dϒk ) Q, HsN


181

where Q > 0 is a constant not depending on ϒ 1 , ϒ 2 and ϒ k−1 . Therefore, by (4.17)–(4.19), we obtain ∞ Dk−1 (ϒ 1 , ϒ k−1 ) × Ik Q ρ=R

Ak (ρ)

× T0 (ϒ 1 , ϒ k−1 ) − T0 (ϒ 2 , ϒ k−1 ) Pk−1 (dϒ k−1 ) ∞ hkρ Dk−1 (ϒ 1 , ϒ k−1 )Pk−1 (dϒ k−1 ) Q Q

ρ=R ∞

Ak (ρ)

hkρ P k − 1, ϒ 1 , Ak (ρ) ,

(4.20)

ρ=R

where A (ρ) is the set of elements in FsN of the form (ϒ 1 , ϒ k−1 ) with ϒ k−1 ∈ Ak (ρ), and hkρ = hkρ (ϒ 1 , ϒ 2 ) = sup T0 (ϒ 1 , ϒ k−1 ) − T0 (ϒ 2 , ϒ k−1 ). k

ϒ k−1 ∈Ak (ρ)

(4) We now estimate hkρ . To this end, we need the following lemma. LEMMA 4.5. There is a constant C > 0 such that for any K 2M and any integer ρ 0 we have T0 (ϒ) K(ρ + 1) 1/2 , (4.21) 0 −1/2 T0 (ϒ 1 ) − T0 (ϒ 2 ) C (CαN )−q eCK(|q|∨ρ+1)|ϒq1 − ϒq2 |,

(4.22)

q=−∞

where ϒ, ϒ , ϒ ∈ FN (K, ρ) ∩ FsN (K). 1

2

Taking this assertion for granted, let us complete the proof of the proposition. By definition, we have (ϒ 1 , ϒ k−1 ) ∈ FsN (3M, ρ) ∩ FsN for ϒ k−1 ∈ Ak (ρ). It follows that (ϒ 2 , ϒ k−1 ) ∈ FN (3K, ρ)∩FsN . Therefore, in view of inequality (4.21) with K replaced by K1 := 3K, we have

sup |T0 (ϒ 1 , ϒ k−1 )| + |T0 (ϒ 2 , ϒ k−1 )| hkρ ϒ k−1 ∈Ak (ρ)

1/2 2 K1 (ρ + 1) .

(4.23)

On the other hand, inequality (4.22) implies that hkρ C

1−k

−1/2 −q CK1 (|q|∨ρ+1)

(CαN

) e

1 2 |ϒq+k−1 − ϒq+k−1 |

q=−∞

C

0

−1/2 −q+k−1 CK1 (|q|+k)+CK1 ρ

(CαN

)

e

|ϒq1 − ϒq2 |

q=−∞

C1 (R) 2−k eCK1 ρ d,

(4.24)

182


where d = d(ϒ 1 , ϒ 2 ), C1 (R) > 0 is a constant depending only on R, and the constant K in (2.14) is chosen to be so large that 2(CK1 + log C + log 2) log αN . Note that the third inequality in (4.24) uses the estimate 0 2q |ϒq1 − ϒq2 |, ϒ 1 , ϒ 2 ∈ FN (K, R). d(ϒ 1 , ϒ 2 ) C (R) q=−∞

Combining (4.23) and (4.24), we derive 1/2 hkρ C1 (R) 2−k eCK1 ρ d ∧ 2K1 (ρ + 1)1/2 .

(4.25)

(5) We can now easily complete the proof of the proposition. We wish to show that Vk β if d(ϒ 1 , ϒ 2 ) r, where r = r(β) > 0 is sufficiently small. In view of inequality (4.11) with m = 3 and the inclusion Ak (ρ) ⊂ FsN \ FsN (3M, ρ − 1) for ρ R + 1, we have (4.26) P k − 1, ϒ 1 , Ak (ρ) CR3 ρ −3 for all k 1, ρ R and ϒ 1 ∈ FsN (2M, R). Substituting (4.25), (4.26) and (4.20) into (4.16) and iterating the resulting inequality, we arrive at ∞ k 1/2 ρ −3 C1 (R) 2−k eCK1 ρ d ∧ 2K1 (ρ + 1)1/2 Vk CR3 Q j =1 ρ=R

0 does not depend on R 0. Let us fix arbitrary integer R1 C1 (R + 1) and estimate the probability of the event |uk | R1 ,

ηk s R1 ,

k = 1, . . . , L1 − 1.

(4.33)

In view of (4.32), (1.5) and the Chebyshev inequality, we have P{(4.33) holds} 1 −

L 1 −1

P{|uk | R1 } + P{ηk s R1 } 1 − p1 ,

k=1

where p1 = p1 (R, R1) → 0 as R1 → ∞ for any fixed R. Furthermore, let us fix sufficiently large integers ρ0 1 and L0 ρ0 , set l1∗ := L1 + L0 , and take an arbitrary l1 l1∗ . Applying Proposition 1.5 to the solution uk , k− k k+ ,

185


where k− = L1 and k+ = k0 = l1 , we conclude that there is a constant C0 > 0, not depending on R, such that with probability no less than p0 = 1 − C0 ρ0−1 , al lT1 M,

L1 T l1 − ρ0 .

(4.34)

Hence, we have shown that P{(4.33) and (4.34) hold} p := p0 + p1 − 1.

(4.35) 2(R12 L1 M −1 + R), then ∈ FsN (2M, ρ0 ). In view

It is a matter of direct verification to showthat if L0 uk inequalities (4.33) and (4.34) imply that ( ηk , k l1 ) of (4.35), it follows that for any U ∈ Fs (2M, R) we have uk s , k l1 ∈ F (2M, ρ0 ) 1 − p1 (R, R1) − C0 ρ0−1 . P ηk

(4.36)

It remains to note that if ρ0 1 is so large that C0 ρ0−1 1/4, then for any fixed R 0 we can choose R1 R such that the right-hand side of (4.36) is no less than 1/2. This completes the proof of (4.30). Proof of (ii). We shall need the following elementary lemma. LEMMA 4.7. Let (xl , l 0) be a sequence of nonnegative numbers such that 0

xl C(|T | + 1)

for T −ρ,

(4.37)

l=T

where ρ 0 is an integer and C > 0 is a constant not depending on T . Then every integer interval = [t1 , t2 ] such that t1 −ρ and t2 0 contains an integer point p such that C|t1 | 1 xl C0 := q − p + 1 l=p t2 − t1 + 1 q

for p q 0.

Proof. Assuming the contrary, for each p ∈ we can find an integer m(p), p m(p) 0, such that

m(p)

xl > C0 (m(p) − p + 1).

(4.38)

l=p

Let us define a finite sequence of integers p1 , p2 , . . . , pn by the following rule: p1 = t1 and pj = m(pj −1 ) + 1 if j 2 and m(pj −1 ) t2 . Setting j = [pj , m(pj )] and using inequality (4.38), we derive 0 l=t1

xl

n j =1 l∈j

xl > C0

n

(m(pj ) − pj + 1) C0 (t2 − t1 + 1).

j =1

This contradicts inequality (4.37) with T = t1 .

✷

186


(1) To establish (4.31), we regard (2.20) as an RDS in X (rather than a Markov chain), and using the isomorphism of (2.18) and (2.20), pass from a random trajectory {ϒ k } to {uk = G(ϒ k )}. More exactly, for ϒ and ϒ 0 as in (4.31), let 0 u s 0 0 s = uˆ = −1 ϒ ∈ F (2M, ρ), U = = −1 U N N ϒ ∈ F (2M, ρ0 ). η0 ηˆ We set FL = FL (K, ρ) ∩ FsL , where L = N or L = ∞, and consider the restriction of H: Fs∞ → FsN to F∞ . In view of (2.16), inequality (2.22) implies that the mapping H: F∞ → FN is uniformly Lipschitz with a Lipschitz constant d not depending on N. Therefore, inequality (4.31) will be proved if we show that

ˆ r/d ε, (4.39) P ul2 ∈ F∞ , dist(ul2 , u) where uk , k 0, is the random trajectory of (2.18) starting from u0 . (2) We fix arbitrary ρ ρ0 and r > 0. Let B > 0 be a sufficiently large constant which will be chosen later. Let an integer T1 = T1 (r, B) 1 and a positive constant δ1 = δ1 (r, B) 1 be such that dist(u , 0) r/d for any element u ∈ Fs∞ whose components satisfy the inequalities |uj | B e−(j +T1 −1) + δ1 ,

1 − T1 j 0.

(4.40)

Since |u| ˆ 2 0q 2M for q −ρ, the sequence xl = |ul |2 , l 0, satisfies the conditions of Lemma 4.7 with C = 2M. Let T2 = T2 (ρ, r, B) be the smallest even integer exceeding (2T1 ) ∨ ρ. Applying Lemma 4.7 with t1 = −T2 and t2 = −T2 /2, we find an integer T = T (ρ, r, B), T1 T T2 , such that 2 −T +l (4.41) |u| ˆ −T 4M for 0 l T . We claim that there is a deterministic trajectory u˜ l = (u0 , u˜ 1 , . . . , u˜ l ), l = 1, . . . , ϕ˜ T , for (2.18) that corresponds to a control η˜ l = ψ˜l ∈ H s and possesses the l following properties: |u˜ l − uˆ l−T | B e−l , l = 0, . . . , T , p/2 l = ψl−T , l = 1, . . . , T , ψ ϕl p 2BαN ,

(4.42) (4.43)

where p 0. Taking this assertion for granted, let us show that (4.39) holds with ∈ Fs (2M, ρ) that l2 = T . It follows from (4.43) and the inclusion U s/2

η˜ l s 2BαN + (2MT )1/2 ,

l = 1, . . . , T .

Therefore, by Lemma 6.2, for any γ > 0 the probability of the event

Fγ := |ηl − η˜ l | γ , l = 1, . . . , T can be estimated from below by a constant ε > 0 depending only on N, B, ρ, r, and γ (but not on ϒ). In view of the continuous dependence of trajectories for (2.11) on the control ηl , for any ω ∈ Fγ we have |ul − u˜ l | δ = δ(γ ),

l = 1, . . . , T ,


187

where ul , l 1, is the trajectory of (2.11) corresponding to ηl , and δ(γ ) → 0 as γ → 0. Combining this with (4.42), we conclude that uj := uj +T − uˆ j , j = 1 − T , . . . , 0, satisfy (4.40) if δ(γ ) δ1 . Therefore, ˆ r/d d(ul2 , u)

for ω ∈ Fγ , γ & 1.

Moreover, it is a matter of direct verification to show that ul2 ∈ F∞ (K, ρ), where K = K(M, B) is sufficiently large. This completes the proof of (4.31). (3) Thus, it remains to establish the existence of a deterministic trajectory u˜ l satisfying (4.42) and (4.43). Let us set l = ψl−T , l = 1, . . . , T , ψ (4.44) ϕl = PN u˜ l−T − S(u˜ l−1 ) , where u˜ 0 is the zeroth component of u0 . Note that the first relation in (4.44) implies that v˜ l = vˆl−T for l = 1, . . . , T . We claim that (4.42) and (4.43) hold with an appropriate constant B > 0. Indeed, (4.43) is a simple consequence of inequality (4.42) whose proof is by induction on l. In view of (4.41) with l = 0 and the inclusion u0 ∈ F∞ (2M, ρ0 ), we have 1/2 + 2M 1/2 := B. |u˜ 0 − uˆ −T | |u˜ 0 | + |uˆ −T | 2M(ρ0 + 1) Let us assume that (4.42) is proved for 0 l k − 1, k 1. It follows from (4.41) and inequality (2.24) in which m = 0, l = k, u1r = u˜ r , and u2r = uˆ r−T that

) exp Ck |u˜ j |2 k−1 + |uˆ j −T |2 k−1 |u˜ 0 − uˆ −T | 0 0

−1/2 k (CαN ) exp 2C(6M + B 2 )k B e−k B, −1/2 k

|u˜ k − uˆ k−T | (CαN

where the integer N 1 is so large that log αN 4C(6M + B 2 ) + 2(1 + log C). This completes the induction and the proof of the proposition.

✷

5. Uniqueness and Mixing for the Original System We recall that the Markov semigroups Pk and Pk∗ associated with Equation (1.1) and the space C(H s , β) of continuous functions with exponential growth at infinity and the corresponding norm f s,β were introduced in Section 1.3. Also recall that we set βd (r) = (1 + r)d , r 0. As before, we assume that ν = 1. For any integer R 0, we denote by H (R) the set of those u ∈ H for which there is u ∈ Fs∞ (2M, R) such that u0 = u, where u0 is the zeroth component of u.

188


THEOREM 5.1. Suppose that condition (1.4) is satisfied for some s > 0. Then there is an integer N 1 such that if bj = 0

for j = 1, . . . , N,

(5.1)

then the Markov semigroup Pk∗ has a unique stationary measure λ ∈ P (H ) satisfying condition (1.34). Moreover, the measure λ is concentrated on H s , and if f ∈ C(H s , βm ) for some m 1, then for any integer R 0, we have Pk f (u) → (λ, f )

as k → ∞ uniformly in u ∈ H (R).

(5.2)

In particular, convergence (5.2) holds for λ-almost all u ∈ H . Finally, if all the constants bj in (1.3) are nonzero, then (5.2) holds uniformly with respect to u ∈ H s , us R, for any R 0. Remark 5.2. The existence and uniqueness of a stationary measure and convergence (5.2) can be established under a weaker assumption. Namely, instead of (0.9), it suffices to assume that ∞ |r|20 pj (r) dr C for all j 1. (5.3) −∞

This assertion can be derived by analysing the arguments in Sections 1–5. We do not dwell on it and only show where the exponent 20 in (5.3) comes from. When verifying condition (H1 ), we used (see (4.26)) inequality (4.11) with m = 3, which, in turn, is based on the fact that the third moment of the random variable Tν (ω) (see Proposition 1.5) is finite. The mth moment of Tν can be estimated by a constant depending only on N2m and E|η1 |4(m+2) (see (1.16) and (1.18)), and N2m admits an estimate in terms of E |ηk |2m (see (1.11)). For m = 3 we obtain the expression E |ηk |20 , which can be estimated by the constant C in (5.3). Proof of Theorem 5.1. The existence of a stationary measure satisfying (1.34) and the fact that λ(H s ) = 1 are established in Proposition 1.11. The uniqueness of such a measure follows from Theorem 2.2 and Corollary 4.3. Let us prove (5.2). (1) We begin with the case f ∈ Cb (H ). Let us define a function f ∈ Cb (H), H = H Z0 , by the formula f (u) = f (u0 ),

u = (ul , l 0).

We recall that Pk and P∗k stand for the Markov semigroups associated with the family (2.17), (2.18). It is clear that Pk f (u) = Pk f (u0 ) for any u ∈ Fs∞ . Let λ ∈ P (Fs∞ ) be the unique stationary measure for P∗k . By Corollary 4.3, we have Pk f (u) → (λ, f ) as k → ∞ uniformly in u ∈ Fs∞ (2M, R). Since the projection u = (ul , l 0) !→ u0 maps λ to λ, we conclude that (λ, f ) = (λ, f ). Therefore (5.2) holds uniformly in u ∈ H (R) for any R 0.


189

(2) To show that (5.2) remains valid for f ∈ C(H, βm ), we use Lemma 1.8. Namely, for L > 0 let hL (r) denote a continuous function that is equal to 1 and 0 for r L and r L+1, respectively. We take an arbitrary function f ∈ C(H, βm ) and represent it in the form f (u) = fL (u) + gL (u),

fL (u) = hL (|u|)f (u).

Since fL ∈ Cb (H ), we conclude that Pk fL (u) → (λ, fL )

as k → ∞ uniformly in u ∈ H (R).

It is easy to see that (λ, fL ) → (λ, f ) as L → ∞. Furthermore, we note that gL 0,βm → 0 as L → ∞, where for any m > m. By Lemma 1.8, the norm of the operators Pk : C(H, βm ) → C(H, βm ) is bounded uniformly in k 1. It follows that Pk gL (u) → 0 as L → ∞ uniformly in k 0 and u ∈ H (R) ⊂ BH (R1 ), R1 = (2M(R + 1))1/2 . We now write Pk f (u) − (λ, f ) Pk fL (u) − (λ, fL ) + Pk gL (u) + (λ, fL − f ).

(5.4)

What has been said above implies that the right-hand side of (5.4) tends to zero as k → ∞. The fact that (5.2) holds also for f ∈ C(H s , βm ) follows from Lemma 1.8. (3) We now assume that bj = 0 for all j 1. To prove that (5.2) holds uniformly in u ∈ H s , us R, it suffices to show that the ball BH s (R) is contained in HR for some R 1. This assertion follows immediately from the definition ✷ of Fs∞ (2M, R). Remark 5.3. If in Theorem 5.1 we assume that condition (B) is also satisfied, then convergence (5.2) holds for functions with exponential growth at infinity (see Main Theorem in the Introduction). Namely, it suffices to assume that f (u) is a l continuous function on H s such that |f (u)| const exp(σ u2κ s ), where l is the smallest integer no less than s, κl is the constant in Theorem 1.4 with ρ = ∞, and σ > 0 is sufficiently small. This assertion can easily be proved by repeating the above arguments and using Remark 1.9, and we shall not dwell on it. As we saw above, convergence (5.2) is a simple consequence of Theorem 4.1. The following assertion shows that under the same conditions we have a much stronger result. Its proof requires some new ideas and will be presented in a subsequent publication.

190


THEOREM 5.4. Under the conditions of Theorem 5.1, if (5.1) is satisfied for a sufficiently large N 1, then for any f ∈ C(H s , βm ), m 1, and R > 0, we have Pk f (u) → (λ, f )

as k → ∞ uniformly in u ∈ BH (R).

(5.5)

Moreover, if condition (B) is also satisfied, then (5.5) holds for any function described in Remark 5.3.

6. Appendix 6.1. PROOF OF THEOREM 1.3 The existence and uniqueness of a solution are obvious, so that we confine ourselves to the proof of (1.11) and (1.12). (1) We begin with the case s = 0. Taking the scalar product of (1.8) and u(t) in H , we obtain |St (u)| e−α1 νt |u0 |,

t 0.

(6.1)

k 1,

(6.2)

Since uk = S(uk−1 ) + ηk ,

we conclude from inequality (6.1) with t = 1 that, for any δ > 0, |uk |m (1 + δ)e−να1 m |uk−1 |m + Cm δ −(m−1) |ηk |m , where the constant Cm > 0 depends on m solely. Choosing q = e−α1 ν and δ = e(m−1)α1 ν − 1, we derive |uk |m q|uk−1 |m + C(m)ν −(m−1) |ηk |m . Taking the average and iterating the resulting inequality, we obtain (1.11). (2) We now consider the case s > 0. We shall need the following lemma, which is proved in Subsection 6.3. LEMMA 6.1. The resolving operator St of the free NS system (1.8) is continuous from H to H s for any t > 0 and s 0. Moreover, for any integer l 2 there is a constant Cl 1 such that if u(t, x) is a solution of (1.8) for t 0, then t θ l u(θ)2l+1 dθ t l u(t)2l + 2ν −l 0 2 0 ν |u | , l = 0, 1, (6.3) Cl ν −l |u0 |2 + ν −5l |u0 |2/κl , l 2, where t 0, and the constant κl is defined in Theorem 1.4. Furthermore, for l = 0 the inequality sign in (6.3) can be replaced by equality.


191

To simplify notation, we confine ourselves to the case s > 1. Let us fix an arbitrary k 1. In view of relation (6.2) and inequality (6.3) with t = 1, we have (the integer l = l(s) is defined in Theorem 1.3) m−1 m S(uk−1 )m uk m s 2 l + ηk s Cml 1 + ν −5lm/2 |uk−1 |m(2l+1) + ηk m s , which implies (1.12). 6.2. PROOF OF THEOREM 1.4 We confine ourselves to the case s = 0. It follows from (6.1) and (6.2) that, for any δ > 0, |uk |2 (1 + δ)e−2α1 ν |uk−1 |2 + (1 + δ −1 )|ηk |2 .

(6.4)

We set q = e−α1 ν , δ = eα1 ν − 1, and σ0 = ρ ∧ (aα1 e−α1 ). Inequality (6.4) implies that σ0 ν|uk |2 σ0 νq|uk−1 |2 + a|ηk |2 , and therefore, in view of independence of uk−1 and ηk , we have 2 2 2 q 2 2 q E eσ0 ν|uk | E ea|ηk | E eσ0 ν|uk−1 | E ea|ηk | E eσ0 |uk−1 | . Arguing by induction on k, we derive (1.14). 6.3. PROOF OF LEMMA 6.1 Inequality (6.3) is proved by induction on l. For l = 0, it is well known (see [CF]). We now fix an arbitrary l = m 1 and assume that inequality (6.3) is established for l < m. Let us take the scalar product in H of Equation (1.8) and the function Lm u. Performing some simple transformations, we derive ∂t t m u2m − mt m−1 u2m + 2νt m u2m+1 + m+1 m−1 (6.5) + 2t m L 2 u, L 2 B(u, u) = 0. If m = 1, then the last term on the left-hand side of (6.5) vanishes, and the required inequality can be established by integration with respect to time. Therefore we assume that m 2. In this case, we have the following estimate, which follows easily from Hölder’s and interpolation inequalities: 4m−1 m+1 1 m+1 L 2 u, L m−1 2 B(u, u) c u 2m u 2m |u| 2 m m+1 ν 2 1−4m um+1 + cm ν u2(m+1) |u|2m , 2

(6.6)

192


where cm and cm are positive constants. Substituting (6.6) into (6.5) and integrating in time, we obtain t t m u(t)2m + ν θ m u(θ)2m+1 dθ 0 t t m−1 2 1−4m θ u(θ)m dθ + cm ν θ m u(θ)2(m+1) |u(θ)|2m dθ. m 0

0

The required inequality follows now from the induction hypothesis. 6.4. PROOF OF LEMMA 2.4 (1) Let ui (t), i = 1, 2, be two solutions of the free NS system (1.8) with initial functions u0i . Then the difference u = u1 − u2 satisfies the equation (recall that ν = 1) u˙ + Lu + B(u, u1 ) + B(u2 , u) = 0.

(6.7)

Let us take the scalar product of this equation with 2u(t) in H . Since 2 (B(u, u1 ), u) c1 |u| u u1 1 u2 + c1 u1 2 |u|2 2 2 and (B(u2 , u), u) = 0, we derive the differential inequality ∂t |u|2 + u2 c12 u1 2 |u|2 . Applying the Gronwall inequality, we obtain t 2 2 2 u1 (θ) dθ |u0 |2 , |u(t)| exp c1 0

(6.8)

u0 = u01 − u02 ,

which coincides with (2.22). Integration of (6.8) now results in t t u(θ)2 dθ |u0 |2 + c2 u1 (θ)2 |u(θ)|2 dθ 1 0 0 t 2 c2 θ u (σ )2 dσ 0 2 2 1 0 1 c1 u1 (θ) e dθ |u | 1 + t 0 u1 (θ)2 dθ |u0 |2 . exp c12

(6.9)

(6.10)

0

(2) We now take the scalar product of (6.7) with 2tLu(t) in H : ∂t tu2 + 2t|Lu|2 = u2 − 2t B(u, u1 ), Lu − 2t B(u2 , u), Lu . (6.11) Let us use the inequalities v2∞ c22 |v| |Lv|,

v2 |v| |Lv|


193

to estimate the second and third terms on the right-hand side of (6.11): (B(u, u1 ), Lu) u∞ u1 |Lu| c2 |u|1/2 |Lu|3/2 |u1 |1/2 |Lu1 |1/2 12 |Lu|2 + c24 |u|2 |u1 |2 |Lu1 |2 , (6.12) 1/2 1/2 1/2 3/2 (B(u2 , u), Lu) u2 ∞ u |Lu| c2 |u2 | |Lu2 | |u| |Lu|

1 |Lu|2 2

+ c24 |u|2 |u2 |2 |Lu2 |2 .

(6.13)

We now note that (see (6.1)) |ui (t)| |u0i |,

t 0, i = 1, 2.

(6.14)

Substituting (6.12)–(6.14) into (6.11) and integrating with respect to t, we derive t 2 u(θ)2 dθ + tu 0 t t θ|u|2 |Lu1 |2 dθ + |u02 |2 θ|u|2 |Lu2 |2 dθ . (6.15) + 2c24 |u01 |2 0

0

To estimate the expression in the brackets on the right-hand side of (6.15), we apply inequalities (6.9) and (6.3) with l = 1: t t t 2 0 2 2 2 0 2 0 2 2 θ|u| |Lui | dθ |u | |ui | exp c1 θ|Lui |2 dθ u1 (θ) dθ |ui | 0 0 0 t 2 0 2 0 4 2 u1 (θ) dθ . (6.16) |u | |ui | exp c1 0

Furthermore, it follows from (6.1) and (6.3) with l = 0 that t t 2 0 2 −2α1 t −1 2 −1 ui dθ c3 (t ∨ 1) ui (θ) dθ. |ui | 2 1 − e 0

(6.17)

0

Substitution of (6.17) into (6.16) results in t 0 2 θ|u|2 |Lui |2 dθ |ui | 0

2 t 2 2 2 ∨ 1) ui (θ) dθ exp c1 u1 (θ) dθ |u0 |2 0 0 t 2 2 2 2 −2 c (t ∨ 1) exp c u1 (θ) + u2 (θ) dθ |u0 |2 . c32 (t −2 3

t

0

1

The required inequality (2.23) follows now from (6.15), (6.10), and (6.18).

(6.18)

194


6.5. LOWER BOUND FOR MEASURES WITH POSITIVE DENSITY LEMMA 6.2. Let γ be the distribution in H of the random variable η(x) =

∞

bj ξj ej (x),

j =1

where bj are real numbers satisfying condition (1.4) and ξj are independent scalar random variables whose distributions have strictly positive, 2 continuous densities pj (r) with respect to the Lebesgue measure such that R r pj (r) dr C for all j 1 and some constant C > 0 not depending on j . Then γ (B) > 0 for any open ball B ⊂ H s . Moreover, for any p > s, R > 0, and r > 0 there is ε = ε(p, R, r) > 0 such that γ (B) ε for any open ball B ⊂ H s of radius r centred at a point u0 ∈ H p , u0 p R. Proof. We recall that HLs and HLs⊥ denote the closed subspaces in H s spanned by the vectors ej , j = 1, . . . , L − 1, and ej , j L, respectively, and that PL and QL are the orthogonal projections in! H onto HL and HL⊥ . It is clear that for any u0 ∈ H s and r > 0 we have √ √ BH s (u0 , r) ⊃ BHLs (v 0 , r/ 2) × BH s⊥ (w 0 , r/ 2), L

where v 0 = PL u0 , w 0 = QL u0 , ϕ = PL η, ψ = QL η, and L 2 is an arbitrary integer. Since ϕ and ψ are independent, we conclude that √

P η ∈ BH s (u0 , r) P ϕ ∈ BHLs (v 0 , r/ 2) × √

(6.19) × P ψ ∈ BHLs⊥ (w 0 , r/ 2) . Let us choose an integer L 2 so large that 0 r w √ , s 2 2

∞ j =L

bj2 αjs
0: 1

1

u(t) β Ct− 3 + 3β |u|X ,

(2.1)

where 4 < β ∞ and u(t)ux (t) ∞ Ct − 3 t− 3 |u|2X . 2

1

(2.2)

Proof. We follow the method of the proof of Lemma 2.2 from [15, 16], so we only give an outline of the proof. By the Sobolev embedding inequality, we have u β C u 1,0 , 4 < β ∞, therefore we only prove (2.1) for t 1. We have the identity ∞ 2 3 u(t, x) = G(t)v(t) = eipx+ip t /3 v(t, ˆ p) dp π 0 √ ∞ 2 q iqχ+iq 3 /3 e − v(t, ˆ κ) dq v(t, ˆ κ) + vˆ t, √ = √ √ 3 π3t t 0 √ x 2π Ai √ v(t, ˆ κ) + R(t, x), (2.3) = √ 3 3 t t where we denote

√ ∞ 2 q iqχ+iq 3 /3 − v(t, ˆ κ) dq, e vˆ t, √ R(t, x) = √ √ 3 π3t t 0 x , κ = −x/t for x 0 χ=√ 3 t

and κ = 0 for x 0, v(t) = G(−t)u(t). Consider the case x 0, i.e. χ 0 and κ = 0. Using the identity eiqχ+iq

3 /3

=

∂ iqχ+iq 3 /3 1 qe , 2 1 + iq(q + χ) ∂q

we integrate by parts with respect to q in the remainder term R(t, x): ∞ q iq(3q 2 + χ) C − v(t, ˆ 0) − v ˆ t, R(t, x) = √ √ 3 3 1 + iq(q 2 + χ) t t 0 iqχ+iq 3 /3 q e dq q ˇ Jvˆ t, √ . −√ 3 3 2 + χ) 1 + iq(q t t By the integration by parts and the Cauchy–Schwarz inequality, we have v(t, ˆ p) C |p| Ju(t) ˆ p) − v(t, ˆ 0) C |p| Jˇ v(t,

(2.4)

(2.5)

ON THE MODIFIED KORTEWEG–DE VRIES EQUATION

203

for all p ∈ R. Therefore, applying the Cauchy inequality, we get from (2.5) ∞ q C − v(t, ˆ 0) + vˆ t, √ |R(t, x)| √ 3 3 t 0 t q dq q ˇ J v ˆ t, √ + √ 3 3 1 + q(q 2 + χ) t t 12 2 ∞ ∞ 2 q dq q C Jˇ vˆ t, √ dq + √ 3 2 3 (1 + q(q 2 + χ))2 t t 0 0 √ ∞ q dq C Ju(t) C (2.6) √ √ . + √ Ju 2 + χ) 1 + q(q χ t t 0 Now let us consider the second case x 0. Denote χ = −µ2 0, so that √ |x| , µ= √ 6 t

µ κ=√ . 3 t

Using the identity eiqχ+iq

3 /3

=

∂ 1 3 (q − µ)eiqχ+iq /3 , 2 1 + i(q − µ) (q + µ) ∂q

(2.7)

we integrate by parts with respect to q in the remainder term R(t, x) q i(q − µ)2 (3q + µ) − v(t, ˆ κ) − vˆ t, √ 3 1 + i(q − µ)2 (q + µ) t 0 3 q eiqχ+iq /3 q −µ ˇ J v ˆ t, dq − − √ √ 3 3 1 + i(q − µ)2 (q + µ) t t v(t, ˆ 0) − v(t, ˆ κ) µ . (2.8) − √ 3 1 + iµ3 π t

C R(t, x) = √ 3 t

∞

Therefore, by the Cauchy–Schwarz inequality, we obtain by (2.8) ∞ |q − µ| q q C + √ Jˇ vˆ t, √ vˆ t, √ − v(t, ˆ κ) × |R(t, x)| √ 3 3 3 3 t 0 t t t C Ju dq + √ × 2 1 + (q − µ) (q + µ) t µ √ ∞ |q − µ|dq C + √ Ju 1 + (q − µ)2 (q + µ) t 0 12 ∞ (q − µ)2 dq C Ju(t) . (2.9) √√ + 2 2 (1 + (q − µ) (q + µ)) t µ 0

204

NAKAO HAYASHI AND PAVEL NAUMKIN 1

Using the estimate |Ai(χ)| C χ− 4 for the Airy-type function (see [10, 11]) we get from (2.3), (2.6) and (2.9) |u(t, x)| C t

− 13

C t

− 13

|x| 1+ √ 3 t |x| 1+ √ 3 t

− 14 − 14

−1 v(t) ˆ ∞ + t 6 Ju(t)

|u|X

(2.10)

√ for all x ∈ R, t > 0. Making a change of the variable χ = x/ 3 t and using inequality (2.10), we get u(t) β C t

1 − 13 + 3β

∞

u X

χ

− β4

β1 dχ

C t− 3 + 3β |u|X 1

1

(2.11)

0

for any β > 4. Hence, (2.1) follows. As in (2.3) we have for the derivative ux √ ∞ q 2 iqχ+iq 3 /3 q dq. ux (t, x) = − √ √ e vˆ t, √ 3 3 t π t2 0 In the domain x 0, using the identity (2.4), analogously to (2.6), we obtain q 2 ∞ ˇ C ∞ J vˆ t, √3 t q dq C q dq + v ˆ ∞ |ux (t, x)| √ 3 2 |1 + iq(q 2 + χ)| t |1 + iq(q 2 + χ)| t 0 0 C 1 C pJˇ v(t, ˆ p) √ u X (2.12) v ˆ ∞+√ √ 3 2 3 2 6 t t t for all t > 0. And in the domain x 0 using the identity (2.7) we get in analogy with (2.9), for all t 1 ∞ q C q |q − µ| ˇ |ux (t, x)| √ + √ J vˆ t, √ × 3 2 vˆ t, √ 3 3 3 t t t t 0 q dq × 1 + (q − µ)2 (q + µ) ∞ q dq C v(t) ˆ ∞ + √ 3 2 1 + (q − µ)2 (q + µ) t 0 ∞ 12 q 2 (q − µ)2 dq C ˇ J v(t, ˆ p) +√ 6 5 (1 + (q − µ)2 (q + µ))2 t 0 √ √ 4 C 4 χ C χ 1 u X . Ju √ √ (2.13) v ˆ ∞+√ 3 2 3 2 6 t t t Now estimate (2.2), for all t 1, follows from (2.10), (2.12) and (2.13), and for the case 0 < t < 1, x 0, we get

205 ∞ |q − µ| ˇ q C q |ux (t, x)| √ + √ J vˆ t, √ × vˆ t, √ 3 3 3 3 t 0 t t t q √ 3 dq t × 1 + (q − µ)2 (q + µ) ∞ 12 12 ∞ dq C 2 2 |v(t, ˆ p)| p dp + √ 6 (1 + (q − µ)2 (q + µ))2 t 0 0 ∞ 12 (q − µ)2 dq C ˇ ˆ p) + √ p Jv(t, (1 + (q − µ)2 (q + µ))2 t 0 C C √ ( Ju 1,0 + u 1,0 ) √ |u|X , t t ON THE MODIFIED KORTEWEG–DE VRIES EQUATION

hence in view of (2.10) and the estimate u ∞ C u 1,0 , we obtain (2.2) for all 0 < t 1. Lemma 2.1 is proved. ✷ In order to show the following lemma, we need some function spaces. We denote Y0 = φ ∈ L2 ; φ Y0 < ∞ , where

1 u (t) Y0 = t− 6 Ju(t) + F G (−t) u (t)∞

and

2 Y = φ ∈ L : φ (x) dx = 0, φ Yν < ∞ , ν

1 , ν ∈ 0, 2

where

ν 1 ν 1 u (t) Yν = t 3 − 6 Ju(t) + sup |p|−ν p 3 t 3 − 6 |F G(−t)u(t)|. p∈R

LEMMA 2.2. Let

u(t, x), ϑ (t, x) ∈ L∞ (0, ∞) , Y0 , 1 1 ∞ δ , w(t, x) ∈ L (0, ∞) , Y , δ = − 3γ , γ ∈ 0, 2 50

be real-valued functions. Then we have the following estimates for all t > 1: u (t) ϑ (t) w (t) C tγ −1 u Y0 ϑ Y0 w Yδ .

(2.14)

Moreover, we have the following asymptotics for large-time t 1 uniformly with respect to x ∈ R: √ 1 x 2π |x| − 4 γ − 12 w r ˆ (t, κ) + O t Ai √ √ 1 + (2.15) w(t, x) = √ δ Y , 3 3 3 t t t

206

NAKAO HAYASHI AND PAVEL NAUMKIN

where

x κ= − t

for x 0 and

κ = 0 for x 0;

the function r(t) = G(−t)w(t) and G(t) is the free Airy evolution group. Proof. By the Cauchy–Schwarz inequality, we get p rˆ (t, p)−rˆ (t, κ) Jˇ rˆ (t, ξ )dξ C t 1−2δ 6 |p − κ| w Yδ .

(2.16)

κ

We use the representation similar to (2.5) and modify the estimates (2.6) and (2.9) via (2.16) as follows ∞ rˆ t, √q − rˆ (t, 0) + √q Jˇ rˆ t, √q dq 3 3 3 C t t t |R(t, x)| √ 3 2 1 + q(q + χ) t 0 2 ∞ 12 ∞ 2 q dq q C Jˇ rˆ t, √ dq + √ 3 2 3 (1 + q(q 2 + χ))2 t t 0 0 ∞ δ √ 1 (q + q)dq Ct γ − 2 w Yδ γ − 12 w Yδ + C t √ 1 + q(q 2 + χ) χ 0 and in the same manner as above ∞ |q − µ| q q C ˇ − rˆ (t, κ) + √ Jrˆ t, √ × |R(t, x)| √ rˆ t, √ 3 3 3 3 t 0 t t t C|ˆr (t, κ) − rˆ (t, 0)| dq + √ × 3 2 1 + (q − µ) (q + µ) t µ √ ∞ δ 1 (|q − µ| + |q − µ|)dq + Ct γ − 2 w Yδ 1 + (q − µ)2 (q + µ) 0 12 ∞ 1 (q − µ)2 dq Ct γ − 2 w Yδ + + µ (1 + (q − µ)2 (q + µ))2 0 1

Ct γ − 2 w Yδ , √ µ hence, using the estimate rˆ (t, κ) = rˆ (t, κ) − rˆ (t, 0)

κ

Jˇ rˆ (t, p)dp |κ|δ κ 3 t γ w Yδ ,

0

we get |w(t, x)| C t

− 13

|x| 1+ √ 3 t

− 14 3γ δ |x| 2 x 2 − 3δ t + δ w Yδ √ 3 t t2


207

and, by virtue of Lemma 2.1 (2.10), we have u (t) ϑ (t) w (t) C u Y0 ϑ Y0 w Yδ × t 3 12 |x|δ x 3γ |x| − 2 − 2δ t 3 + δ √ dx × 1+ √ 3 3 t t t C t− 6 − 3 u Y0 ϑ Y0 w Yδ . 5

δ

Thus, estimate (2.14) and asymptotics (2.15) are true. Lemma 2.2 is proved. In the next lemma we estimate the following integral: 2it 3 e 3 p Q 0(t, p, ξ )dξ1 dξ2 , M(t, p) = p 3 where Q = − 12 1 − ξ13 − ξ23 − ξ33 , t > 1, p > 0, the vector ξ = (ξ1 , ξ2 , ξ3 ) with the relation ξ1 + ξ2 + ξ3 = 1, the function ˆ pξ2 )w(t, ˆ pξ1 )ϑ(t, ˆ pξ3 ) − 0(t, p, ξ ) = ψ3 (ξ ) v(t,

p p p ϑˆ t, wˆ t, ψ1 (ξ ) − − vˆ t, 3 3 3 ˆ p)w(t, − v(t, ˆ p)ϑ(t, ˆ −p)ψ2 (ξ ) and ψj ∈ C2 (R3 ), j = 1, 2, 3 are such that ψ1 (ξ ) = 1 as ξ1 − 13 + ξ2 − 13
15 , ψ2 (ξ ) = 1 as |ξ1 − 1| + |ξ2 − 1|
15 , ψ3 (ξ ) = 1 as ξ1 > 12 , ξ2 >

1 2

and ψ3 (ξ ) = 0 as ξ1 < 16 , or ξ2 < 16 .

✷

208


Denote

1 |φ|0 = sup t − 6 φ (t) + φ(t) ∞ , t 1

ν 1 ν −1 |φ|ν = sup t 3 − 6 φ (t) + sup|p|−ν p 3 t 3 6 φ(t, p) . t 1

p∈R

(If ν > 0, then by |φ|ν < ∞, we assume also that φ(t, 0) = 0 for all t 1.) ˆ wˆ be such that the norms |v| ˆ σ , |w| LEMMA 2.3. Let the functions v, ˆ ϑ, ˆ α , |ϑ| ˆ δ< ∞, where 1 1 − 3γ , 0, 0 , 0, 2 − 3γ , 0 , or (α, σ, δ) = (0, 0, 0), 2 0, 0, 12 − 3γ , 1 ). Then the following estimate is valid for all t > 1, p > 0: where γ ∈ (0, 50

|M(t, p)|

ˆ σ |w| ˆ α |ϑ| ˆ δ p ω |v| γ

1

(p 3 t)1− 2 p 3 t 12 −2γ

,

where ω = 3 + α + σ + δ. Proof. We make a change of variables of integration ξ1 =

1 3

so that ξ3 =

+ z − y, 1 3

ξ2 =

1 3

+z+y

− 2z and

Q = − 49 + 3z2 (1 − z) + y 2 (1 + 3z). Denote also τ = (2t/3)p 3 . Then we have eiτ Q 0(t, p, ξ )dξ1 dξ2 . M(t, p) = p 3 Let us consider first the case τ ∈ (0, 1). We integrate by parts via the identity eiτ Q =

∂ iτ Q 1 ye 2 1 + 2iτy (1 + 3z) ∂y

to get M = M1 + M2 − M3 , where M1 (t, p) = p

3

and

eiτ Q 0(t, p, ξ )

Mj +1 (t, p) = p 3

eiτ Q

4iτy 2 (1 + 3z)dy dz (1 + 2iτy 2 (1 + 3z))2

0ξj (t, p, ξ )y dy dz 1 + 2iτy 2 (1 + 3z)

,

j = 2, 3.

209


√

In the first summand, we make a change of variables y 1 + 3z = η and integrate by parts with respect to z and using the identity eiτ Q = we obtain

∂ iτ Q 1 ze , 2 1 + 3iτ z (2 − 3z) ∂z

4iτ η2 dη dz √ (1 + 2iτ η2 )2 1 + 3z 6iτ z2 (4 − 9z) 3z 0(t, p, ξ ) + 0(t, p, ξ ) + eiτ Q = p3 1 + 3iτ z2 (2 − 3z) 2(1 + 3z) 3yz 3yz 0ξ1 (t, p, ξ ) + z + 0ξ2 (t, p, ξ ) − + z− 2(1 + 3z) 2(1 + 3z) 4iτ η2 dη dz √ . − 2z0ξ3 (t, p, ξ ) (1 + 2iτ η2 )2 1 + 3z(1 + 3iτ z2 (2 − 3z))

M1 = p 3

eiτ Q 0(t, p, ξ )

Since 1 ˆ σ |w| ˆ α |ϑ| ˆ δ |0(t, p, ξ )| C6(ξ )τ γ p α+σ +δ z 2 −3γ |v|

and p|Jˇ v(t, ˆ pξ1 )| Cp α

τ 1−2α √ 6 | p Jˇ v(t, ˆ pξ1 )|, t

we get ˆ σ |w| ˆ δ × |0ξ1 (t, p, ξ )| C6(ξ )τ γ p α+σ +δ z 2 −3γ |ϑ| 1−2α τ 6 √ ˇ p J v(t, ˆ pξ1 ) + |v| ˆα , × t 1

(2.17)

where 6(ξ ) = 1 if ξ1 >

1 6

and ξ2 > 16 and ˇ 6(ξ ) = 0 otherwise, Jφ(t, q) = ∂q φ(t, q).

The derivatives 0ξ2 and 0ξ3 are estimated in the same way. Then, by the Cauchy– Schwarz inequality, we have 3 |0(t, p, ξ )| + (|z| + |y|) 0ξ1 (t, p, ξ ) + |M1 | Cp dy dz + 0ξ2 (t, p, ξ ) + 0ξ3 (t, p, ξ ) (1 + τ |y|3 )(1 + τ |z|3 ) √ z dy dz ω ˆ + ˆ α |ϑ|σ |w| ˆ δ Cp |v| 3 3 |y| 16 +z (1 + τ |y| )(1 + τ |z| )

210


3 z 2 dz dy + 1 1 + τ |z|3 1 1 + τ |y|3 z>− 6 |y±z|< 6 1 z3−3γ dz 2 dy + 1 + τ |y|3 z|y|− 16 (1 + τ |z|3 )2 +

ˆ σ |w| Cτ 2 −1 p ω |v| ˆ α |ϑ| ˆ δ. γ

We estimate the second summand M2 by the Cauchy–Schwarz inequality to get |y| dy dz 3 |M2 | Cp |0ξ1 (t, p, ξ )| 1 + τ |z|y 2 12 y 2 dz ω ˆ Cp |v| ˆ α |ϑ|σ |w| ˆ δ + dy 2 2 z>|y|− 61 (1 + τ |z|y ) 3 γ z 2 dz ˆ σ |w| dy Cτ 2 −1 p ω |v| ˆ α |ϑ| ˆ δ. + 3 z>− 16 1 + τ |z| |y±z|< 61 The integral M3 is estimated analogously. Thus, in the case τ ∈ (0, 1), we have |M(t, p)|

Cp ω γ

1

τ 1− 2 τ 12

ˆ σ |w| |v| ˆ α |ϑ| ˆ δ.

(2.18)

Now consider the case τ 1. We have two stationary points: (1) y = 0, z = 0 and (2) y = 0, z = 23 . As above, we integrate by parts with respect to y to get M = M1 + M2 − M3 . Denote the function Z ∈ C2 (R) such that Z = z if z 16 , Z 16 if 16 z 12 and Z = 23 − z if z 12 . Then, in the first integral M1 , we √ change the variable of integration y 1 + 3z = η, integrate by parts with respect to z and use the identity eiτ Q =

1 ∂ iτ Q Ze Z − 9iτ Zz z − 23 ∂z

to get Z − 9iτ Zz z − 23 Z 0(t, p, ξ ) + e M1 = p Z − 9iτ Zz z − 23 3yZ 3Z 0(t, p, ξ ) + Z − 0ξ1 (t, p, ξ ) + + 2(1 + 3z) 2 (1 + 3z) 3yZ 0ξ2 (t, p, ξ ) + 2Z0ξ3 (t, p, ξ ) × + Z+ 2 (1 + 3z) 4iτ η2 dη dz × √ . (1 + 2iτ η2 )2 1 + 3z Z − 9iτ Zz z − 23

3

iτ Q

211


From the structure of the function 0, we have 1 ˆ σ |w| |0(t, p, ξ )| Cτ 6 p α+σ +δ |y| + |Z| |v| ˆ α |ϑ| ˆ δ and we estimate the derivatives 0ξ1 , 0ξ2 and 0ξ3 as above. Then, since 2 2 C 1 + τ Zz z − C 1 + τ Z2 , Z − 9iτ Zz z − 3 3 we get

|M1 | Cp

3 |y| 16 +z

dy dz |0(t, p, ξ )| + 2 2 (1 + τy )(1 + τ Z )

+ (|Z| + |y|) |0ξ1 (t, p, ξ )| + |0ξ2 (t, p, ξ )| + |0ξ3 (t, p, ξ )| √ √ ( |y| + |Z|)dy dz 1 ω ˆ σ |w| + ˆ α |ϑ| ˆ δ Cτ 6 p |v| 2 )(1 + τ Z2 z) 1 (1 + τy |y| 6 +z 12 z Z2 dz dy + + 1 + τy 2 (1 + τ Z2 z)2 12 z dz y 2 dy + + 1 + τ Z2 z (1 + τy 2 )2 |Z| z dz dy + 2 2 z>− 16 1 + τ |z|Z |y±z|< 61 1 + τy ˆ σ |w| ˆ α |ϑ| ˆ δ. Cτ − 12 p ω |v| 13

(2.19)

Consider the integral M2 . We make a change of variables of integration ξ1 = 1 − 23 ζ,

ξ2 =

ζ + η, 3

ξ3 =

ζ − η, 3

that is y=

ζ +η−1 , 2

z=

3η − ζ + 1 . 6

Then Q = ζ η2 + 23 ζ 2 − 19 ζ 3 − ζ. Thus we have M2 = p

3

0ξ1 (t, p, ξ )(ζ + η − 1)dη dζ

iτ Q

e

(2 + iτ (ζ + η − 1)2 )(1 + 12 (3η − ζ + 1))

We divide the domain of integration into two parts: (1) η integrate by parts with respect to η with the identity eiτ Q =

1 ∂ iτ Q He , A ∂η

1 2

.

and (2) η < 12 . We

212


where A = 1 + 2iτ ηH ζ , H = η − 1 if η p3 4

1 2

and H = η if η < 12 , to get

iτ ζ dζ eiτ Q 0ξ1 (t, p, ξ ) + (4 + τ 2 ζ 2 )B (ζ + η − 1)Bη (ζ + η − 1)Aη 3 iτ Q e 0ξ1 (t, p, ξ ) −1 + + + +p A B H dζ dη , + (ζ + η − 1) 0ξ1 ξ3 (t, p, ξ ) − 0ξ1 ξ2 (t, p, ξ ) AB

M2 = −

where B = 2 + iτ (ζ + η − 1)2 1 + 12 (3η − ζ + 1) , Bη = 2iτ (ζ + η − 1) 1 + 12 (3η − ζ + 1) + 32 iτ (ζ + η − 1)2 . Aη = 2iτ (η + H )ζ,

If the domain of integration is 1 2

− 3η ζ 54 ,

η − 14 ,

then we have 1 + 12 (3η − ζ + 1)

1 2

+

5 4

ζ − ζ + 32 η + − 16 12 , 3

which implies H A C |A| , |B| C 1 + τ (ζ + η − 1)2 , η (ζ + η − 1) H B C |B| (|H | + |ζ + η − 1|) . η

Therefore, |M2 | Cp

3

0 (t, p, ξ ) 1 ξ1 η=

2

dζ 2 + (1 + τ |ζ |) 1 + τ ζ − 12

(|H | + |ζ + η − 1|)|0ξ1 (t, p, ξ )| + + |H | |ζ + η − 1| |0ξ1 ξ3 (t, p, ξ )| + |0ξ1 ξ2 (t, p, ξ )| × dζ dη . × (1 + τ |ηH ζ |)(1 + τ (ζ + η − 1)2 )

+ Cp

3

As in (2.17), we have 0 (t, p, ξ ) C6(ξ )τ γ p α+σ +δ z 12 −3γ |ϑ| ˆ σ |w| ˆ δ × ξ1

τ 1−2α √ 6 | p Jˇ v(t, ˆ pξ1 )| × |v| ˆα+ t


and

0 (t, p, ξ ) ξ1 ξ2 C6(ξ )τ p γ

α+σ +δ

z

1 2 −3γ

213

ˆ σ+ |w| ˆ δ |v| ˆ α |ϑ|

τ 1−2σ

τ 1−2α √ √ ˆ 6 6 ˆ σ+ | pJˇ v(t, ˆ pξ1 )||ϑ| | p Jˇ ϑ(t, pξ2 )||v| ˆ α+ t t

τ 1−α−σ √ √ ˆ 3 | pJˇ v(t, ˆ pξ1 )|| pJˇ ϑ(t, pξ2 )| , + t +

the derivative |0ξ1 ξ3 (t, p, ξ )| is estimated in the same way. Hence, we obtain ˆ σ |w| ˆ α |ϑ| ˆ δ |M2 | Cp ω |v|

6

Ij ,

j =1

where I1 = I2 = I3 = I4 = I5 = I6 =

5 12 4 dζ dζ 1 +τ6 , 1 2 1 2 2 2 −1 (1 + τ |ζ |)(1 + τ (ζ − 2 ) ) −1 (1 + τ |ζ |) (1 + τ (ζ − 2 ) ) √ ∞ 5 4 |ζ + η − 1| ζ , dη dζ 1 (1 + τ |ηH ζ |)(1 + τ (ζ + η − 1)2 ) − 14 2 −3η √ ∞ 5 4 |H | ζ , dη dζ 1 (1 + τ |ηH ζ |)(1 + τ (ζ + η − 1)2 ) − 14 2 −3η 12 ∞ 5 4 (1 + H 2 )(ζ + η − 1)2 ζ 1 τ6 dη dζ , 1 (1 + τ |ηH ζ |)2 (1 + τ (ζ + η − 1)2 )2 − 14 2 −3η 12 ∞ 5 4 H 2 ζ 1 τ6 dη dζ , 1 (1 + τ |ηH ζ |)2 (1 + τ (ζ + η − 1)2 )2 − 14 2 −3η ∞ 5 12 4 H 2 (ζ + η − 1)2 ζ 1−6γ 1 τ3 dη dζ . 1 (1 + τ |ηH ζ |)2 (1 + τ (ζ + η − 1)2 )2 − 14 2 −3η

5 4

We estimate now each integral. We have 1 5 C 4 C 4 dζ dζ + + I1 τ −1 (1 + τ |ζ |) τ 41 1 + τ (ζ − 12 )2 1 12 5 12 4 4 C C dζ dζ C + 5 . + 5 3 2 1 1 (1 + τ (ζ − 2 )2 )2 τ2 τ 6 −1 (1 + τ |ζ |) τ6 4 Now we consider the integral √ 1 5 2 4 C ζ dη dζ + I2 √ 1 1 −γ 1 τ − 14 (τ η2 |ζ |) 2 (τ (ζ + η − 1)2 ) 2 −γ 2 −3η

214


C +√ τ

√

5 4

dη

C

1 2

dζ

1 2

1 2 −3η 5 4

dη

τ 2 −2γ 3

+

∞

dζ

C τ C

+

η1−2γ |ζ | 2 −γ |ζ + η − 1|1−2γ √ ∞ ζ dη dζ 1 1 η1−γ |η − 1|1−γ |ζ |1−γ 2 2 −3η

− 14

3 2 −2γ

ζ (τ η |η − 1| |ζ |)1−γ √ ζ

dη

3

τ 2 −2γ

1

1 2 −3η 5 4

η1−3γ C 3 . 1−γ 1−γ |η| |η − 1| τ 2 −2γ

In the same manner, we get I3 C

1 2

√ |η| ζ

5 4

+ 1 (τ η2 |ζ |)1−γ (τ (ζ + η − 1)2 ) 2 −γ √ ∞ |η − 1| ζ dη dζ +C 1 1 1 (τ η|η − 1||ζ |)1−γ (τ (ζ + η − 1)2 ) 2 −γ 2 2 −3η √ 5 1 2 4 C ζ 3 dη dζ 1−2γ 1−γ + 1 η |ζ | |ζ + η − 1|1−2γ τ 2 −2γ − 14 2 −3η √ ∞ 5 4 |η − 1|γ ζ C dη dζ 1−γ 1−γ + 3 1 η |ζ | |ζ + η − 1|1−2γ τ 2 −2γ 12 2 −3η C C η1−3γ 3 3 . dη 1−γ 1−γ |η| |η − 1| τ 2 −2γ τ 2 −2γ dη

− 14

dζ

1 2 −3η 5 4

Now we have I4 Cτ

− 13

∞

5 4

dη − 14

Cτ − 3 1

∞

1 2 −3η 5 4

dη

(τ |η|H |ζ |)1−γ (τ (ζ + η − 1)2 ) 2 −γ 12 ∞ 5 1 4 η 2 −6γ dζ dη 1 (|η|H |ζ |)1−γ |ζ + η − 1|1−2γ − 14 2 −3η γ −1 γ −1 1 1 13 |η| 2 H 2 |η − 1|γ − 2 η 2 −6γ dη Cτ γ − 12 ,

13

13

Cτ γ − 12

1

1 2 −3η

− 14

Cτ γ − 12

12 η1−6γ dζ (1 + τ |ηH ζ |)2 (1 + τ (ζ + η − 1)2 ) 12 1 η 2 −6γ dζ

(2.20)

analogously dividing the domain of integration in seven parts then we obtain I5 Cτ

1 6

1 2

dη 0

12

5 4 1−η 2

F1 dζ

+ Cτ

1 6

1 2

dη − 14

12

5 4 1 2 −3η

F2 dζ

+

215


+ Cτ

1 6

1 2

dη 0

+ Cτ + Cτ

1 6

1 6

12

1−η 2

1

dη 1 2 −3η 1−η 2

1 2

F3 dζ

1 2 −3η 1−η 2

12 F5 dζ

∞

dη 1 2 −3η

1

+ Cτ

1 6

+ Cτ

1 6

12 F4 dζ

1

dη

12

5 4

∞

dη

12

5 4

F5 dζ

1−η 2

1

+

F4 dζ

1−η 2

1 2

+

Cτ 2γ − 13 , 12

where F1 = F2 =

η2 ζ (τ η2 |ζ |)2−2γ (τ (ζ + η − 1)2 ) 2 −γ η2 ζ 1

,

, 1 (τ η2 |ζ |)2γ −2 (τ (ζ + η − 1)2 ) 2 −γ η2 ζ , F3 = (τ η2 |ζ |)1−γ τ 2 (ζ + η − 1)4 (η − 1)2 ζ , F4 = 1 (τ η(η − 1)2 )2−2γ (τ (ζ + η − 1)2 ) 2 −γ (η − 1)2 ζ . F5 = 3 (τ η|η − 1||ζ |)1−γ (τ (η − 1)2 ) 2 For the last integral we get 12 H 2 ζ 1−6γ dζ dη I6 Cτ 1 (1 + τ |ηH ζ |)2 (1 + τ (ζ + η − 1)2 ) − 14 2 −3η 12 12 1 0 5 5 2 4 4 − 16 − 16 Cτ dη F6 dζ + Cτ dη F6 dζ + − 16

∞

1−η 2

0

+ Cτ

− 16

1 2

+ Cτ + Cτ

− 16

1

1 2 −3η 1−η 2

12

dη 1

+ Cτ

F9 dζ 12

∞

+ Cτ

F7 dζ

1 2 −3η 1−η 2

dη 1 2

− 14

12

1−η 2

dη 0

− 16

5 4

1 2 −3η

F8 dζ

where F6 =

η2 ζ 1−6γ (τ η2 |ζ |) 2 −γ (τ (ζ + η − 1)2 ) 2 −γ 3

1

,

− 16

− 16

1 2 −3η

1

dη

1 2

dη 1 7

Cτ γ − 6 ,

1−η 2

∞

12

5 4

F8 dζ

12

5 4 1−η 2

+

F10 dζ

+

216


F7 = F8 =

η2 ζ 1−6γ , (τ η2 |ζ |)1−γ τ (ζ + η − 1)2 (η − 1)2 ζ 1−6γ

(τ η(η − 1)2 ) 2 −γ (τ (ζ + η − 1)2 ) 2 −γ (η − 1)2 ζ 1−6γ , F9 = (τ η|η − 1||ζ |)1−γ (τ (η − 1)2 )1−γ (η − 1)2 ζ 1−6γ . F10 = (τ η|η − 1||ζ |)1−γ τ (η − 1)2 3

1

,

Thus, we obtain ˆ σ |w| ˆ α |ϑ| ˆ δ. |M2 | Cτ 2γ − 12 p ω |v| 13

(2.21)

In the integral M3 , we make a change of variables ξ1 =

2 ξ2 = 1 − ζ, 3

ζ + η, 3

ξ3 =

ζ −η 3

that is y=−

ζ +η−1 , 2

z=

3η − ζ + 1 6

and then all the estimates are the same as for M2 , so we get the estimate 13 ˆ σ |w| ˆ α |ϑ| ˆ δ. |M3 | Cτ 2γ − 12 p ω |v|

(2.22)

From the estimates (2.18)–(2.22) the result of the lemma follows. Lemma 2.3 is proved. ✷ In the next lemma, we consider the asymptotic representation for the nonlinearity N (t, p) = p

3

e

2it 3 3 p Q

v(t, ˆ pξ1 )v(t, ˆ pξ2 )w(t, ˆ pξ3 ) dξ1 dξ2 ,

where Q = − 12 1 − ξ13 − ξ23 − ξ33 ,

t > 0,

p > 0,

ξ1 + ξ2 + ξ3 = 1.

ˆ δ < ∞, where LEMMA 2.4. Let the functions v, ˆ wˆ be such that the norms |v| ˆ 0 , |w| δ ∈ 0, 12 − 3γ ,

1 . γ ∈ 0, 50


217

Then the following representation is valid for all t > 1, p > 0: √

p p π 3p 3 3 2 − 8it 27 p v wˆ t, + e ˆ t, N (t, p) = − γ γ 3 3 (p 3 t)1− 2 p 3 t 2 iπp 3 2 2 w(t, ˆ p) + w(t, ˆ p) v ˆ (t, p) + ˆ p)| + γ γ 2|v(t, (p 3 t)1− 2 p 3 t 2 ˆ 20 |w| ˆ δ p 3+δ |v| . +O γ 1 (p 3 t)1− 2 p 3 t 12 −2γ Proof. There are four stationary points in the integral N : (1) ξ1 (2) ξ1 (3) ξ1 (4) ξ1

= 13 , ξ2 = 13 , ξ3 = 13 , = 1, ξ2 = 1, ξ3 = −1, = 1, ξ2 = −1, ξ3 = 1, = −1, ξ2 = 1, ξ3 = 1.

In view of the symmetry with respect to variables ξ1 , ξ2 , ξ3 we can write the representation N = 5j =1 Nj , where

p p 2it 3 wˆ t, e 3 p Q ψ1 (ξ ) dξ1 dξ2 , N1 (t, p) = p vˆ t, 3 3 ˆ p) + ˆ p)|2 w(t, N2 (t, p) = p 3 2|v(t, 2it 3 ˆ p)vˆ 2 (t, p) e 3 p Q ψ2 (ξ ) dξ1 dξ2 , + w(t, 2it 3 e 3 p Q 0j (t, p, ξ ) dξ1 dξ2 , j = 3, 4, 5, Nj (t, p) = p 3 3 2

the functions 0j (t, p, ξ ) are

p p wˆ t, ψ1 (ξ ) − ˆ pξ1 )v(t, ˆ pξ2 )w(t, ˆ pξ3 ) − vˆ 2 t, 03 (t, p, ξ ) = v(t, 3 3 ˆ p)ψ2 (ξ ) ψ3 (ξ ) , − vˆ 2 (t, p)w(t,

p p wˆ t, ψ1 (ξ ) − ˆ pξ3 )v(t, ˆ pξ1 )w(t, ˆ pξ2 ) − vˆ 2 t, 04 (t, p, ξ ) = v(t, 3 3 ˆ p)ψ2 (ξ ) ψ4 (ξ ), − |v(t, ˆ p)|2 w(t,

p p wˆ t, ψ1 (ξ ) − ˆ pξ2 )v(t, ˆ pξ3 )w(t, ˆ pξ1 ) − vˆ 2 t, 05 (t, p, ξ ) = v(t, 3 3 ˆ p)ψ2 (ξ ) ψ5 (ξ ), − |v(t, ˆ p)|2 w(t,

218


where ξ = (ξ1 , ξ2 , ξ3 ) and the functions ψj ∈ C2 (R3 ), j = 1, 2, 3, 4, 5 are such that 1 ψ1 (ξ ) = 1 ifξ1 − 13 + ξ2 − 13 < 10 and ψ1 (ξ ) = 0 if ξ1 − 13 + ξ2 − 13 > 15 ; ψ2 (ξ ) = 1 if |ξ1 − 1| + |ξ2 − 1| < ψj (ξ ) = 1 as ξ1 > 12 , ξ2 >

1 2

1 10

and ψ2 (ξ ) = 0 if |ξ1 − 1| + |ξ2 − 1| > 15 ;

and ψj (ξ ) = 0 as ξ1 < 16 , or ξ2
1, we get √

p p π 3 − 8it p3 1 3 2 27 N1 (t, p) = − 3 e wˆ t, +O v ˆ t, p p t p6 t 2 3 3 and

N2 (t, p) =

3 1 iπ +O ˆ p)|2 w(t, ˆ p) + p 3 w(t, ˆ p)vˆ 2 (t, p) . 2p |v(t, 3 6 2 p t p t

For the summands Nj , j = 3, 4, 5, we can write an estimate via Lemma 2.2 Nj (t, p)

p 3+δ |v| ˆ 20 |w| ˆ δ γ

1

(p 3 t)1− 2 p 3 t 12 −2γ

,

hence the result of the lemma follows. Lemma 2.4 is now proved.

✷

3. Proof of Theorems We first state the local existence theorem. THEOREM 3.1. Let u0 ∈ H1,1 (R). Then there exists a finite time interval [0, T ] with T > 0 such that there exists a unique solution u ∈ C([0, T ]; X) of the Cauchy problem (1.1) satisfying the estimate supt ∈[0,T ] ||u||X 2 u0 X . Moreover, if we assume that the norm of the initial data u0 1,1 is sufficiently small, then there exists a finite time interval [0, T ] with T > 1 and a unique solution u ∈ C([0, T ]; X) of the Cauchy problem (1.1) such that supt ∈[0,T ] ||u||X 2 u0 X . For the proof of Theorem 3.1, see [4–6, 12, 17–19, 25, 31]. Proof of Theorem 1.1. Let u be the local solution of the Cauchy problem (1.1) described in Theorem 3.1. Then we prove the following estimate: u(t) X < 20ε

(3.1)

219


1 ), ε = u0 1,1 . We prove the theorem by for any t ∈ [0, T ], where γ ∈ (0, 50 the method of contradiction. We assume that T1 > 1 is the maximal time such that estimate (3.1) is valid for t ∈ [0, T1 ), but is violated at t = T1. We first note ˆ 0)| = |uˆ 0 (0)| take place. We that the two conservation laws u = u0 and |u(t, differentiate Equation (1.1) with respect to x to get Lux = −a(t)(u3 )xx , where Lu = (∂t + 13 ∂x3 )u. Multiplying both sides of this equation by ux and integrating by parts, we obtain

d ux 2 C uux ∞ ux 2 . dt Using estimates (2.1), (2.2) of Lemma 2.1 and Theorem 3.1, we get u(t) β Cε t− 3 + 3β , 1

1

2

1

u(t)ux (t) ∞ Cε 2 t − 3 t− 3 ,

(3.2)

where 4 < β ∞. By the energy method we easily see that u 1,0 εtγ . A direct computation shows that I a(t)u3 x = 3ta (t)u3 + 3a(t)u2 Iux . Therefore applying the operator I to Equation (1.1) we find x Lu dx LIu = ILu + 3 −∞ = −3ta (t)u3 − a(t)I u3 x − 3a(t)u3 = −3ta (t)u3 − 3a(t)u2 (Iu)x . Hence, we get d 1 Iu 2 C uux Iu 2 + C t− 6 u3 . dt

(3.3)

Similarly, we find LIux = ILux + 3Lu = −3ta (t) u3 x − a(t)I u3 xx − 3a(t) u3 x = −3ta (t) u3 x − 3a(t) u2 (Iux )x + uux Iux + 2u2 ux . Multiplying both sides of the equation by Iux and integrating by parts, we obtain d Iux 2 C uux ∞ ( Iux + u ) Iux . dt

(3.4)

Applying (3.2) to (3.3)–(3.4) and using the Gronwall inequality for the resulting inequalities, we obtain the estimate Iu + Iux 2εtγ ,

(3.5)

220


thus by (3.2), (3.5) and Lemma 2.1, we get 1 Ju 1,0 Iu + Iux + u + Ct u3 + Ct u2 ux 4ε t 6 . (3.6) Multiplying both sides of (1.1) by G(−t), we get (G(−t)u(t))t + a(t)G(−t) u3 x = 0. Taking the Fourier transformation, we get p dζ1 dζ2 eit Q v(t, ˆ ζ1 )v(t, ˆ ζ2 )v(t, ˆ ζ3 ) = 0, vˆt (t, p) + i a(t) 2π where ζ3 = p − ζ1 − ζ2 ,

Q = − 13 p 3 − ζ13 − ζ23 − ζ33 ,

(3.7)

v (t) = G(−t)u(t).

ˆ p) since the solution u(t, x) is real, therefore it is sufWe have v(t, ˆ −p) = v(t, ficient to consider only the case p > 0. Changing the variables of integration ζj = pξj and applying Lemma 2.2 to (3.7), we get the following equation for the function v(t, ˆ p) for all p > 0, t > 1 √

p 3a(t)p 3 3 3 − 8it 27 p v − ˆ t, vˆt (t, p) = γ γ e 3 2(p 3 t)1− 2 p 3 t 2 3ia(t)p 3

ˆ p)|2 v(t, ˆ p) + γ γ |v(t, 2(p 3 t)1− 2 p 3 t 2 ˆ 30 p 3 |v| . +O γ 1 (p 3 t)1− 2 p 3 t 12 −2γ

−

(3.8)

To get rid of the second summand in the right-hand side of Equation (3.8), we make a change of the dependent variable vˆ = f Ev , with 3i t a(τ )p 3 dτ 2 |v(τ, ˆ p)| . Ev (t, p) = exp − γ γ 2 1 (p 3 τ )1− 2 p 3 τ 2 Then integrating the resulting equation with respect to t, we obtain t

p a(τ )p 3 dτ 8iτ 3 Ev (τ )e− 27 p vˆ 3 τ, f (t) = f (1) − C γ γ + 3 (p 3 τ )1− 2 p 3 τ 2 1 t 3 p dτ . + O ε3 γ 1 1 (p 3 τ )1− 2 p 3 τ 12 −2γ Therefore we get F (G(−t)u(t)) ∞ = f ∞ t

p 8iτ 3 10ε + C Ev (τ )e− 27 p vˆ 3 τ, 3 1

a(τ )p 3 dτ γ

γ

(p 3 τ )1− 2 p 3 τ 2

.

(3.9)


221

To estimate the last integral we integrate by parts using the identity 8iτ

e− 27 p =

1

3

1−

8iτ 3 p 27

d − 8iτ p3 τ e 27 . dτ

Then we have, for all 1 s t, p > 0, t

p a(τ )p 3 dτ 8iτ 3 Ev (τ )e− 27 p vˆ 3 τ, γ γ 3 (p 3 τ )1− 2 p 3 τ 2 s t 8iτ 3 a(τ )Ev (τ )e− 27 p vˆ 3 τ, p3 = − 1 − 8iτ p3 27 s 8i 3 3 t 8iτ 3 p τp a(τ )vˆ 3 τ, p3 27 Ev (τ )e− 27 p − γ γ + 8iτ 3 8iτ 1− 1 − 27 p 1 − 27 p 3 (p 3 τ ) 2 p 3 τ 2 s

p p 3a(τ )τp 3 2 vˆτ τ, − ˆ τ, + γ γ v 3 3 (p 3 τ )1− 2 p 3 τ 2

p 3iπ a 2 (τ )p 3 τp 3 |v(τ, ˆ p)|2 + − 3 2−γ 3 2γ vˆ 3 τ, (p τ ) p τ 3

a(τ ) p 3 d 3 dτ. vˆ τ, + τp γ γ dτ (p 3 τ )1− 2 p 3 τ 2 3

(3.10)

From Equation (3.8) we have the estimate vˆt (t, p) ∞ Cεt −1 . Hence, by (3.10), t

p a(τ )p 3 dτ 3 3 p Ev (τ )e− 8iτ 27 vˆ τ, γ γ 1− 3 (p 3 τ ) 2 p 3 τ 2 s t Cε 3 p 3 dτ 3 . (3.11) Cε γ 1− γ2 p 3 s p 3 τ 1+ 2 s (p 3 τ ) Estimates (3.6), (3.9) and (3.11) give us estimate (3.1). By virtue of estimate (3.1), we can continue the local solution u(t, x) for all t > 0. Theorem 1.1 is proved. ✷ Proof of Theorem 1.2. We denote w(t, x) = u(t, x) − S(t, x), where x 1 ϕ √ S(t, x) = √ 3 3 t t is the self-similar solution described in the Introduction. Let us prove the following estimate: w(t) Yδ < 30ε for any t ∈ [0, T ], where 1 , ε = u0 1,1 . δ ∈ 0, 12 − 3γ , γ ∈ 0, 50

(3.12)

222


In the same way, as in the proof of (3.1), we assume that T1 > 1 is the maximal time such that estimate (3.12) is true for all t ∈ [0, T1 ), but it is violated at time t = T1 . In view of the self-similar structure of S(t, x), we have IS = 0 and, hence, x dxL S = 3tAS 3 . JS = I − 3t −∞

From Equation (1.1), we get Lw = A w 3 − 3uw 2 + 3u2 w x − (a (t) − A) u3 x . Via (3.5) and Lemmas 2.1 and 2.2, we obtain 1 Jw Iu + Ct w 3 + uw 2 + u2 w + Ct − 6 u3 12εt γ for all t 1. Since w (t, x) dx = rˆ (t, 0) = 0

(3.13)

for all t > 1,

where r = G(−t)w, we have |ˆr (t, p)| = |ˆr (t, p) − rˆ (t, 0)| p √ |Jˇ rˆ (t, p)|dp p Jw 0 1

δ

1

δ

|p|δ p 3 t 6 − 3 t −γ Jw Cε|p|δ p 3 t 6 − 3 , hence, in view of (3.13), estimate (3.12) follows. Now as in (3.7) we obtain p ˆ ζ2 )ϑ(t, ˆ ζ3 ) = 0, ˆ ζ1 )ϑ(t, dζ1 dζ2 eit Q ϑ(t, ϑˆ t (t, p) + iA 2π ˆ where ϑ(t) = F G(−t)S(t). We now make a change of dependent variables vˆ = ˆ f Ev , ϑ = gEϑ , where t 3 p 3 dτ 2 |v(τ, ˆ p)| , Ev (t, p) = exp − iA γ γ 2 (p 3 τ )1− 2 p 3 τ 2 1 t 3 p 3 dτ 2 ˆ |2π ϑ(τ, p)| . Eϑ (t, p) = exp − iA γ γ 2 (p 3 τ )1− 2 p 3 τ 2 1 Then, for the difference h = f − g, we get ht (t, p)

ˆ ζ1 )v(t, ˆ ζ2 )v(t, ˆ ζ3 ) − dζ1 dζ2 eit Q v(t, = ip (A − a (t)) Ev ˆ ζ2 )v(t, ˆ ζ3 ) − ˆ ζ1 )v(t, dζ1 dζ2 eit Q v(t, − iApEv

223


ˆ ζ2 )ϑ(t, ˆ ζ3 ) ˆ ζ1 )ϑ(t, − ϑ(t, 3iπ Ap 3 2 2ˆ ˆ − v(τ, ˆ p) − | ϑ(τ, p)| ϑ(τ, p) − | v(τ, ˆ p)| γ γ (p 3 τ )1− 2 p 3 τ 2 ˆ ζ2 )ϑ(t, ˆ ζ3 ) − ˆ ζ1 )ϑ(t, dζ1 dζ2 eit Q ϑ(t, − ipA(Ev − Eϑ ) 3iπ Ap 3 2 ˆ ˆ − γ γ |ϑ(τ, p)| ϑ(τ, p) , (p 3 τ )1− 2 p 3 τ 2 Hence, as in the proof of Theorem 1.1 applying Lemma 2.4 and estimates (3.1), (3.10), (3.11) and (3.12), we get |h(t) − h(s)|

Cε 3 |p|δ 1

p 3 s 12 −3γ

|f (t) − f (s)|

,

Cε 3 1

p 3 s 12 −3γ

and |g(t) − g(s)|

Cε 3

(3.14)

p 3 s 12 −3γ 1

for all t > s > 1. Therefore, there exist limits F (p) = lim f (t, p)

H (p) = lim h(t, p), t →∞

t →∞

and G (p) = lim g(t, p) t →∞

with the estimates |H (p)| Cε 3 |p|δ+3γ − 12 , 1

|F (p)| Cε 3

and

|G (p)| Cε 3 .

Then, using the estimates, |h (t, p) − H (p)| |g (t, p) − G (p)|

Cε 3 |p|δ 1

,

1

,

p 3 t 12 −3γ Cε 3 p 3 t 12 −3γ

|f (t, p) − F (p)|

Cε 3

we obtain rˆ = f Ev − gEϑ = f (Ev − Eϑ ) + hEϑ = H Eϑ + F (Ev − Eϑ ) + (h − H ) Eϑ − (F − f ) (Ev − Eϑ ) 1 = Eϑ H + F (Ev Eϑ − 1) + O ε 3 t 4γ − 12 . Note that

1

p 3 t 12 −3γ γ |Ev − Eϑ | Cε 2 |p|δ p 3 t

ˆ p)|2 = |g (t, p)|2 = |G (p)|2 + O ε 2 p 3 t3γ − 121 . |ϑ(t,

,

224


We now denote 3 @1 (t) = − iA 2

t

ˆ p)|2 ˆ |ϑ(τ, p)|2 − |ϑ(t,

1

p 3 dτ γ

γ

(p 3 τ )1− 2 p 3 τ 2

and @2 (t) =

3 iA 2

t

ˆ p)|2 + ˆ ˆ p)|2 − |ϑ(t, |ϑ(τ, p)|2 − |v(τ,

1

+ |v(t, ˆ p)|2

p 3 dτ γ

γ

(p 3 τ )1− 2 p 3 τ 2

.

Since ˆ p)|2 = |g|2 , |ϑ(t,

|v(t, ˆ p)|2 = |f |2 ,

we get @1 (t) − @1 (s) t p 3 dτ 3 ˆ p)|2 ˆ |ϑ(τ, p)|2 − |ϑ(t, = − iA γ γ + 2 (p 3 τ )1− 2 p 3 τ 2 s s p 3 dτ 3 ˆ p)|2 − |ϑ(s, ˆ p)|2 + iA |ϑ(t, γ 1− γ2 2 p 3 τ 2 1 (p 3 τ ) and 3 @2 (t) − @2 (s) = − iA 2

t

|f (τ, p)|2 − |g(τ, p)|2 −

s

− |f (t, p)|2 + |g(t, p)|2

p 3 dτ γ

γ

(p 3 τ )1− 2 p 3 τ 2

+

3 + iA |f (t, p)|2 − |g(t, p)|2 − 2 s p 3 dτ − |f (s, p)|2 + |g(s, p)|2 γ 1− γ 1 (p 3 τ ) 2 p 3 τ 2 for all 1 < s < τ < t. Using estimates (3.14) we get 4γ − 121 , @j (t) − @j (s) ∞ Cε 2 p 3 s

j = 1, 2.

(3.15)

Therefore by (3.15), we see that there exist unique functions 0j ∈ L∞ , such that i0j = limt →∞ @j (t) and 4γ − 121 , i0j − @j (t) ∞ Cε p 3 t

j = 1, 2.

(3.16)


225

By (3.15), (3.16), we find t p 3 dτ 3 ˆ |ϑ(τ, p)|2 − iA γ γ 2 (p 3 τ )1− 2 p 3 τ 2 1 3 1 = iA |G (p)|2 log tp 3 + i03 + O ε 2 (tp 3 )4γ − 12 2 and

t 3 p 3 dτ |g(τ )|2 − |f (τ )|2 − iA γ γ 2 (p 3 τ )1− 2 p 3 τ 2 1 3 1 = iA |H |2 − 2H F log tp 3 + i04 + O ε 2 t 4γ − 12 2

with some functions 03 (p) and 04 (p) ∈ L∞ . Hence 2 3 4γ − 1 3 2 3 12 Eϑ = exp − iA|G(p)| log tp + i03 + O ε p t 2 and

2 4γ − 1 3 2 3 12 . Eϑ Ev = exp iA |H | − 2H F log tp + i04 + O ε t 2

Thus 3

rˆ = e− 2 iA|G(p)| log tp 1 + O ε 3 t 4γ − 12 2

3 +i0 3

3 2 3 H + F e 2 iA(|H | −2H F ) log tp +i04 − 1 +

1 3 3 = H1 eiB1 log tp + H2 eiB2 log tp + O ε 3 t 4γ − 12 .

The asymptotic formula (1.2) follows now from the asymptotics (2.15) of Lemma 2.2. This completes the proof of Theorem 1.2. ✷

References 1. 2. 3.

4. 5.

Ablowitz, M. J. and Segur, H.: Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. Bona, J. L. and Saut, J.-C.: Dispersive blow-up of solutions of generalized Korteweg–de Vries equation, J. Differential Equations 103 (1993), 3–57. de Bouard, A., Hayashi, N. and Kato, K.: Gevrey regularizing effect for the (generalized) Korteweg–de Vries equation and nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Anal. Non Linéaire 12 (1995), 673–725. Christ, F. M. and Weinstein, M. I.: Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation, J. Funct. Anal. 100 (1991), 87–109. Constantin, P. and Saut, J.-C.: Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), 413–446.

226 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19.

20. 21. 22. 23. 24. 25. 26. 27. 28. 29.


Craig, W., Kapeller, K. and Strauss, W. A.: Gain of regularity for solutions of KdV type, Ann. Inst. H. Poincaré, anal. non linéaire 9 (1992), 147–186. Deift, P. and Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Ann. Math. 137 (1992), 295–368. Dix, D. B.: Large-time Behavior of Solutions of Linear Dispersive Equations, Lecture Notes in Math. 1668, Springer, Berlin, 1997. Dix, D.: Temporal asymptotic behavior of solutions of the Benjamin–Ono equation, J. Differential Equations 90 (1991), 238–287. Fedoryuk, M. V.: Asymptotic methods in analysis, Encycl. of Math. Sciences 13, SpringerVerlag, New York, 1987, pp. 83–191. Fedoryuk, M. V.: Asymptotics: Integrals and Series, Nauka, Moscow, 1987. Ginibre, J., Tsutsumi, Y. and Velo, G.: Existence and uniqueness of solutions for the generalized Korteweg–de Vries equation, Math. Z. 203 (1990), 9–36. Hayashi, N.: Analyticity of solutions of the Korteweg–de Vries equation, SIAM J. Math. Anal. 22 (1991), 1738–1745. Hayashi, N. and Naumkin, P. I.: Large time asymptotics of solutions to the generalized Benjamin–Ono equation, Trans. Amer. Math. Soc. 351(1) (1999), 109–130. Hayashi, N. and Naumkin, P. I.: Large time asymptotics of solutions to the generalized Korteweg–de Vries equation, J. Funct. Anal. 159 (1998), 110–136. Hayashi, N. and Naumkin, P. I.: Large time behavior of solutions for the modified Korteweg–de Vries equation, Internat. Math. Res. Notices 8 (1999), 395–418. Kato, T.: On the Cauchy problem for the (generalized) Korteweg–de Vries equation, In: V. Guillemin (ed.), Advances in Mathematics Supplementary Studies, Stud. in Appl. Math. 8, Berlin, 1983, pp. 93–128. Kenig, C. E., Ponce, G. and Vega, L.: On the (generalized) Korteweg–de Vries equation, Duke Math. J. 59 (1989), 585–610. Kenig, C. E., Ponce, G. and Vega, L.: Well-posedness and scattering results for the generalized Korteweg–de Vries equation via contraction principle, Comm. Pure Appl. Math. 46 (1993), 527–620. Klainerman, S.: Long time behavior of solutions to nonlinear evolution equations, Arch. Rat. Mech. Anal. 78 (1982), 73–89. Klainerman, S. and Ponce, G.: Global small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), 133–141. Kruzhkov, S. N. and Faminskii, A. V.: Generalized solutions of the Cauchy problem for the Korteweg–de Vries equation, Math. USSR, Sb. 48 (1984), 391–421. Naumkin, P. I. and Shishmarev, I. A.: Asymptotic behavior as t → ∞ of solutions of the generalized Korteweg–de Vries equation, Math. RAS, Sb. 187(5) (1996), 695–733. Ponce, G. and Vega, L.: Nonlinear small data scattering for the generalized Korteweg–de Vries equation, J. Funct. Anal. 90 (1990), 445–457. Saut, J.-C.: Sur quelque généralisations de l’ equation de Korteweg–de Vries, J. Math. Pure Appl. 58 (1979), 21–61. Sidi, A., Sulem, C. and Sulem, P. L.: On the long time behavior of a generalized KdV equation, Acta Appl. Math. 7 (1986), 35–47. Shatah, J.: Global existence of small solutions to nonlinear evolution equations, J. Differential Equations 46 (1982), 409–425. Staffilani, G.: On the generalized Korteweg–de Vries equation, Differential Integral Equations 10 (1997), 777–796. Strauss, W. A.: Dispersion of low-energy waves for two conservative equations, Arch. Rat. Mech. Anal. 55 (1974), 86–92.


30. 31.

227

Strauss, W. A.: Nonlinear scattering theory at low energy, J. Funct. Anal. 41 (1981), 110–133. Tsutsumi, M.: On global solutions of the generalized Korteweg–de Vries equation, Publ. Res. Inst. Math. Sci. 7 (1972), 329–344.


229

Inverse Spectral Results for AKNS Systems with Partial Information on the Potentials R. DEL RIO1 and B. GRÉBERT2

1 IIMAS-UNAM, Circuito Escolar, Ciudad Universitaria, 04510 México, D.F., México.

e-mail: [email protected] 2 UMR 6629, Département de Mathématiques, Université de Nantes, 2 rue de la Houssinière, 44322 Nantes Cedex 03, France. e-mail: [email protected] (Received: 1 March 2001; in final form: 24 August 2001) Abstract. For the AKNS operator on L2 ([0, 1], C2 ) it is well known that the data of two spectra uniquely determine the corresponding potential ϕ a.e. on [0, 1] (Borg’s type Theorem). We prove that, in the case where ϕ is a-priori known on [a, 1], then only a part (depending on a) of two spectra determine ϕ on [0, 1]. Our results include generalizations for Dirac systems of classical results obtained by Hochstadt and Lieberman for the Sturm–Liouville case, where they showed that half of the potential and one spectrum determine all the potential functions. An important ingredient in our strategy is the link between the rate of growth of an entire function and the distribution of its zeros. Mathematics Subject Classifications (2000): 34A55, 34B05, 34L40, 47E05, 47A10, 47A75. Key words: AKNS systems, determination of coefficients, inverse problems, selfadjoint operator, spectral theory.

1. Introduction We study problems related to classical results by Borg [2] and Hochstadt and Lieberman [10]. A vast amount of literature exists on this type of inverse problems for the Sturm–Liouville operator, cf. [4] and references quoted therein. In this note we want to address similar results for AKNS systems. Actually, we use a different approach than in [4], in particular we do not use the Titchmarsh–Weyl function and Marchenko’s theorem. Historically, Gesztesy and Simon first deviated from the Hochstadt–Lieberman approach by linking the length on which the potential was known to a portion of eigenvalue spectra needed to recover the potential on the whole interval in question, see [5, 6]. For related interesting results, see [3]. For ϕ ∈ L2 ([0, 1], C) we define the AKNS operator on L2 ([0, 1], C2 ) by −q(x) p(x) 0 −1 d + , (1) H (ϕ) := p(x) q(x) 1 0 dx where ϕ = q − ip and q and p are real valued.

230

´ R. DEL RIO AND B. GREBERT

Notice that H (ϕ) is unitarily equivalent to the Zakharov–Shabat operator, d 0 ϕ 1 0 + , L(ϕ) := i ϕ¯ 0 0 −1 dx

(2)

where ϕ¯ is the complex conjugate of ϕ. For each α ∈ [0, π ) we consider σ (ϕ, α) the spectrum of the selfadjoint opera tor H (ϕ) with domain, F = ZY ∈ H 1 ([0, 1], C2 ) such that Z(0) = 0,

cos α Y (1) − sin α Z(1) = 0.

(3)

Following [7] (cf. also [8]), σ (ϕ, α) is a sequence of real numbers (µn (ϕ, α))n∈Z satisfying µn < µn+1 (n ∈ Z) and µn = α + nπ − π/2 + l 2 (n). By ϕ|[a,b] , we shall denote the restriction of ϕ to the interval [a, b], that is ϕ|[a,b] (x) = ϕ(x) if x ∈ [a, b]. Our main result is the following theorem: THEOREM 1. Let ϕ ∈ L2 ([0, 1], C), α, β ∈ [0, π ) with α = β, 0 a 1, l, k ∈ N ∪ {∞} with 1l + 1k 2a. Then {µln (α), µkn (β) | n ∈ Z} and ϕ|[a,1] uniquely determine ϕ a.e. on [0, 1] and α, β. For particular values of a, k, l, we obtain for the AKNS systems − Borg type Theorem: two spectra uniquely determine ϕ on [0, 1] (a = 1, l = k = 1). − Hochstadt–Liebermann type Theorem: one spectrum and ϕ on [1/2, 1] uniquely determine ϕ on [0, 1] (a = 1/2, l = 1, k = ∞) (cf. [1] for an other proof of this result). Actually our Theorem includes many more general results such as, for example, − Half of one spectrum and ϕ on [1/4, 1] uniquely determine ϕ on [0, 1] (a = 1/4, l = 2, k = ∞). − Half of two spectra and ϕ on [1/2, 1] uniquely determine ϕ on [0, 1] (a = 1/2, l = k = 2). For the shake of simplicity, we only consider the case of two different boundary conditions. However, the same method of proof applies to more general situations, for instance, considering three spectra, with obvious notation the condition would be 1 1 1 + + 2a. k1 k2 k3 In the same way we prove the following theorem: a = b + l 2 (n) means that 2 n n n∈Z |an − bn | < ∞. If l (resp. k) equals ∞ we shall understand that {µ (α) | n ∈ Z} (resp. {µ (β) | n ∈ Z}) is ln kn

empty.

231

INVERSE SPECTRAL RESULTS FOR AKNS

β, 0 a 1, THEOREM 2. Let ϕ ∈ L2 ([0, 1], C), α, β ∈ [0, π ) with α = 1 1 l, k ∈ N ∪ {∞} with l + k 4a. Then {µln (α), µkn (β) | n 0} and ϕ|[a,1] uniquely determine ϕ a.e. on [0, 1] and α, β. Roughly speaking, Theorem 2 means that the ‘positive’ part of one spectrum (µn )n0 gives the same information as (µ2n )n∈Z . Of course, the data ϕ|[a,1] in Theorems 1 and 2 can be replaced by ϕ|[0,1−a]. Nevertheless, the interval where ϕ is a-priori known must contain 0 or 1. Actually, even the data of Re ϕ on [0, 1], Im ϕ on [1/2 − ε, 1/2 + ε] (ε ∈ (0, 1/2) arbitrary) and one spectrum do not uniquely determine ϕ on [0, 1]. Let us construct a counterexample: From [8, Propositions 1.3 and 1.4] we learn that for any π π 2 ˜ ϕ ∈ L ([0, 1], C) and n ∈ Z, µn ϕ, = µn ϕ, , 2 2 where ϕ(x) ˜ = ϕ(1 ¯ − x). Let ϕ = q − ip with q even, p(1 − x) = −p(x) for x ∈ [1/2 − ε, 1/2 + ε] and p(1 − x) = p(x) for x ∈ [0, 1/2 − ε). Define ψ by ψ(x) := q(x) + ip(1 − x). Then, σ (ϕ, π/2) = σ (ψ, π/2), Im ϕ = Im ψ

Re ϕ = Reψ

in [0, 1],

on [1/2 − ε, 1/2 + ε],

nevertheless ϕ ≡ ψ. Similar construction shows that the data of Re ϕ on [0, 1], Im ϕ on [0, 1]\(1/2− ε, 1/2 + ε) (ε ∈ (0, 1/2) arbitrary) and one spectrum do not uniquely determine ϕ on [0, 1]. Our fundamental strategy can be described as follows. Let ϕ and ψ satisfy the conditions of Theorem 1. (a) We construct an entire function f (λ, ϕ, ψ) (cf. Section 2.1) which vanishes at each µln (α) and µkn (β), n ∈ Z. (b) The partial information on the potential allows us to bound the exponential type of f obtaining |f (λ)| = o(e2a|Im λ| ) as |λ| → ∞ (cf. Section 2.2). (c) We use the general principle that the growth of an entire function is related to the distribution of its zeros to prove that steps (a) and (b) and the condition 1 + 1k 2a imply f ≡ 0 (cf. Section 2.3). l (d) f ≡ 0 imply ϕ ≡ ψ (cf. Section 2.4). This is a generalization of the fact that the rate of growth of a polynomial is given by its degree

or, equivalently, by its number of roots.

232


Notice that the spectral problem is well posed for ϕ ∈ L1 ([0, 1], C2 ). However, in this case the asymptotic µn = α + nπ − π/2 + l 2 (n) is not satisfied and then we cannot use Lemma 5 (see Section 2.3 on infinite products). Furthermore, working with L2 kernels greatly facilitates the proof of Proposition 9 below. Therefore, the condition ϕ ∈ L2 ([0, 1], C2 ) in Theorems 1 and 2 is possibly not optimal but is needed for our approach. 2. Proofs 2.1. CONSTRUCTION OF THE ENTIRE FUNCTION

f

For ϕ ∈ L2 ([0, 1], C) and λ ∈ C, let Y (·, λ, ϕ) F (·, λ, ϕ) ≡ ∈ H 1 ([0, 1], C2 ) Z(·, λ, ϕ) the unique vector-valued function satisfying H (ϕ)F = λF and F (0, λ, ϕ) = be 1 . 0 For each 0 x 1 and ϕ ∈ L2 ([0, 1]), C), λ → F (x, λ, ϕ) is entire (cf. [7] or [8] for the construction of F ). Notice that the spectrum σ (ϕ, α) is the root’s set of cos α Y (1, λ, ϕ) − sin α Z(1, λ, ϕ) = 0. Let

G≡

G1 (x, λ, ϕ) G2 (x, λ, ϕ)

(4)

with G1 (x, λ, ϕ) = Z(x, λ, ϕ) − iY (x, λ, ϕ), G2 (x, λ, ϕ) = Z(x, λ, ϕ) + iY (x, λ, ϕ).

(5)

One has L(ϕ)G(·, λ, ϕ) = λG(·, λ, ϕ)

(6)

and, furthermore, for λ ∈ R, ¯ 1 (x, λ, ϕ) = G2 (x, λ, ϕ). G Let ϕ, ψ ∈ L2 ([0, 1], C), from (6) the cross-product, A(λ) := G(·, λ, ψ), (L(ϕ) − λ)G(·, λ, ϕ) − − G(·, λ, ϕ), (L(ψ) − λ)G(·, λ, ψ) vanishes for all λ ∈ C.

(7)

233


On the other hand, by direct calculation using (7), one obtains for λ ∈ R A(λ) = G(ψ), L(ϕ)G(ϕ) − G(ϕ), L(ψ)G(ψ) 1 1 ¯ dx + G2 (ψ)G2 (ϕ)(ϕ − ψ) dx + G1 (ψ)G1 (ϕ)(ϕ¯ − ψ) = 0 0 1 d −G1 (ψ)G2 (ϕ) + G2 (ψ)G1 (ϕ) dx. +i 0 dx Thus defining for λ ∈ C, 1 G2 (x, λ, ψ)G2 (x, λ, ϕ)(ϕ(x) − ψ(x)) dx + f (λ, ϕ, ψ) := 0 1 ¯ + G1 (x, λ, ψ)G1 (x, λ, ϕ)(ϕ(x) ¯ − ψ(x)) dx,

(8)

0

one gets for λ ∈ R,

1 f (λ, ϕ, ψ) = −i W (G(ϕ), G(ψ)) 0 .

(9)

LEMMA 3. Let ϕ, ψ ∈ L2 ([0, 1], C) and α ∈ [0, π ). Assume that µ ∈ σ (ϕ, α) ∩ σ (ψ, α). Then f (µ, ϕ, ψ) = 0. Proof. In view of formula (9), one has to prove that [W (G(·, µ, ϕ), G(·, µ, ψ))]10 = 0. Since G(0, µ) = −i , one has +i W (G(0, µ, ϕ), G(0, µ, ψ)) = 0. On the other hand W G(1, µ, ϕ), G(1, µ, ψ) = 2i Z(1, µ, ϕ)Y (1, µ, ψ) − Y (1, µ, ϕ)Z(1, µ, ψ) = 0, where we used that (cf. (4)) cos α Y (1) − sin α Z(1) = 0.

✷

For simplicity, for i = 1, 2, we introduce gi (x, λ) ≡ gi (x, λ, ϕ, ψ) := Gi (x, λ, ϕ)Gi (x, λ, ψ). Then formula (8) becomes, with r = ϕ − ψ, 1 g2 (x, λ)r(x) + g1 (x, λ)¯r (x) dx. f (λ, ϕ, ψ) =

(10)

(11)

0

Notice that since λ → F (x, λ, ϕ) is entire, λ → f (λ, ϕ, ψ) is entire too. 2.2. ORDER AND TYPE OF THE ENTIRE FUNCTION

f

In this section we shall prove that the entire function f defined in (11) satisfies the following lemma:

234


LEMMA 4. Let 0 a 1. Assume ϕ(x) = ψ(x) for x ∈ [a, 1], then f (λ, ϕ, ψ) = o(e2|Im λ|a ) when |λ| → ∞. Proof. From [7] (see also [8]) we learn that, uniformly for x ∈ [0, 1], one has when |λ| → ∞, cos λx F (x, λ) − = o(e|Im λ|x ). (12) − sin λx Therefore, we get from (10) and (5) 2 g2 (x, λ, ϕ, ψ) = ieiλx + o(e|Im λ|x ) = −e2iλx + o(e2|Im λ|x ) and g1 (x, λ, ϕ, ψ) = −e−2iλx + o(e2|Im λ|x ). Using (11), we obtain (with r = ϕ − ψ) 1 2iλx + o(e2|Im λ|x ) r(x) dx + −e f (λ, ϕ, ψ) = 0 1 −2iλx + o(e2|Im λ|x ) r¯ (x) dx. −e + 0

Thus, if ϕ ≡ ψ on [a, 1], a f (λ) = − r(x)e2iλx dx − 0

a

r¯ (x)e−2iλx dx + o(e2|Im λ|a ),

(13)

0

where we used that the error term o(e2|Im λ|x ) is uniform in x ∈ [0, 1] and that r ∈ L1 ([0, 1]). For α ∈ C 1 ([0, 1]) one obtains integrating by parts,

a 2|Im λ|a

e 2iλx

. α(x)e dx = O

|λ| 0 Now for α ∈ L2 ([0, 1]) and ε > 0 let αε in C 1 ([0, 1]) such that αε − αL2 < ε/2. There exists A > 0 such that, for |λ| > A,

a

ε

2iλx

αε (x)e dx

e2|Im λa|

2 0

and thus

a

2iλx

α(x)e dx

0

a 2iλx

αε (x)e 0

dx

+

a

|α − αε |e2|Im λ|x dx

0

εe2|Im λa| . Therefore, one has proved (cf. [13, Problem 3, p. 15]) that a α(x)e2iλx dx = o(e2|Im λ|a ) 0

for α ∈ L2 ([0, 1]). Thus, using (13), f (λ) = o(e2a|Im λ| ).

✷


235

Remark.. We have proved that f is an entire function of order not greater than 1 and type not greater than 2a. We are going to use that such a function cannot have ‘many’ zeros (cf. [11]).

2.3. INFINITE PRODUCT REPRESENTATION We begin with three auxiliary Lemmas on infinite products. Given a sequence of complex numbers (ak )k∈Z , we say that the product k∈Z ak is convergent if the limit limN→∞ |k|N ak exists. In such a case we write ak := lim ak . k∈Z

N→∞

|k|N

LEMMA 5. Let (zn )n∈Z a complex sequence satisfying zn = nπ + l 2 (n). Then the formula zn − λ h(λ) := (λ − z0 ) nπ n∈Z\{0} defines and entire function satisfying uniformly for (n + 1/6)π |λ| (n + 5/6)π , h(λ) = sin λ(1 + o(1)),

n → +∞.

(14)

Lemma 5 is proved in [8, Lemma I-16] (cf. [7] and [13]), but the uniformity of (14) is proved only for (n + 1/4)π |λ| (n + 3/4)π . The generalization to (n + 1/6)π |λ| (n + 5/6)π is straightforward (actually, the only important fact is that |λ| is far away from nπ (n 0)). From Lemma 5 follows: LEMMA 6. Let (zn )n∈Z a sequence of complex numbers satisfying zn = nπ + l 2 (n) and k 1. Then the formula znk − λ hk (λ) := (λ − z0 ) nkπ n∈Z\{0} defines an entire function satisfying uniformly on k(n + 1/6)π |λ| k(n + 5/6)π λ (1 + o(1)), n → +∞. hk (λ) = k sin k Proof. For n ∈ Z we define, z˜ n = znk /k. One has k z˜ n − λ . hk (λ) = (λ − k z˜ 0 ) nkπ n∈Z\{0}

236


By Lemma 5, the function h(λ) = (λ − z˜ 0 )

z˜ n − λ nπ n∈Z\{0}

satisfies h(λ) = sin λ(1 + o(1)),

n → +∞

uniformly on (n + 1/6)π |λ| (n + 5/6)π . Notice that hk (λ) = kh(λ/k), hence λ (1 + o(1)), n → +∞ hk (λ) = k sin k uniformly on k(n + 1/6)π |λ| k(n + 5/6)π .

✷

As an application of Lemma 6, one has LEMMA 7. Let π π − aj < 2 2

(j = 1, 2),

k1 1,

k2 1

and (µn )n∈Z , (νn )n∈Z two sequences of real numbers satisfying µn = nπ + l 2 (n),

νn = nπ + l 2 (n).

Then the function defined by h(λ) := (λ − µ0 − a1 )(λ − ν0 − a2 )

µk n + a1 − λ νk n + a2 − λ 1 × 2 k nπ k2 nπ 1 n∈Z\{0}

is entire. Furthermore, there exists (γp )p1 a sequence of positive real numbers with γp → ∞, and C > 0 such that uniformly on |λ| = γp and p 1, 1 1 + |Im λ| . |h(λ)| C exp k1 k2 Proof. By Lemma 6, the functions h1 (λ) := (λ − µ0 − a1 )

µk n + a1 − λ 1 k1 nπ n∈Z\{0}

and h2 (λ) := (λ − ν0 − a2 )

νk n + a2 − λ 2 k2 nπ n∈Z\{0}

237


are entire and satisfy as n → ∞ λ − a1 (1 + o(1)) h1 (λ) = k1 sin k1

(15)

uniformly on k1 (n + 1/6)π |λ − a1 | k1 (n + 5/6)π and λ − a2 h2 (λ) = k2 sin (1 + o(1)) k2

(16)

uniformly on k2 (n + 1/6)π |λ − a2 | k2 (n + 5/6)π . Moreover, for j = 1, 2, there exist Cj > 0 such that, uniformly on

{λ | kj (n + 1/6)π |λ − aj | kj (n + 5/6)π }, Ij := n∈Z

one has (cf. [13, Lemma 1, p. 27])

sin λ − aj > Cj exp |Im(λ − aj )| .

kj kj

(17)

Noticing that h(λ) = h1 (λ)h2 (λ) and, in view of (15)–(17), it remains to prove that there exists a sequence (γp )p1 with γp > 0 (p 1) and γp −→ ∞ such that p→∞

for each p 1 γp ∈ I1 ∩ I2 ∩ R. As Ij ∩ R is the union of segments whose wide is 2/3 kj π and the distance between two consecutive segments is 1/3 kj π , the existence of such sequence ✷ (γp )p1 is clear. We can now state the main result of this section. Recall that σ (ϕ, α) ≡ (µn (ϕ, α))n∈Z is the spectrum of H (ϕ) with domain, F = ZY ∈ H 1 (0, 1) such that Z(0) = 0 and cos α Y (1) − sin α Z(1) = 0. PROPOSITION 8. Let ϕ, ψ ∈ L2 ([0, 1], C), 0 a 1, 0 α, β < π with α = β and k1 , k2 1 with 1/k1 + 1/k2 2a. Assume that (i) ϕ ≡ ψ on [a, 1], (ii) µk1 n (ϕ, α) = µk1 n (ψ, α), n ∈ Z, (iii) µk2 n (ϕ, β) = µk2 n (ψ, β), n ∈ Z. Then f (·, ϕ, ψ) ≡ 0. Proof. As mentioned in the introduction, following [8] (see also [7]), one deduces from Rouché’s Theorem and formula (12) π (18) µn (ϕ, α) = nπ + α − + l 2 (n) 2 and µn (ϕ, β) = nπ + β −

π + l 2 (n). 2

(19)

238


Therefore, by Lemma 7, the entire function h(λ) := (λ − µ0 )(λ − ν0 )

µk n − λ νk n − λ 1 2 , k nπ k 1 2 nπ n∈Z\{0}

with µj := µj (ϕ, α), νj := µj (ϕ, β) (j ∈ Z), satisfies for some constant C > 0 1 1 + |Im λ| C exp(2a|Im λ|) (20) |h(λ)| C exp k1 k2 uniformly on |λ| = γp and p 1, where (γp )p1 is a sequence of positive real numbers with γp −→ ∞. p→∞

Furthermore, as α = β, σ (ϕ, α) ∩ σ (ϕ, β) = ∅. Thus (µk1 n )n∈Z and (νk2 n )n∈Z are simple roots of h. On the other hand, by Lemma 3 and Hypotheses (ii) (iii), f (µk1 n ) = f (νk2 n ) = 0 for all n ∈ Z. Besides, by Lemma 4 and Hypothesis (i), f (λ) = o(e2a|Im λ| )

(21)

when |λ| → ∞. Therefore λ → f (λ)/h(λ) is entire and combining (20), (21) we get |f (λ)/h(λ)| = o(1) as p → ∞ uniformly on |λ| = γp . Hence, by the maximum principle, we conclude that f ≡ 0. ✷

2.4. INTEGRAL REPRESENTATION The main result of this section is PROPOSITION 9. Let ϕ, ψ ∈ L2 ([0, 1], C). Assume that f (λ, ϕ, ψ) = 0 for λ ∈ R, then ϕ ≡ ψ. We follow the same strategy as in [11, Appendix 4]. We first establish an integral representation of g1 and g2 (cf. formula (10)). LEMMA 10. There exists a kernel K ∈ L2 ([−1, 1]2 , C) such that for x ∈ [−1, 1] and λ ∈ R, x u)e−2iλu du (22) K(x, g1 (x, λ) = −e−2iλx + −x

and

g2 (x, λ) = −e

2iλx

+

x

K(x, u)e2iλu du. −x

(23)


239

Proof. Let M(·, λ, ϕ) be the fundamental (2 × 2 matrix) solution of H (ϕ)M = λM satisfying M(0, λ, ϕ) = Id2×2 and R(x, λ) be given by cos λx sin λx R(x, λ) = . − sin λx cos λx From [12, p. 514] (cf. also [11]) one learns that there exists a Kernel A ∈ L2 ([0, 1]2 , M2×2 (R)) (where M2×2 (R) denotes the space of 2×2 matrix with real entries) such that x R(x − 2y, λ)A(x, y) dy. (24) M(x, λ) = R(x, λ) + 0

By definition, F (x, λ, ϕ) is the first column of M(x, λ, ϕ). Therefore, by (5), G2 (x, λ) = (i, 1)M(x, λ) 10 and by a straightforward calculation, one gets x eiλ(x−2y)Kϕ (x, y) dy, G2 (x, λ, ϕ) = ieiλx +

(25)

0

where

Kϕ (x, y) = (i, 1)A(x, y) 10 .

2 (x, λ, ϕ), one also has Since for λ ∈ R, G1 (x, λ, ϕ) = G x −iλx ϕ (x, y) dy. + e−iλ(x−2y)K G1 (x, λ, ϕ) = −ie

(26)

0

Inserting (25) in (10) one gets g2 (x, λ) = −e2iλx + h1 (x, λ) + h2 (x, λ), where

x

h1 (x, λ) = i

(Kϕ (x, t) + Kψ (x, t))e2iλ(x−t ) dt

(27)

(28)

0

and

h2 (x, λ) =

x

x

dt 0

dsKϕ (x, t)Kψ (x, s)e2iλ(x−t −s) .

(29)

0

By a change of variable u = x − t − s and v = t − s in (29), one obtains 1 v−u+x −v − u + x 2iλu Kψ x, e Kϕ x, du dv, h2 (x, λ) = 2 D 2 2 In [12] the authors do not use the same spectral variable and we have to transform λ in λ/2 in

our formula (24).

240


where D = D1 ∪ D2 and with D1 := {(u, v) | −x u 0; u v −u} and D2 := {(u, v) | 0 u x; −u v u}. Thus

h2 (x, λ) =

with

x

−x

e2iλu K1 (x, u) du

(30)

v−u+x −v − u + x Kψ x, 1D (u, v) dv Kϕ x, K1 (x, u) := 2 2 −x

x

and where 1D denotes the characteristic function of the set D. Similarly by the change of variable u = x − t in (28), one has x Kϕ (x, x − u) + Kψ (x, x − u) e2iλu du. h1 (x, λ) = i 0

Thus h1 (x, λ) =

x

−x

K2 (x, u)e2iλu du,

(31)

where K2 (x, u) := i1[0,x] (u) Kϕ (x, x − u) + Kψ (x, x − u) . Combining (27), (30) and (31), one gets (23) with K(x, u) = K1 (x, u) + K2 (x, u). We deduce (22) from (23) recalling that g1 (x, λ) = g¯2 (x, λ) for λ ∈ R.

✷

Proof of Proposition 9. Recall that, with r = ϕ − ψ (cf. (11)), 1 g2 (x, λ)r(x) + g1 (x, λ)¯r (x) dx. f (λ) ≡ f (λ, ϕ, ψ) = 0

By Lemma 10, for λ ∈ R one gets x 1 2iλx 2iλu + K(x, u)e du r(x) dx + −e f (λ) = 0 −x y 1 −2iλy −2iλv v)e + dv r¯ (y) dy. K(y, −e + 0

−y

(32)

241


By the change of variable x = −y and u = −v in the second term of the right-hand side of (32), one obtains x 1 2iλx 2iλu + K(x, u)e du r(x) dx + −e f (λ) = 0 −x x 0 2iλx 2iλu − du r¯ (−x) dx. (33) K(−x, −u)e −e + −1

−x

Thus, with m(x) := r(x)

for x ∈ [0, 1]

and

m(x) := r¯ (−x)

for x ∈ [−1, 0]

and with B(x, u) := K(x, u),

for x ∈ [0, 1]

(34)

and B(x, u) := K(−x, −u), (33) leads to 1 2iλx + −e f (λ) = =

−1 1

for x ∈ [−1, 0],

|x|

2iλu

B(x, u)e

du m(x) dx

−|x|

2iλx

e −1

−m(x) +

−|x|

−1

B(u, x)m(u) du +

1

B(u, x)m(u) du dx.

|x|

Since f (λ) = 0 for all λ ∈ R and {e2iλx | λ ∈ R} spans L2 ([−1, 1], C), the last formula implies that, for x ∈ [−1, 1], 1 −|x| B(u, x)m(u) du + B(u, x)m(u) du. m(x) = −1

|x|

In particular, for x ∈ [0, 1], one gets by definition of B and m, 1 −x)¯r (u) + K(u, x)r(u) du. K(u, r(x) = x

Therefore, defining P ∈ L2 ([−1, 1]) by −v)|, P (u, v) := |K(u, v)| + |K(u, one obtains

1

|r(x)|

P (u, x)|r(u)| du, x

(35)

242


for x ∈ [0, 1]. Iterating formula (35) leads to, for each n 1, 1 1 1 du1 du2 . . . dun P (u1 , x) . . . P (un , un−1 )|r(un )|. |r(x)| x

u1

un−1

Interchanging the order of integration we obtain 1 u2 un |r(x)| dun dun−1 . . . du1 P (u1 , x) . . . P (un , un−1 )|r(un )|. x

x

(36)

x

Now setting K1 (u, x) = P (u, x) and defining u Kj (u, x) := dv Kj −1 (v, x)P (u, v), x

the inequality (36) can be written as 1 duKn (u, x)|r(u)|. |r(x)|

(37)

x

Defining for 0 x u 1 1 |P (z, x)|2 dz, B(x) = x u |P (u, z)|2 dz, A(u) = 0 x A(u) du, ρ(x) = 0

one obtains by a straightforward recurrence and the Cauchy–Schwarz inequality |Kn (u, x)|2 A(u)B(x) Therefore 1 |Kn (u, x)|2 du x

ρ(u)n−2 . (n − 2)!

B(x) (n − 2)!

1

du A(u)ρ(u)n−2 =

0

B(x) ρ(1)n−1 . (n − 1)!

Hence 1 |Kn (u, x)|2 du x

1 (n − 1)!

1

0

1

|P (z, x)| dz · 2

2

du 0

n−1

1

dz|P (u, z)| 0

The following argument is analogous to parts of [9, Theorem 6, Ch. 2].

.

(38)

243


Using the Cauchy–Schwarz in (37), we get 1 1 2 2 du|Kn (u, x)| · |r(u)|2 du. |r(x)| x

(39)

x

By integrating (39) and using (38) we find r2L2

1 2 n→∞ P 2n L2 rL2 −→ 0. (n − 1)!

It follows then that r(x) = ϕ(x) − ψ(x) = 0.

✷

2.5. PROOFS OF THEOREMS 1 AND 2 Theorem 1 is a direct consequence of Proposition 8 and Proposition 9. The proof of Theorem 2 is similar; we only have to make the following changes: − We replace f (λ, ϕ, ψ) by f˜(λ, ϕ, ψ) = f (−λ, ϕ, ψ)f (λ, ϕ, ψ). Thus, since µln (ϕ, α) = µln (ψ, α) and µkn (ϕ, β) = µkn (ψ, β) for all n ∈ N one has by Lemma 3 that for any n ∈ N f˜(µln (ϕ, α)) = f˜(µkn (ϕ, β)) = 0 and f˜(−µln (ϕ, α)) = f˜(−µkn (ϕ, β)) = 0. − By Lemma 4 one has f˜(λ) = o(e|Im λ|4a )

as |λ| → +∞.

− In Proposition 8 we replace h by h˜ where ˜ h(x) = (λ − µ0 )(λ − ν0 )

(µ2 − λ2 )(ν 2 − λ2 ) ln kn 2 (knπ )2 (lnπ ) n>0

with µj = µj (ϕ, α) and νj = µj (ϕ, β) (j ∈ N). ˜ The function λ → f˜(λ)/h(λ) is still entire. − By Lemma 7 there exist C > 0 and (γp )p1 with γp −→ ∞ such that uniformly on |λ| = γp

p→∞

1 1 + |Im λ|. |h(x)| C exp l k

244


˜ − As in Proposition 8 we obtain, using 1/ l + 1/k 4a, that |f˜(λ)/h(λ)| = o(1) for p → ∞ uniformly on |λ| = γp . Hence by the maximum principle f˜ ≡ 0, i.e. f ≡ 0. − We apply Proposition 9 to conclude to ϕ = ψ. Acknowledgements B.G. would like to acknowledge the support of the ACI project (French Government) and the hospitality of the IIMAS-UNAM institute. R. del R. gratefully acknowledges support by projects IN-102998 PAPIIT-UNAM and 27487E CONACyT (Mexican Government) and the hospitality of the Department of Mathematics of the University of Nantes. References 1. 2. 3.

4. 5.

6. 7. 8. 9. 10. 11. 12. 13.

Amour, L.: Extension on isospectral sets for the AKNS systems, Inverse Problems 12 (1999), 115–120. Borg, G.: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math. 78 (1946), 1–96. Clark, S. and Gesztesy, F.: Weyl–Titchmarsh M-function asymptotics, local, uniqueness results, trace formulas, and Borg-type theorems for Dirac operators, http://www.ma.utexas.edu/ mp_arc-bin/mpa?yn=01-61. del Rio, R., Gesztesy, F. and Simon, B.: Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions, Internat. Math. Res. Notices 15 (1997), 751–758. del Rio, R., Gesztesy, F. and Simon, B.: Corrections and addendum to inverse spectral analysis with partial information on the potential, III. Updating boundary conditions, Internat. Math. Res. Notices 11 (1999), 623–625. Gesztesy, F. and Simon, B.: Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, Trans. Amer. Math. Soc. 352 (1999), 2765–2787. Grébert, B. and Guillot, J. C.: Gaps of one dimensional periodic AKNS systems, Forum Math. 5 (1993), 459–504. Grébert, B. and Kappeler, T.: Normal form theory for the NLS equation, Preprint, Univ. Nantes. Hochstadt, H.: Integral Equations, Pure Appl. Math., Wiley, New York, 1973. Hochstadt, H. and Lieberman, B.: An inverse Sturm–Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (1978), 676–680. Levin, B. Ja.: Distribution of Zeros of Entire Functions, Trans. Math. Monogr. 5, Amer. Math. Soc., Providence, 1964. McKean, H. P. and Vaninsky, K. L.: Action-angle variables for the cubic Schrödinger equation, Comm. Pure Appl. Math. 50 (1997), 489–562. Pöschel, J. and Trubowitz, E.: Inverse Spectral Theory, Academic Press, New York, 1987.


245

Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds DAVIDE GUZZETTI Research Institute for Mathematical Sciences (RIMS), Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan. e-mail: [email protected] (Received: 10 April 2001; in final form: 21 September 2001) Abstract. We study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we explicitly compute a parametric form of the solutions of the WDVV equations in terms of Painlevé VI transcendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric form to explicitly construct polynomial and algebraic solutions and to derive the generating function of Gromov–Witten invariants of the quantum cohomology of the two-dimensional projective space. The procedure is a relevant application of the theory of isomonodromic deformations. Mathematics Subject Classifications (2000): 53D45, 34M55, 81T45. Key words: WDVV equation, Frobenius manifold, isomonodromic deformation, Painlevé equation, monodromy, boundary-value problem.

1. Introduction In this paper we face the problem of analyzing the global structure of three-dimensional Frobenius manifold and the analytic properties of the corresponding solutions of WDVV equations. The procedure followed is an application of the theory of isomonodromic deformations and Painlevé equations. The three-dimensional case is analyzed here. It is already highly nontrivial and it is the first step towards a generalization to higher dimensions. The WDVV equations of associativity were introduced by Eric Witten [34], R. Dijkgraaf, E. Verlinde and H. Verlinde [8]. They are differential equations satisfied by the primary free energy F (t) in two-dimensional topological field theory. F (t) is a function of the coupling constants t := (t 1 , t 2 , . . . , t n ) t i ∈ C. Let ∂α := ∂/∂tα . Given a nondegenerate symmetric matrix ηαβ , α, β = 1, . . . , n, and numbers q1 , q2 , . . . , qn , r1 , r2 , . . . , rn , d, (rα = 0 if qα = 1, α = 1, . . . , n), the WDVV equations are ∂α ∂β ∂λ F ηλµ ∂µ ∂γ ∂δ F = the same with α, δ exchanged, ∂1 ∂α ∂β F = ηαβ , E(F ) = (3 − d)F + (at most) quadratic terms,

(1) (2) (3)

246

DAVIDE GUZZETTI

where the matrix (ηαβ ) is the inverse of the matrix (ηαβ ) and the differential operator E is E :=

n

E α ∂α ,

E α := (1 − qα )t α + rα ,

α = 1, . . . , n,

α=1

and will be called Euler vector field. The theory of Frobenius manifolds was introduced by B. Dubrovin [9] to formulate the WDVV equations in geometrical terms. It has links to many branches of mathematics like singularity theory and reflection groups [11, 14, 30, 31], algebraic and enumerative geometry [24, 26], isomonodromic deformations theory, boundary-value problems, and Painlevé equations [12]. γ If we define cαβγ (t) := ∂α ∂β ∂γ F (t), cαβ (t) := ηγ µ cαβµ (t) (sum over repeated indices is always omitted in the paper), and we consider a vector space A = span(e1 , . . . , en ), then we obtain a family of commutative algebras At with the γ multiplication eα · eβ := cαβ (t)eγ . Equation (1) is equivalent to associativity and (2) implies that e1 is the unity. DEFINITION. A Frobenius manifold is a smooth/analytic manifold M over C whose tangent space Tt M at any t ∈ M is an associative, commutative algebra with unity e. Moreover, there exists a nondegenerate bilinear form , defining a flat metric (flat means that the curvature associated to the Levi–Civita connection is zero). We denote the product by · and the covariant derivative of ·, · by ∇. We require that the tensors c(u, v, w) := u · v, w ,

and

∇y c(u, v, w), u, v, w, y ∈ Tt M,

be symmetric. Let t 1 , . . . , t n be (local) flat coordinates for t ∈ M. Let eα := ∂α be the canonical basis in Tt M, ηαβ := ∂α , ∂β ,

cαβγ (t) := ∂α · ∂β , ∂γ .

The symmetry of c becomes the complete symmetry of ∂δ cαβγ (t) in the indices. This implies the existence of a function F (t) such that ∂α ∂β ∂γ F (t) = cαβγ (t), satisfying the WDVV (1). Equation (2) follows from the axiom ∇e = 0 which yields e = ∂1 . Some more axioms are needed to formulate the quasi-homogeneity condition (3) and we refer the reader to [11–13]. In this way, the WDVV equations are reformulated in geometrical terms. We first consider the problem of the local structure of Frobenius manifolds (which has its counterpart in the local classification of solutions of WDVV equa(z) tions). A Frobenius manifold is characterized by a family of flat connections ∇ parameterized by a complex number z, such that for z = 0 the connection is

247

INVERSE PROBLEM FOR FROBENIUS MANIFOLDS

associated to , . For this reason ∇(z), are called deformed connections. Let d u, v ∈ Tt M, dz ∈ Tz C; the family is defined on M × C as u v := ∇u v + zu · v, ∇ 1 ∂ d v := v + E · v − µv, ˆ ∇ dz ∂z z d d = 0, u d = 0, ∇ ∇ dz dz dz where E is the Euler vector field and µˆ := I − (d/2) − ∇E is an operator acting on v. In flat coordinates t = (t 1 , . . . , t n ), µˆ becomes d , 2 provided that ∇E is diagonalizable. This will be assumed in the paper. A flat t˜ = 0, which is a linear system coordinate t˜(t, z) is a solution of ∇d µˆ = diag(µ1 , . . . , µn ),

µα = qα −

∂α ξ = zCα (t)ξ, µˆ ξ, ∂z ξ = U(t) + z

(4) (5)

where ξ is a column vector of components ξ α = ηαµ and

∂ t˜ , ∂t µ

α = 1, . . . , n

β (t) , Cα (t) := cαγ

β U := E µ cµγ (t)

are n × n matrices. We restrict to semi-simple Frobenius manifolds, namely analytic Frobenius manifolds such that the matrix U can be diagonalized with distinct eigenvalues on an open dense subset M of M. Then, there exists an invertible matrix φ0 = φ0 (t) such that φ0 Uφ0−1 = diag(u1 , . . . , un ) =: U,

ui = uj

for i = j on M.

The systems (4) and (5) become ∂y = [zEi + Vi ]y, ∂ui V ∂y = U+ y, ∂z z

(6) (7)

where the row-vector y is y := φ0 ξ , Ei is a diagonal matrix such that (Ei )ii = 1 and (Ei )j k = 0 otherwise, and Vi :=

∂φ0 −1 φ , ∂ui 0

V := φ0 µφ ˆ 0−1 .

248

DAVIDE GUZZETTI

As it is proved in [11, 12], u1 , . . . , un are local coordinates on M. The two bases ∂ , ν = 1, . . . , n ∂t ν

and

∂ , i = 1, . . . , n ∂ui

are related by φ0 according to the linear combination (φ0 )iν ∂ ∂ = . ν ∂t (φ ) ∂u 0 i1 i i=1 n

Locally we obtain a change of coordinates, t α = t α (u), then φ0 = φ0 (u), V = V (u). The local Frobenius structure of M is given by parametric formulae t α = t α (u),

F = F (u),

(8)

where t α (u), F (u) are certain meromorphic functions of (u1 , . . . , un ), ui = uj , which can be obtained from φ0 (u) and V (u). Their explicit construction is the object of the present paper. The dependence of the system on u is isomonodromic. This means that the monodromy data of the system (7), to be introduced below, do not change for a small deformation of u. Therefore, the coefficients of the system in every local chart of M are naturally labeled by the monodromy data. To calculate the functions (8) in every local chart one has to reconstruct the system (7) from its monodromy data. This is the inverse problem. We briefly explain what are the monodromy data of the system (7) and why they do not depend on u (locally). For details, the reader is referred to [12]. At z = 0 the system (7) has a fundamental matrix solution (i.e. an invertible n × n matrix solution) of the form ∞ p φp (u) z zµˆ zR , (9) Y0 (z, u) = p=0

where Rαβ = 0 if µα − µβ = k > 0, k ∈ N. At z = ∞ there is a formal n × n matrix solution of (7) given by F1 (u) F2 (u) + + · · · ezU , YF = I + z z2 where Fj (u)’s are n × n matrices. It is a well-known result that there exist fundamental matrix solutions with asymptotic expansion YF as z → ∞ [2]. Let l be a generic oriented line passing through the origin. Let l+ be the positive half-line and l− the negative one. Let .L and .R be two sectors in the complex plane to the left and to the right of l, respectively. There exist unique fundamental matrix solutions YL and YR having the asymptotic expansion YF for x → ∞ in .L and .R , respectively [2]. They are related by an invertible connection matrix S, called Stokes matrix, such that YL (z) = YR (z)S for z ∈ l+ . As it is proved in [12] we also have YL (z) = YR (z)S T on l− .


249

Finally, there exists a n × n invertible connection matrix C such that Y0 = YR C on .R . DEFINITION. The matrices R, C, µˆ and the Stokes matrix S of the system (7) are the monodromy data of the Frobenius manifold in a neighborhood of the point ˆ u = (u1 , . . . , un ). It is also necessary to specify which is the first eigenvalue of µ, because the dimension of the manifold is d = −2µ1 (a more precise definition of monodromy data is in [12]). The definition makes sense because the data do not change if u undergoes a small deformation. This problem is discussed in [12]. We also refer the reader to [21] for a general discussion of isomonodromic deformations. Here we just observe that since a fundamental matrix solution Y (z, u) of (7) also satisfies (6), then the monodromy data can not depend on u (locally). In fact, (∂Y /∂ui )Y −1 = zEi + Vi is single-valued in z. The compatibility of (6) and (7) is equivalent to [U, Vk ] = [Ek , V ], ∂V = [Vk , V ]. ∂uk

(10) (11)

Note that (10) determines Vk uniquely, provided that ui = uj for i = j , namely (Vk )ij =

δki − δkj Vij . ui − uj

We finally recall that by construction φ0 satisfies ∂φ0 = Vk φ0 , ∂uk

k = 1, . . . , n.

(12)

According to the results of [21], (11) and (12) are necessary and sufficient conditions for the deformation u to be isomonodromic. The inverse problem can be formulated as a boundary-value problem (b.v.p.). (0) (0) = u(0) Let’s fix u = u(0) = (u(0) j for i = j . Suppose we 1 , . . . , un ) such that ui give µ, µ1 , R, an admissible line l, S and C.1 Some more technical conditions must be added, but we refer to [12]. Let D be a disk specified by |z| < ρ for some small ρ. Let PL and PR be the intersection of the complement of the disk with .L and .R , respectively. We denote by ∂DR and ∂DL the lines on the boundary of D on the side of PR and PL respectively; we denote by l˜+ and l˜− the portion of l+ and l− on the common boundary of PR and PL . Let’s consider the following discontinuous b.v.p.: we want to construct a piecewise holomorphic n × n matrix function   6R (z), z ∈ PR , 6(z) = 6L (z), z ∈ PL ,  60 (z), z ∈ D, 1 We remark that due to symmetries of Equation (7) the matrix C is determined by S, R and µ ˆ up

to some ambiguity that does not affect the corresponding Frobenius structure. Therefore, the relevant monodromy data are S, R, µˆ and µ1 .

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DAVIDE GUZZETTI

continuous on the boundary of PR , PL , D respectively, such that 6L (ζ ) = 6R (ζ )eζ U Se−ζ U , ζ ∈ l˜+ , 6L (ζ ) = 6R (ζ )eζ U S T e−ζ U , ζ ∈ l˜− , 60 (ζ ) = 6R (ζ )eζ U Cζ −R ζ −µˆ , ζ ∈ ∂DR , 60 (ζ ) = 6L (ζ )eζ U S −1 Cζ −R ζ −µˆ , ζ ∈ ∂DL , 6L/R (z) → I if z → ∞ in PL/R . The reader may observe that L/R (z) := 6L/R (z)ezU , Y

(0)(z) := 60 (z, u)zµˆ zR Y

have precisely the monodromy properties of the solutions of (7). THEOREM ([12, 25, 27]). If the above boundary-value problem has solution for (0) (0) (0) a given u(0) = (u(0) 1 , . . . , un ) such that ui = uj for i = j , then: (i) it is unique. (ii) The solution exists and it is analytic for u in a neighborhood of u(0) . (iii) The solution has analytic continuation as a meromorphic function on the universal covering of Cn \{diagonals}, where ‘diagonals’ stand for the union of all the sets {u ∈ Cn | ui = uj , i = j }. (0) of the b.v.p. solves the system (6), (7).1 This means that L/R , Y A solution Y we can locally reconstruct V (u), φ0 (u) and (8) from the local solution of the b.v.p. It follows that every local chart of the atlas covering the manifold is labeled by monodromy data. Moreover, V (u), φ0 (u) and (8) can be continued analytically as meromorphic functions on the universal covering of Cn \diagonals. Let Sn be the symmetric group of n elements. Local coordinates (u1 , . . . , un ) are defined up to permutation. Thus, the analytic continuation of the local structure of M is described by the braid group Bn , namely the fundamental group of (Cn \diagonals)/Sn . There exists an action of the braid group itself on the monodromy data, corresponding to the change of coordinate chart. The group is generated by n − 1 elements β1 , . . . , βn−1 such that βi is represented as a deformation consisting of a permutation of ui , ui+1 moving counter-clockwise (clockwise or counter-clockwise is a matter of convention). 1 We show that a solution Y L/R , Y (0) of the b.v.p. solves the system (6), (7). We have 6R (z) = F1 1 p I + z + O( 2 ) as z → ∞ in PR . We also have 60 (z) = ∞ p=0 φp z as z → 0. Therefore z

R −1 ∂Y = U + [F1 , U ] + O 1 Y , z → ∞, ∂z R z z2 (0) (0) −1 ∂Y ) = 1 [φ0 µφ ˆ 0−1 + O(z)], z → 0. (Y ∂z z Since C is independent of u the right-hand side of the two equalities above are equal. Also S is

251


If u1 , . . . , un are in lexicographical order w.r.t. l, so that S is upper triangular, the braid βi acts on S as follows [12]: S → S βi = Ai (S)SAi (S), where k = 1, . . . , n, n = i, i + 1, (Ai (S))kk = 1, (Ai (S))i+1,i+1 = −si,i+1 , (Ai (S))i,i+1 = (Ai (S))i+1,i = 1 and all the other entries are zero. For a generic braid β the action S → S β is decomposed into a sequence of elementary transformations as above. In this way, we are able to describe the analytic continuation of the local structure in terms of monodromy data. Not all the braids are actually to be considered. Suppose we do the following gauge y → Jy, J = diag(±1, . . . , ±1), on the system (7). Therefore J U J −1 ≡ U but S is transformed to J SJ −1 , where some entries change sign. The formulae which define a local chart of the manifold in terms of monodromy data, which we are going to describe later, are not affected by this transformation. The analytic continuation of the local structure on the universal covering of (Cn \diagonals)/Sn is therefore described by the elements of the quotient group Bn /{β ∈ Bn | S β = J SJ }.

(13)

From these considerations it is proved in [12] that: ˆ R, S, C), the local Frobenius THEOREM ([12]). Given monodromy data (µ1 , µ, structure obtained from the solution of the b.v.p. extends to an open dense subset of the covering of (Cn \diagonals)/Sn w.r.t. the covering transformations (13). above satisfy independent of u, therefore the matrices Y ∂y V ˆ 0−1 . = U+ y, V (u) := [F1 (u), U ] ≡ φ0 µφ ∂z z In the same way

R −1 ∂Y = zEi + [F1 , Ei ] + 1 , z → ∞, Y ∂ui R z (0) ∂Y (0) )−1 = ∂φ0 φ −1 + O(z), z → 0. (Y ∂ui ∂ui 0 ’s satisfy The right-hand sides are equal, therefore the Y ∂y = [zEi + Vi ]y, ∂ui

Vi (u) := [F1 (u), Ei ] ≡

∂φ0 −1 φ . ∂ui 0

We conclude that from the solution of the boundary value problem we obtain solutions to (7), (6).

252

DAVIDE GUZZETTI

Let’s start from a Frobenius manifold M of dimension d. Let M be the open submanifold where U(t) has distinct eigenvalues. If we compute its monodromy data (µ1 = −(d/2), µ, ˆ R, S, C) at a point u(0) ∈ M and we construct the Frobenius structure from the analytic continuation of the corresponding b.v.p. on the covering of (Cn \diagonals)/Sn w.r.t. the quotient (13), then there is an equivalence of Frobenius structures between this last manifold and M. We now turn to the problem of understanding the global structure of a Frobenius manifold. In order to do it we have to study (8) when two or more distinct coordinates ui , uj , etc., merge. φ0 (u), V (u) and (8) are multi-valued meromorphic occurs when u goes around a functions of u = (u1 , . . . , un ) and the branching n loop around the set of diagonals ij {u ∈ C | ui = uj , i = j }. φ0 (u), V (u) and (8) have singular behavior if ui → uj (i = j ). We call such behavior critical behavior. Although it is impossible to solve the boundary-value problem exactly, except for special cases occurring for 2 × 2 systems, we may hopefully compute the asymptotic/critical behavior of the solution, using the isomonodromic deformation method. We will face the problem in the first nontrivial case, namely for three-dimensional Frobenius manifolds. Instead of analyzing the boundary-value problem directly, we exploit the isomonodromic dependence of the system (7) on u, which implies that the solution of the inverse problem must satisfy the nonlinear equations (11), (12). For threedimensional Frobenius manifolds, (11), (12) are reduced in [11] to a special case of the Painlevé VI equation1 : 2 1 1 dy 1 1 dy 1 1 1 d2 y + + + + + = − 2 dx 2 y y−1 y−x dx x x − 1 y − x dx x(x − 1) 1 y(y − 1)(y − x) 2 (2µ − 1) + , + 2 x 2 (x − 1)2 (y − x)2 u3 − u1 . (14) µ ∈ C, x = u2 − u1 The parameter µ is µ1 and the matrix µˆ =diag(µ, 0, −µ). We are going to show that the entries of V (u) and :(u) are rational functions of x, y(x), dy/dx. If ui → 1 The six classical Painlevé equations were discovered by Painlevé [28] and Gambier [16], who classified all the second-order ordinary differential equations of the type

d2 y dy = R x, y, , dx dx 2 where R is rational in dy/dx, x and y. The Painlevé equations satisfy the Painlevé property of absence of movable critical singularities. The general solution of the VIth Painlevé equation can be analytically continued to a meromorphic function on the universal covering of P1 \{0, 1, ∞}. For generic values of the integration constants and of the parameters in the equation, the solution can not be expressed via elementary or classical transcendental functions. For this reason, the solution is called a Painlevé transcendent.


253

uj , the critical behavior of V (u), φ0 (u) and (8) is a consequence of the critical behavior of the transcendent y(x) close to the critical points x = 0, 1, ∞. This will be described in the paper. 1.1. RESULTS OF THE PAPER (1) Let F0 (t) := 12 [(t 1 )2 t 3 + t 1 (t 2 )2 ]. We prove in Theorem 5.1 of Section 5 that for generic µ the parametric representation (8) becomes t 3 (u) = τ3 (x, µ)(u2 − u1 )1+2µ, t 2 (u) = τ2 (x, µ)(u2 − u1 )1+µ , F (u) = F0 (t) + F (x, µ)(u2 − u1 )3+2µ ,

(15) (16)

where τ2 (x, µ), τ3 (x, µ), F (x, µ) are certain rational functions of µ, x, y(x), dy/dx 1+µ and a quadrature of y, which we will compute explicitly. The ratio t 2 /(t 3 ) 1+2µ is independent of (u2 − u1 ). Therefore, the closed form F = F (t) must be

3+2µ t2 3 1+2µ ϕ , F (t) = F0 (t) + (t ) 1+µ (t 3 ) 1+2µ where the function ϕ has to be determined by the inversion of (15), (16). For the value µ = −1, corresponding to the Frobenius manifold called quantum cohomology of the projective space CP2 , denoted QH ∗ (CP2 ), we prove in Theorem 5.2 of Section 5 that the coordinate t 2 (u) is x 2 dζ τ (ζ ), (17) t (u) = 3 ln(u2 − u1 ) + 3 where τ (x) is also computed explicitly as a rational function of x, y(x), dy/dx. 2 The coordinate t 3 and F are the limit for µ → −1 of (15), (16). Now et (t 3 )3 is independent of (u2 − u1 ) and so F (t) = F0 (t) +

1 t 2 3 3 ϕ e (t ) . t3

To our knowledge, this is the first time the explicit parameterization (15)–(17) is given. We stress that in Section 5 the formulas will be completely explicit. Although the proof is mainly a computational problem (the theoretical problem being already solved by the reduction to the Painlevé VI equation [11]), it is very hard. Moreover, the knowledge of this explicit form is necessary to proceed to the inversion of the parametric formulae t = t (u), F = F (u) close to the diagonals ui = uj , in order to investigate the global structure of the Frobenius manifold and to obtain F = F (t) in closed form. (2) As we discussed above, the local structure of the manifold and of F (t) is labeled by the monodromy data. The formulae (15)–(17) make this explicit. This follows from the fact that the two integration constants which govern the critical

254

DAVIDE GUZZETTI

behavior of y(x) – and thus of the corresponding solution of (11), (12) – and the parameter µ, are contained in the three entries (x0 , x1 , x∞ ) of the Stokes’s matrix 1 x∞ x0 2 − x0 x1 x∞ = 4 sin2 (π µ), S = 0 1 x1 , such that x02 + x12 + x∞ 0 0 1 of the system (7). It is known that there exists a class of transcendents whose critical behaviour is  (0)   a (0) x 1−σ (1 + O(|x|δ )), x → 0, (1) (18) y(x) = 1 − a (1) (1 − x)1−σ (1 + O(|1 − x|δ )), x → 1,   a (∞) x −σ (∞) (1 + O(|x|−δ )), x → ∞, where 0 < δ < 1 is a small positive number, a (i) and σ (i) are complex numbers such that a (i) = 0 and 0 σ (i) 1, σ (i) = 1. The above behavior depends on the entries of S, which determine the constants a (i) , σ (i) through the formulae

2 2 π (i) σ , (19) xi = 4 sin 2 iG(σ (0) , µ)2 −iπσ (0) 2 −iπσ (0) 2 2(1 + e ) − f (x , x , x )(x + e x ) × a (0) = 0 1 ∞ ∞ 1 2 sin(π σ (0)) (20) ×f (x0 , x1 , x∞ ), where f (x0 , x1 , x∞ ) :=

4 − x02 , 2 − x02 − 2 cos(2π µ)

4σ D( σ 2+1 )2 1 , µ) = 2 D(1 − µ + σ (0) )D(µ + (0)

(0)

G(σ

(0)

2

σ (0) ) 2

.

The parameters a (1) , a (∞) are obtained like a (0) , provided that we do the substitutions (x0 , x1 , x∞ ) → (x1 , x0 , x0 x1 − x∞ ),

σ (0) → σ (1)

and (x0 , x1 , x∞ ) → (x∞ , −x1 , x0 − x1 x∞ ),

σ (0) → σ (∞) ,

respectively in the formula for a (0) . Note that σ (i) = 1 ⇔ xi = ±2. The critical behavior of the Painlevé transcendents was obtained in [15, 20] for generic values of the entries of the Stokes’matrix, with the exception of real xi such that |xi | 2, i = 0, 1, ∞. We generalized the result to any xi = ±2 [see D. Guzzetti: On the critical behavior, the connection problem and the elliptic representation of a Painlevé VI equation (2001), to appear. See also: Inverse Problem for

255


Semisimple Frobenius Manifolds, Monodromy Data and the Painleve’ VI Equation, Ph.D. thesis and SISSA preprint 101/2000/FM (2000).– Formulae (19) and (20) are found in these papers]. We proved in the above-mentioned paper that for the special case σ (i) = 1 (i) (σ = 1) – therefore for xi real, |xi | > 2 – the solution y(x) behaves like (18) only along spirals, but it is oscillatory when x → i, i = 0, 1, ∞, along a radial path. For example, if x → 0 we have y(x) = O(x) +

sin2 ν2

x → 0,

1 + O(x) eiπ ν1 m , x iν ln x − ν ln 16 + + ∞ c (ν) 0m m=1 16iν (21) πν1 2

where σ

(0)

= 1 − iν,

ν ∈ R\{0} and

a

(0)

1 eiπν1 =− . 4 16iν−1

The series in the denominator converges and defines a holomorphic and bounded function in a suitable domain where y(x) has no movable poles. The critical behavior of Painlevé transcendents is also analyzed in [33], though the the relation to monodromy data is not considered. In Section 7 we reduce the formulae (15), (16) to closed form for the five algebraic solutions of the Painlevé equation. In this case the transcendent behaves like (18) with rational exponents, then t and F in (15), (16) are expanded in Puiseux series in x, 1−x or 1/x. The expansion can be inverted, in order to obtain F = F (t) in closed form as an expansion in t. We prove that we obtain here the three polynomial solutions of the WDVV equations corresponding to the Frobenius structure on the orbit space of Coxeter groups [11, 30, 31], plus two algebraic solutions. We also apply the procedure to QH ∗ (CP2 ). This time, σ (i) = 1, because the Stokes matrix is 1 3 3 S= 0 1 3 , 0 0 1 as it is proved in [12, 17]. Therefore, the transcendent has the oscillatory behavior (21) and the reduction of (15)–(17) to closed form is hard. To avoid this difficulty we expanded the transcendent in Taylor series close to a regular point xreg , we plugged the expansion into (15)–(17) and we obtained t and F as a Taylor series in (x − xreg ). We inverted the series and we got a closed form F = F (t). We prove in Section 8 that the closed form we obtain through (15)–(17) coincides with the solution of the WDVV equations which generates the numbers Nk of rational curves CP1 → CP2 of degree k passing through 3k − 1 generic points [24]. Namely Nk 3 3k−1 kt 2 e . (22) F (t 1 , t 2 , t 3 ) = 12 (t 1 )2 t 3 + t 1 (t 2 )2 + ∞ k=1 (3k−1)! (t )

256

DAVIDE GUZZETTI

Therefore, we have constructed a procedure to compute the Nk ’s, which is an application of the theory of isomonodromic deformations. 2 It is known [7] that (22) is convergent in a neighborhood of (t 3 )3 et = 0, but the global analytic properties of F (t) are unknown. The inverse reconstruction of the corresponding Frobenius manifold starting from its monodromy data may shed some light on these properties. To this purpose, we still have to manage to invert the parametric formulae (15)–(17) if x converges to a critical point. Particularly, we hope to better understand the connection between the monodromy data of the quantum cohomology and the number of rational curves. This problem will be the object of further investigations. The entire procedure developed here is a significant application of the theory of isomonodromic deformations to a problem of mathematical physics (construction of solutions of WDVV equations) and to pure mathematics (investigation of the global structure of a Frobenius manifold). 2. Inverse Reconstruction of a Frobenius Manifold In this section we review the construction of the local parametric solution (8) of the WDVV equations in terms of the coefficients φp of (9). The result is discussed in [12] and it is the main formula which allows to reduce the problem of solving the WDVV equations to problems of isomonodromic deformations of linear systems dt˜ = 0 is of differential equations. As a first step we note that the condition ∇ α β satisfied both by a flat coordinate t˜ and by t˜α := ηαβ t˜ (sum over β). Thus, we choose a fundamental matrix solution of (4), (5) of the form:

∞ α ∂ t˜β αγ p Hp (t)z zµˆ zR , H0 = I, = G = ∂ t˜β ≡ η ∂t γ p=0 close to z = 0. If we restrict to the system (4) only, we can choose as a fundamental solution H (z, t) :=

∞

Hp (t)zp

p=0

on M (not on M × C) have the expansion and so the flat coordinates of ∇ t˜α =

∞

hα,p (t)zp ,

p=0

where the functions hα,p must satisfy hα,0 = tα ≡ ηαβ t β , ∂γ ∂β hα,p+1 = cγI β ∂I hα,p ,

p = 0, 1, 2, . . . .

(23) (24)

257


We stress that the normalization H0 = I is precisely what is necessary to have t˜α (z = 0) = tα and it corresponds exactly to Y0 = φ0 G in (9). Observe that hα0 = tα ≡ ηαβ t β implies ∂β hα,0 = ηβα ≡ cβα1 .

(25)

Denote by ∇f := (ηαβ ∂β f )∂α the gradient of the function f . We claim that tα = ∇hα,0 , ∇h1,1 ≡ ηµν ∂µ hα,0 ∂ν h1,1

(26)

are flat coordinates and F (t) = 12 ∇hα,1 , ∇h1,1 ηαβ ∇hβ,0 , ∇h1,1 − − ∇h1,1 , ∇h1,2 − ∇h1,3 , ∇h1,0

(27)

solves the WDVV equations. To prove it, it is enough to check by direct differentiation that ∂α tβ = ηαβ and ∂α ∂β ∂γ F (t) = cαβγ (t), using (23)–(25) and. . . some patience. In the following, we denote the entry (i, j ) of a matrix Ak by Aij,k . Recall that ∂ ∂µ = µ , ∂t

∂ ∂i = ∂ui

and

∂µ =

n φiµ,0 i=1

φi1,0

∂i .

Therefore Yiα = 1/φi1,0 ∂i t˜α . It follows that 1/φi1,0 ∂i hα,p = φiα,p and thus tα (u) =

n

φiα,0 φi1,1 ,

(28)

i=1

n n 1 α β t t φiα,0 φiβ,1 − φi1,1 φi1,2 + φi1,3 φi1,0 . F (t (u)) = 2 i=1 i=1

(29)

It is now clear that we can locally reconstruct a Frobenius manifold from the matrices φ0 (u), φ1 (u), φ2 (u), φ3 (u) of (9). They are obtained as solutions of the b.v.p. and thus they depend on the monodromy data. Their analytic continuation extends the Frobenius structure on (Cn \diagonals)/Sn , as explained in the Introduction. To understand the global structure of the manifold we need to study the critical behavior of φp (u) and of (28), (29) as ui → uj . Instead of solving the b.v.p. directly, we exploit the isomonodromic deformation theory. Let again consider a solution of the b.v.p. of the form ∞ p φp (u)z zµˆ zR . Y0 (z, u) = p=0

It also satisfies (7) and (6) as we explained in the Introduction. Since ∂R/∂ui = 0, if we plug Y0 into (6) we get ∂φp = Ei φp−1 + Vi φp . ∂ui

(30)

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DAVIDE GUZZETTI

Let 6(z, u) := it implies

∞ p=0

φp (u)zp . The condition 6(−z, u)T 6(−z, u) = η holds1 and m

φ0T φ0 = η,

φpT φm−p = 0

for any m > 0.

(31)

p=0

We conclude that φp (u) can be obtained either solving the b.v.p. or solving (30) with the condition (31). 3. Inverse Reconstruction of Two-Dimensional Frobenius Manifolds Let n = 2. In this section we explain the inverse reconstruction of a semi-simple Frobenius manifold for n = 2 through the formulae (28), (29). The two-dimensional case is exactly solved by elementary methods, so our purpose here is didactic: we clarify the procedure which will be followed in the nontrivial three-dimensional case. 3.1. EXACT SOLUTION IN DIMENSION 2 AND MONODROMY DATA The coefficients of the system (5) are necessarily:

0 i σ2 , U = diag(u1 , u2 ). V (u) = −i σ2 0 Here u = (u1 , u2 ), and V is independent of u. It has the diagonal form

σ σ σ σ −1 ,− , ⇒ µ1 = , µ2 = − , d = −σ, µˆ = φ0 V φ0 = diag 2 2 2 2 where

φ0 (u) =

1 2f (u) 1 2if (u)

f (u) if (u)

,

φ0T φ0

= η :=

0 1

1 0

.

1 The symmetries ηµ ˆ + µˆ T η = 0, UT η = ηU imply that ξ1 (−z, t)T ηξ2 (z, t) is independent of

z for any two solutions ξ1 (z, t), ξ1 (z, t) of (5). We choose a fundamental matrix solution of (4), (5) of the form:   ∞ Hp (t)zp zµˆ zR , H0 = I, G= p=0

close to z = 0. Let H (z, t) := φ0 (u)H (z, t (u)), therefore 6(−z, u)T 6(z, u) = η.

∞

p T p=0 Hp (t)z . Then H (−z, t) ηH (z, t) = η. Now, 6(z, u) =

259


The computation of the Stokes’ matrix of dY 0 i σ2 u1 0 + = Y 0 u2 −i σ2 0 dz

(32)

requires to keep into account the two oriented half-lines R12 = {z = −iρ(u1 − u2 ), ρ > 0},

R21 = −R12 .

Let l be an oriented line through the origin, having R12 to the left. Then YL (z, u) = YR (z, u)S on l+ and YL (z, u) = YR (z, u)S T on l− . The Stokes matrix is

1 s S= , s ∈ C. 0 1 At the origin we have the solution ∞ φk (u)zk zµˆ zR . Y0 (z, u) =

(33)

k=0

It is connected to YR through the invertible matrix C according to: Y0 (z, u) = YR (z, u)C, z ∈ .R (recall that .R is the half plane to the right of l). Then σ 0 2

2πi

0 − σ2

S T S −1 = Ce

e2πiR C −1 .

From the trace, s 2 = 2(1 − cos(π σ )). The above monodromy data R, µ, ˆ S define the boundary-value problem to reconstruct the system (5). The standard technique to solve a two-dimensional boundary value problem is to reduce it to a system of differential equations, which is (32) in our case, and then to reduce the system to a second-order differential equation. It turns out that the equation is (after a change of dependent and independent variables) a Whittaker equation. Therefore, the solution of the b.v.p. is given in terms of Whittaker functions Wκ,µ . Let H := u1 − u2 ; the fundamental solutions are YR (z, u)  =

ei 2 (H z)− 2 ez π

1

u1 +u2 2

i σ2 ei 2 (H z)− 2 ez π

1

W 1 , σ e−iπ H z 2 2

u1 +u2 2

W− 1 , σ e−iπ H z 2 2

−i σ2 (H z)− 2 ez 1

(H z)− 2 ez 1

u1 +u2 2

u1 +u2 2

W− 1 , σ (H z) 2 2

 

W 1 , σ (H z) 2 2

for arg(R12) < arg(z) < arg(R12 )+2π , where arg(R12 ) := −(π/2)−arg(u1 −u2 ).   u1 +u2 1 (YR (z, u))11 i σ2 (H z)− 2 ez 2 W− 1 , σ e−2iπ H z 2 2  YL (z, u) =  −2iπ u1 +u2 1 −2 z 2 W1,σ e Hz (YR (z, u))12 −(H z) e 2 2

for arg(R12 ) + π < arg(z) < arg(R12 ) + 3π .

260

DAVIDE GUZZETTI

For the choice of YR and YL above, also the sign of s can be determined. According to our computations from the expansion of YR and YL at z = 0, it is s = 2 sin(π σ /2). We stress that the only monodromy data are σ and the nonzero entry of R. The purpose of this didactic chapter is to show that (28) and (29) bring solutions F (t) explicitly parameterized by σ and R. 3.2. PRELIMINARY COMPUTATIONS The functions φp (u) to be plugged into (28), (29) may be derived from the above representations in terms of Whittaker functions. We prefer to proceed in a different way, namely by imposing the conditions of isomonodromicity (30) and the constraint (31) to the solution (33). This is the procedure we will also follow in the three-dimensional case. The function f (u) in φ0 (u) is arbitrary, but subject to the condition of isomonodromicity (30) for p = 0, namely ∂i φ0 = Vi φ0 , where V1 = V /(u1 − u2 ), V2 = −V1 . Let U := φ0−1 U φ0 . We will use h(u) to denote an arbitrary function of u. Let’s also denote the entry (i, j ) of a matrix Ak by Aij,k or by (Ak )ij according to the convenience. Let us decompose R = R1 + R2 + R3 + · · ·, where Rij,k = 0 only if µi − µj = k > 0 integer. In order to compute φp (u) of (33) we decompose it (and define Hp (u)) as follows: φp (u) := φ0 Hp (u),

p = 0, 1, 2, . . . .

Plugging the above into (32) we obtain (1∗∗ )

φ0 is given,

(2∗∗ )

µ1 = ± 12 , µ1 = µ1 =

(3∗∗ )

(4∗∗ )

Hij,1 =

Uij , 1 + µj − µi

1 , 2 − 12 ,

H12,1 = h1 (u),

3 , 2 − 32 ,

H12,3 = h3 (u),

R = 0,

R12,1 = U12 ,

H21,1 = h1 (u), R21,1 = U21 , (UH1 − H1 R1 )ij µ = ±1, Hij,2 = , R2 = 0, 2 + µj − µi µ = 1, H12,2 = h2 (u), R12,2 = (UH1 )12 , R1 = 0, µ = −1, H21,2 = h2 (u), R21,2 = (UH1 )21 , R1 = 0, (UH2 − H1 R2 − H2 R1 )ij µ = ± 32 , Hij,3 = , R3 = 0, 3 + µj − µi

µ = µ =

H21,3 = h3 (u),

R12,3 = (UH2 )12 , R21,3 = (UH2 )21 ,

R1 = R2 = 0, R1 = R2 = 0.

For any value of σ the isomonodromicity condition ∂i φ0 = Vi φ0 reads σ f (u) ∂f (u) =− , ∂u1 2 u1 − u2

∂f (u) σ f (u) = . ∂u2 2 u1 − u2

261


In other words, ∂f (u) ∂f (u) =− ∂u1 ∂u2 and thus f (u) ≡ f (u1 − u2 ). Let H := u1 − u2 . Therefore f (H ) σ f (H ) σ =− ⇒ f (H ) = CH − 2 , dH 2 H

C a constant.

We are ready to compute t = t (u), F = F (t (u)) from the formulae t1 =

2

t2 =

φi2,0 φi1,1 ,

i=1

1 α β t t F = 2

2 i=1

2

2 φiα,0 φiβ,1 − φi1,1 φi1,2 + φi1,3 φi1,0

i=1

(34)

φi1,0 φi1,1 ,

(35)

i=1

and to reduce them to closed form. 3.3. THE GENERIC CASE We start from the generic case of σ not integer. The result of the application of formulae (34), (35) is t1 =

u1 + u2 , 2

t2 =

u1 − u2 1 4(1 + σ ) f (u)2

and F (t (u)) = 12 (t 1 )2 t 2 + But now observe that u1 − u2 , f (u)2 = 4(1 + σ )t 2

2(1 + σ )3 (t 2 )3 f (u)4 . (1 − σ )(σ + 3) f (u)2 ≡ f (H )2 = C 2 H −σ , u1 − u2 = H.

The above three expressions imply H = C1 (t 2 )1/(1+σ ), where C1 = [4(1 + σ )C 2 ]1/(1+σ ). Therefore f (u)4 = C2 (t 2 )−2σ/(1+σ ), where C2 is a constant from C1 (we do not need to compute it explicitly in terms of C1 or C). Finally, 1 σ +3 F (t) = (t 1 )2 t 2 + C3 (t 2 ) σ +1 . 2 Here C3 is another constant, from C2 .

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DAVIDE GUZZETTI

3.4. THE CASES

µ1 = 32 , µ1 = 1, µ1 = −1

(1) Case µ1 = 32 , σ = 3. Formula (34) gives the same result of the generic case (with σ = 3) because h3 (u) appears only in φ3 (u) and does not affect t: u1 − u2 u1 + u2 1 1 2 , t = . t = 2 4(1 + σ ) f (u)2 σ =3 Although h3 (u) appears in φ3 (u), it does not in F :

2(1 + σ )3 1 1 2 2 2 3 4 F (t (u)) = (t ) t + (t ) f (u) . 2 (1 − σ )(σ + 3) σ =3

We may proceed as in the generic case. Actually, now the computation of f (u) is straightforward because

1 0 − 16 0 r (u1 − u2 )3 f (u)2 R3 = ≡ , r = constant. 0 0 0 0 Namely 1 (u1 − u2 )3 f (u)2 = r. − 16

On the other hand, from t 2 we have u1 − u2 = 16t 2 f (u)2 and thus 1

f (u)4 =

(−r) 2 3

16(t 2 ) 2

and, finally, 1

3

3

F (t) = 12 (t 1 )2 t 2 − 32 (−r) 2 (t 2 ) 2 ≡ 12 (t 1 )2 t 2 + C(t 2 ) 2 , where C is an arbitrary constant, depending on r. (2) Case µ1 = 1, σ = 2. Again, the arbitrary function h2 (u) does not appear in t (u) and F (t (u)): u1 − u2 u1 + u2 1 1 2 , t = , t = 2 4(1 + σ ) f (u)2 σ =2 2(1 + σ )3 2 3 4 1 1 2 2 (t ) f (u) . F (t (u)) = 2 (t ) t + (1 − σ )(σ + 3) σ =2 Now we proceed like in the generic case and we find the generic result with σ = 2. (3) Case µ1 = −1, σ = −2. Now the formulae (34), (35) yield u1 + u2 u2 − u1 , t 2 = 14 , 2 f (u)2 F (t (u)) = 32 (t 1 )2 t 2 − 23 (t 2 )3 f (u)4 − t 1 h2 (u).

t1 =

263


The condition 0=

φ0T φ2

−

φ1T φ1

+

φ2T φ0

=

2h2 (u) + 0

u21 −u22 4f (u)2

0 0

implies h2 (u) =

1 u22 − u21 ≡ t 1t 2. 8 f (u)2

Therefore F (t (u)) = 12 (t 1 )2 t 2 − 23 (t 2 )3 f (u)4 . Now we proceed as in the generic case, using f (H ) = CH −σ/2 = CH and we find the generic result with σ = −2. 3.5. THE CASE µ1 = − 12 We analyze the case µ1 = − 12 , σ = −1. The formula (34) gives t1 =

u1 + u2 , 2

t 2 = h1 (u).

By putting h1 (u) = t 2 we get, from (35), F (t (u)) = 12 (t 1 )2 t 2 +

u −u 1 (u1 − t 1 )3 1 ( 1 2 2 )3 1 1 2 2 = (t ) t + . 2 16 f (u)2 16 f (u)2

It is straightforward to obtain f (u) from

0 0 0 0 ≡ , R1 = u1 −u2 0 r 0 4f (u)2 namely, f (u)2 =

1 u1 − u2 = H. 4r 4r

The last thing we need is to determine H as a function of t 1 , t 2 . We can’t use the condition 6(−z)T 6(z) = η, because direct computation shows that φ0T φ2 − φ1T φ1 + φ2T φ0 = 0, φ0T φ1 − φ1T φ0 = 0, φ0T φ3 − φ1T φ2 + φ2T φ1 − φ3T φ0 = 0 are identically satisfied. We make use of the isomonodromicity conditions ∂φ1 = E1 φ0 + V1 φ1 , ∂u1

∂φ1 = E2 φ0 + V2 φ1 , ∂u2

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DAVIDE GUZZETTI

which become ∂h1 (u) 1 = , ∂u1 4f (u)2

∂h1 (u) ∂h1 (u) =− . ∂u2 ∂u1

Thus h1 (u) ≡ h1 (u1 − u2 ) and r dh1 (H ) = ⇒ t 2 ≡ h1 (H ) = r ln(H ) + D, dH H D a constant. Thus f (u)4 =

H2 t2 = Ce2 r , 2 16r

where C is a constant (C = exp(−2D)). We get the final result F (t) = 12 (t 1 )2 t 2 + t2

Ce2 r . 3.6. THE CASE µ1 =

1 2

Let µ1 = 12 , σ = 1. t is like in the generic case t1 =

u1 + u2 , 2

t2 =

u1 − u2 , 8f (u)2

while F contains h1 (u) f (t (u)) = 12 (t 1 )2 t 2 + 12 (t 2 )2 h1 (u) − 3(t 2 )3 f (u)4 . We can determine f (u) as in the generic case, or better we observe that

0 r 0 (u1 − u2 )f (u)2 ≡ . R1 = 0 0 0 0 Thus f (u)2 = r/(u1 − u2 ). We determine h1 (u). The condition 6(−z)T 6(z) = η does not help, because it is automatically satisfied. We use the isomonodromicity conditions ∂φ1 = E1 φ0 + V1 φ1 , ∂u1

∂φ1 = E2 φ0 + V2 φ1 ∂u2

which become ∂h1 (u) = f (u)2 , ∂u1

∂h1 (u) ∂h1 (u) =− . ∂u2 ∂u1


265

Therefore h1 (u) ≡ h1 (u1 − u2 ). Then, keeping into account that f (u)2 = r/H , we obtain, dh1 (H ) r = ⇒ h1 (H ) = r ln(H ) + D, dH H D being a constant. Finally, recall that t2 =

H H2 , ≡ 8f (u)2 8r

hence f (u)4 = r/8t 2 , which contributes a linear term to F (t), and h1 (u) = (r/2) ln(t 2 ) + B, where B = (r/2) ln(8r) + C is an arbitrary constant. Finally, F (t) = 12 (t 1 )2 t 2 + 4r (t 2 )2 ln(t 2 ) as we wanted. 3.7. THE CASE µ1 = − 32 Finally, let’s take µ1 = − 32 , σ = −3. From (34), (35) we have u1 + u2 u2 − u1 , t2 = , 2 8f (u)2 F (t (u)) = 34 (t 1 )2 t 2 + (t 2 )3 f (u)4 − 12 h3 (u).

t1 =

f (u) is obtainable as in the generic case, but it is straightforward to use

(u2 − u1 )3 0 3 0 0 0 . ≡ ⇒ f (u)2 = R3 = (u2 −u1 ) 0 r 0 64r 64f (u)2 To obtain h3 (u) we can’t rely on 6(−z)T 6(z) = η, which turns out to be identically satisfied. We use again the conditions ∂φ3 = Ei φ2 + Vi φ3 . ∂ui

(36)

It is convenient to introduce G(u) := 14 (t 1 )2 t 2 − 12 h3 (u). The above (36) becomes r ∂G , = ∂u1 2(u2 − u1 )

∂G ∂G =− , ∂u2 ∂u1

which implies G(u) = G(u1 − u2 ) and r r dG =− ⇒ G(H ) = − ln(H ) + C, dH 2H 2

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DAVIDE GUZZETTI

C is constant. Finally, recall that t2 =

H2 r ⇒ G(H (t 2 )) = ln(t 2 ) + C1 . 8r 4

Thus F (t) = 12 (t 1 )2 t 2 +

r ln(t 2 ), 4

having dropped the constant terms. 3.8. CONCLUSIONS The solution of the boundary value problem for the monodromy data σ and the nonzero entry r of the matrix R was obtained solving Equations (30) with the constraints (31). We have obtained the solutions of the WDVV equations from (34), (35). They can also be derived from elementary considerations (see [11]). Here it becomes clear that they depend explicitly on the monodromy data σ , r: For σ = ±1, −3,

3+σ

F (t) = 12 (t 1 )2 t 2 + C(t 2 ) 1+σ ,

where C is a constant. σ = −1,

t2

F (t) = 12 (t 1 )2 t 2 + Ce2 r , F (t) = 12 (t 1 )2 t 2 + C(t 2 )2 ln(t 2 ),

σ = 1, σ = −3,

F (t) = 12 (t 1 )2 t 2 + C ln(t 2 ).

4. The Three-Dimensional Case: Computation of φ0 and V in Terms of Painlevé Transcendents Let n = 3. In this section we explicitly compute φ0 and V in terms of a Painlevé VI transcendent y(x), and conversely we give a formula for y(x) in terms of the entries of φ0 and V . We can bring η = (ηαβ ) to the form [11]: 0 0 1 η= 0 1 0 . 1 0 0 Let

V (u) =

0 M3 −M2

−M3 0 M1

M2 −M1 0

,

267


which is similar to

−(M21 + M22 + M23 ) constant,

d µ=− . 2 By simple linear algebra, we find the eigenvectors of V . φ0 is precisely the matrix whose columns are the eigenvectors. We have to impose also the condition φ0T φ0 = η and we find the most general form for φ0 :   M1 √i M1 M2 −µM13 G(u) √i M1 M2 +µM13 1 iµ 2µ (M2 +M2 ) 2 2µ (M2 +M2 ) 2 G(u) 1 3 1 3  2 12 2 12 1    M2 i i 2 2 √ √ φ0 =  − M1 + M3 G(u) iµ − 2µ M1 + M3 G(u)  , 2µ   M3 √i M2 M3 +µM11 G(u) √i M2 M3 −µM11 1 iµ G(u) 2µ 2µ µˆ = diag(µ, 0, −µ),

µ=

(M21 +M23 ) 2

(M21 +M23 ) 2

where G(u) is so far an arbitrary function of u = (u1 , u2 , u3 ). To determine it we must impose the isomonodromicity condition (30) for p = 0 ∂φ0 = Vi (u)φ0 . ∂ui

(37)

We observe that φi2,0 = Mi /(iµ), i = 1, 2, 3. If we compute the entries of (37) ’s we recover the equation ∂i V = [Vi , V ]. In particular, we note that for the φi2,0 ∂ V = i i i ui ∂i V = 0. Thus V (u1 , u2 , u3 ) ≡ V (x), where x=

u3 − u1 . u2 − u1

Therefore, ∂i V = [Vi , V ] becomes: 1 dM1 = M2 M3 , dx x

1 dM2 = M1 M3 , dx 1−x

1 dM3 = M1 M2 . dx x(x − 1)

(38)

Equations (37), (38) are reduced in [11] to a special case of the VIth Painlevé equation, with the following choice of the parameters (in the standard notation of [18]): α=

(2µ − 1)2 , 2

β = γ = 0,

δ = 12 .

Namely:

2 1 1 dy 1 1 dy 1 1 1 d2 y + + + + + = − 2 dx 2 y y−1 y−x dx x x − 1 y − x dx x(x − 1) 1 y(y − 1)(y − x) 2 (2µ − 1) + , µ ∈ C. (39) + 2 x 2 (x − 1)2 (y − x)2

In the following, this equation will be referred to as PVIµ . Let H := u2 − u1 , let y = y(x) be a Painlevé transcendent of PVIµ , and let " #x $ )−ζ k0 exp (2µ − 1) dζ y(ζ ζ(ζ −1) , k0 ∈ C\{0}. k = k(x, H ) := H 2µ−1

268

DAVIDE GUZZETTI

LEMMA. The following φ0 is the general solution of (37): √ √ √ √ √ √ k y k y −1 k y−x φ13,0 = i √ √ , φ23,0 = i √ √ , φ33,0 = − √ √ √ , H x H 1−x H x 1−x √ √ A 1 y−1 y−x +µ , √ φ12,0 = µ (y − 1)(y − x) x √ √ A 1 y y−x φ22,0 = +µ , √ µ y(y − x) 1−x √ √ A i y y−1 +µ , φ32,0 = √ √ µ x 1 − x y(y − 1) √ √ H y 2µ i 2 + µ (y − 1 − x) , φ11,0 = √ √ A B+ 2µ2 k x y √ √ H y−1 2µ i 2 A B+ + µ (y + 1 − x) , φ21,0 = √ √ 2µ2 k 1 − x y−1 √ √ 2µ 1 H y−x 2 A B+ + µ (y − 1 + x) , φ31,0 = − 2 √ √ √ 2µ k x 1 − x y−x where

1 dy x(x − 1) − y(y − 1) , A = A(x) := 2 dx A . B = B(x) := y(y − 1)(y − x)

We can also rewrite φ0 as follows   E11 E12 E13 f f φ0 =  Ef21 E22 E23 f  , E31 E32 E33 f f where

√ √ k y−1 , f = f (x, H ) := i √ √ H 1−x Mi , i = 1, 2, 3, Ei2 := iµ M1 M2 − µM3 M1 M2 + µM3 , E13 := − , E11 := 2 2µ M21 + M23 M21 + M23 , 2µ2 M2 M3 + µM1 := , 2µ2

E21 := − E31

E23 := 1, E33 := −

M2 M3 − µM1 , M21 + M23

269


and

√ √ A y −1 y−x +µ , √ M1 = i (y − 1)(y − x) x √ √ y y−x A +µ , M2 = i √ y(y − x) 1−x √ √ y y−1 A +µ . M3 = − √ √ x 1 − x y(y − 1)

The branches (signs) in the √ square roots above are arbitrary. A change of the sign of one root (for example of H ) implies a change of two signs in (M1 , M2 , M3 ), or the change (φi1,0 , φi3,0 ) → −(φi1,0 , φi3,0 ). The reader may verify that all these changes do not affect the equations for φ0 and V . Proof. The first proof of the lemma is direct substitution of the above φ0 into (37). Direct computation shows that (37) is satisfied if and only if y(x) satisfies the Painlevé equation PVIµ . (Also (38) is satisfied by the Mj ’s above if and only if y(x) satisfies PVIµ .) The second proof is constructive. We derived φ0 from the link between the matrix φ0 and the 2 × 2 Fuchsian system associated to Painlevé VI in the theory of isomonodromic deformations developed in [22]. The construction of the Fuchsian system in terms of φ0 can be found in [11]. This construction also implies that φ0 above is the general solution of (37). The Fuchsian system is Ai (u) ∂X = −µ X, ∂λ λ − ui i=1 3

where Ai :=

φi1,0 φi3,0 2 φi1,0

λ ∈ C,

2 −φi3,0 φi1,0 φi3,0

(40)

,

A1 + A2 + A3 =

1 0

0 −1

.

(41)

The system depends isomonodromically on u and it is solved by introducing the following coordinates q(u), p(u) in the space of matrices Ai modulo diagonal conjugation (see [11]): q is the root of 3 3 Ai Ai = 0 and p := . q − ui q − ui i=1 i=1 12

11

The entries of the Ai ’s are re-expressed as follows: φi1,0 φi3,0 = −

q − ui × 2µ2 P (ui )

3 2µ P (q)p + µ2 (q + 2ui − uj , × P (q)p 2 + q − ui j =1 2 = −k φ13,0

q − ui , P (ui )

(42) (43)

270

DAVIDE GUZZETTI 2 φi1,0 =−

q − ui × 4µP (ui )k

2µ P (q)p + µ2 (q + 2ui − uj × P (q)p 2 + q − ui j =1 3

2 (44)

.

Here k is a parameter, P (z) = (z − u1 )(z − u2 )(z − u3 ). A solution of (40) must also satisfy ∂ ∂ui

X1 X2

Ai =µ λ − ui

X1 X2

(45)

.

This is precisely the equation which implies that the dependence on u is isomonodromic. The compatibility of (40), (45) is P (q) 1 ∂q = 2p + , ∂ui P (ui ) q − ui

(46)

P (q)p 2 + (2q + ui − 3j =1 uj )p + µ(1 − µ) ∂p , =− ∂ui P (ui ) q − ui ∂ ln k . = (2µ − 1) ∂ui P (ui )

(47)

In the variables x=

u3 − u1 , u2 − u1

y=

q − u1 , u2 − u1

the system (46), (47) becomes PVIµ . From a solution of PVIµ one can reconstruct

u3 − u1 + u1 , q = (u2 − u1 )y u2 − u1

u3 − u1 1 1 1 P (u3 ) y , − p = 2 P (q) u2 − u1 2 q − u3 and from the very definition of q we have: xR(x) , y(x) = x[1 + R(x)] − 1

R(x) :=

φ13,0 φ23,0

2 =

M1 M2 + µM3 µ2 + M22

2 . (48)

This makes it possible to compute y(x) from a solution M1 , M2 , M3 of (38). The explicit form for φi1,0 and φi3,0 is derived taking squares roots of (43) and (44). The sign is chosen in order to satisfy (42). We then determine φi2,0 from the equality

271


(φ12,0 , φ22,0 , φ32,0 ) = ±i(φ21,0 φ33,0 − φ23,0 φ31,0 , φ13,0 φ31,0 − −φ11,0 φ33,0 , φ11,0 φ23,0 − φ13,0 φ21,0 ).

✷

5. Explicit Computation of the Flat Coordinates and of F for n = 3 Let t = (t 1 , t 2 , t 3 ), with higher indices. We compute the parametric form t = t (x, H ) and F = F (x, H ) using t1 =

3

φi3,0 φi1,1 ,

i=1

t2 =

3

φi2,0 φi1,1 ,

t3 =

i=1

3

φi1,0 φi1,1 ,

(49)

i=1

3 3 1 α β t t φiα,0 φiβ,1 − (φi1,1 φi1,2 + φi1,3 φi1,0 ) . F = 2 i=1 i=1

(50)

We recall that µ1 = µ, µ2 = 0, µ3 = −µ. Let us compute φ1 , φ2 , φ3 . We decompose φp := φ0 Hp , p = 0, 1, 2, . . . . Hi appears in the fundamental matrix solution of (5): G(z, u) = (I + H1 z + H2 z2 + H3 z3 + · · ·)zµˆ zR ,

z → 0.

By plugging G(z, u) into (5) we computed the Hi ’s. We give their explicit expression below. In the formulae which follow we denote by h(k) ij (u) arbitrary functions of u = (u1 , u2 , u3 ), to be determined later; they appear any time 2µ ∈ Z and R is not zero. Let U = φ0−1 U φ0 , where φ0 is given by the lemma. (1∗∗∗ ) Computation of H1 . Uij , R1 = 0. 1 + µj − µi H13,1 = h(1) R13,1 = U13 , 13 (u), Uij if (i, j ) = (1, 3). Hij,1 = 1 + µj − µi H31,1 = h(1) R31,1 = U31 , 31 (u), Uij if (i, j ) = (3, 1). Hij,1 = 1 + µj − µi

Generic case: Hij,1 = µ = 12 : µ = − 12 :

µ = 1:

H12,1 = h(1) 12 (u),

R12,1 = U12 ,

H23,1 = h(1) 23 (u),

R23,1 = U23 ,

Hij,1 =

Uij 1 + µj − µi

if (i, j ) ∈ {(1, 2), (2, 3)}.

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DAVIDE GUZZETTI

µ = −1: H21,1 = h(1) 21 (u),

R21,1 = U21 ,

H32,1 = h(1) 32 (u),

R32,1 = U32 ,

Hij,1 =

Uij 1 + µj − µi

if (i, j ) ∈ {(2, 1), (3, 2)}.

(2∗∗∗ ) Computation of H2 . Let U2 : = UH1 − H1 R1 . Uij,2 , 2 + µj − µi

Generic case:

Hij,2 =

µ = 1:

H13,2 = h(2) 13 (u), Hij,2 =

µ = −1:

µ = 2:

Uij,2 2 + µj − µi

if (i, j ) = (1, 3). R31,2 = U31,2 ,


if (i, j ) = (3, 1).

H12,2 = h(2) 12 (u),

R12,2 = U12,2 ,

H23,2 = h(2) 23 (u),

R23,2 = U23,2 ,

Hij,2 = µ = −2:

R13,2 = U13,2 ,

H31,2 = h(2) 31 (u), Hij,2 =

R2 = 0.


H21,2 = h(2) 21 (u), H23,2 = h(2) 32 (u), Hij,2 =

if (i, j ) ∈ {(1, 2), (2, 3)}. R21,2 = U21,2 ,

R32,2 = U32,2 ,


if (i, j ) ∈ {(2, 1), (3, 2)}.

(3∗∗∗ ) Computation of H3 . Let U3 : = UH2 − H2 R1 − H1 R2 . Generic case: Hij,3 = µ = 32 :

H13,3 = h(3) 13 (u), Hij,3 =

µ = − 32 :

Uij,3 , 3 + µj − µi Uij,3 3 + µj − µi

H31,3 = h(3) 31 (u), Hij,3 =


R3 = 0.

R13,3 = U13,3 , if (i, j ) = (1, 3). R31,3 = U31,3 , if (i, j ) = (3, 1).

273


µ = 3:

H12,3 = h(3) 12 (u),

R12,3 = U12,3 ,

H23,3 = h(3) 23 (u),

R23,3 = U23,3 ,

Hij,3 = µ = −3:


if (i, j ) ∈ {(1, 2), (2, 3)}.

H21,3 = h(3) 21 (u),

R21,3 = U21,3 ,

H32,3 = h(3) 32 (u),

R32,3 = U32,3 ,

Hij,3 =


if (i, j ) ∈ {(2, 1), (3, 2)}.

5.1. THE GENERIC CASE µ = ± 12 , ±1, ± 32 , ±2, ±3 Let µ = ± 12 , ±1, ± 32 , ±2, ±3 and let  E11 φ0 = 

f E21 f E31 f

E12 E22 E32

 E13 f E23 f  , E33 f

where Eij = Ei,j (x) and f (x, H ) are given in the lemma of Section 4 in terms of y(x). Direct computation gives the entries of H1 , H2 , H3 , then φ1 , φ2 , φ3 and finally t and F from (49), (50). They are rational functions of x, y(x), y (x), k(x, H ). The computation is very hard and long: we need to substitute the entries of φ0 , φ1 , φ2 and φ3 in (49), (50) and do proper simplifications. We finally obtain the following THEOREM 5.1. The flat coordinates (t 1 , t 2 , t 3 ) and the free energy F of a threedimensional semi-simple Frobenius manifold such that µ = −d/2 is not equal to ± 12 , ±1, ± 32 , ±2, ±3, are given by the parametric formulae: t 1 = u1 + a(x)H, H 1 b(x) , t2 = 1+µ f (x, H ) H 1 c(x) , t3 = 1 + 2µ f (x, H )2 (µ + 4)b(x)b1 (x)c(x) a1 (x)c(x)2 + + F = F0 (t) + 2(1 − 2µ)(3 + 2µ) 2(1 − µ)(2 + µ)(3 + 2µ) H3 b(x)2 (b2 (x) − a(x)) , + (2 + µ)(3 + 2µ) f (x, H )2 1 1 F0 (t) := t 1 (t 2 )2 + (t 1 )2 t 3 , 2 2

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DAVIDE GUZZETTI

where a(x) := E21 E23 + xE31 E33 , b(x) := E22 E21 + xE32 E31 , b1 (x) := E23 E22 + xE33 E32 , 2 2 + xE33 , a1 (x) := E23 2 2 b2 (x) := E22 + xE32 , 2 2 + xE31 , c(x) := E21 H = u2 − u1 , and f (x, H ), Eij (x) are given in the lemma. In particular, t and F are rational functions of x, y(x), dy(x)/dx, k(x, H ). Note that F − F0 is independent of u1 , namely it is independent of t 1 . 5.2. THE CASE OF THE QUANTUM COHOMOLOGY OF PROJECTIVE SPACES : µ = −1

Let µ = −1. This is a nongeneric case, corresponding to the Frobenius manifold called the Quantum Cohomology of CP2 . In this case the unknown functions h(1) 21 , (1) (2) h32 , h31 have to be determined. It is known [12, 17] that 0 0 0 R2 = 0. R1 = 3 0 0 , 0 3 0 The direct computation gives 0 0 0 −1 0 0 , R1 = b(x)Hf −1 0 0 b(x)Hf 0 0 0 R2 = 0 0 0 , (1) − h ) 0 0 b(x)Hf −1 (h(1) 21 32 which implies f (x, H ) =

H b(x), 3

(1) h(1) 21 = h32 ,

(51)

where h(1) 32 is determined using the differential equation ∂φ1 = Ei φ0 + Vi φ1 ∂ui which implies E12 E11 ∂h(1) 32 , = ∂u1 f

∂h(1) E22 E21 32 , = ∂u2 f

∂h(1) E32 E31 32 . = ∂u3 f

275


Therefore ∂h(1) ∂h(1) ∂h(1) 32 + 32 + 32 = 0. ∂u1 ∂u2 ∂u3 The last equation follows from E12 E11 + E22 E21 + E32 E31 = 0, which is a consequence of φ0T φ0 = η. Therefore h(1) 32 is a function of x = (u3 − u1 )/(u2 − u1 ) and H = u2 − u1 . Keeping into account (51) and the relations ∂x ∂x x −1 x 1 ∂x , = =− , = , ∂u1 H ∂u2 H ∂u3 H ∂H ∂H ∂H = 0, = 1, = −1, ∂u1 ∂u2 ∂u3 we obtain 3 ∂h(1) 32 = , E22 ∂x x + EE21 31 E32 which are integrated as h(1) 32

= 3 ln(H ) + 3

∂h(1) 32 = 3, ∂H

x

dζ

1 ζ+

E21 E22 E31 E32

.

(52)

Before determining h(2) 31 , it is worth computing t through (49). We again need to substitute the entries of φ0 and φ1 in the formulae (49) and do nontrivial simplifications. We obtain t 2 = h(1) t 1 = u1 + a(x)H, 32 , H c(x) 1 . t 3 = −c(x) 2 = −9 f b(x)2 H

(53) (54)

1 3 We observe that h(2) 31 does not appear in t. We also observe that both t and t coincide with the limits for µ → −1 of the same coordinates computed in the generic case. Instead, such a limit does not exist for t 2 . Now we turn to the differential equation ∂φ2 /∂ui = Ei φ1 + Vi φ2 which gives the following differential equations for h(2) 31 :

∂h(2) ∂t 3 ∂t 2 ∂t 1 31 = t1 + t2 + t3 , ∂ui ∂ui ∂ui ∂ui

i = 1, 2, 3.

1 2 2 1 3 They are immediately integrated: h(2) 31 = 2 (t ) + t t . Now all the entries of φp , p = 0, 1, 2, 3, are known and we can substitute them into (50). We obtain: F = F0 (t) + 9 16 a1 (x)c(x)2 + 34 b(x)b1 (x)c(x) + (b2 (x) − a(x))b(x)2 × H . (55) × b(x)2

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DAVIDE GUZZETTI

Remarkably, this coincides with the limit, for µ → −1, of the generic case. We have proved the following theorem: THEOREM 5.2. The flat coordinates (t 1 , t 2 , t 3 ) and the free energy F for the Quantum Cohomology of CP2 are given by the parametric formulae (52), (53), (54) and (55).

6. F (t) in Closed Form (1) Generic case µ = ± 12 , ±1, ± 32 , ±2, ±3. If we keep into account the dependence of f (x, H ) and k(x, H ) on H , we see that both t and F − F0 can be factorized in a part depending only on x and another one depending only on H t 2 (x, H ) = τ2 (x)H 1+µ , t 3 (x, H ) = τ3 (x)H 1+2µ , F (x, H ) = F0 (t) + F (x)H 3+2µ , where τ2 (x), τ3 (x) and F (x) are explicitly given as rational functions of x, y(x), dy(x)/dx and quadratures by the formulae of the Theorem 5.1. Hence, the ratio t2 1+µ

(t 3 ) 1+2µ is independent of H and the closed form F = F (t) must be

3+2µ t2 3 1+2µ ϕ , F (t) = F0 (t) + (t ) 1+µ (t 3 ) 1+2µ where the function ϕ(ζ ) has to be determined. We’ll obtain closed forms F = F (t) following the steps below. (i) First we choose a critical point x = 0, 1, ∞ of PVIµ and we expand y(x) close to the critical point, with parameters σ (i) , a (i) given by (19), (20) in terms of the entries of the Stokes’ matrix S. In the paper we consider only the case of rational σ (i) , therefore any expansion is a Taylor or Puiseux series. The coefficients of the expansion, which are rational in a (i) and σ (i), are classical functions of the entries of S (actually, a (i) and σ (i) are rational, trigonometric or D functions of the monodromy data). The most efficient way to do the expansion is to compute the expansions of M1 (x), M2 (x), M3 (x). The algorithm we use is an expansion of the Mi ’s in a small parameter [see the Appendix in Section 9]. The effective variable in the expansion is a variable s → 0  if x → 0, x s := 1 − x if x → 1,  1 if x → ∞. x


277

(ii) We plug the above expansions into τi (x) and F (x), obtaining an expansion in s. In particular t2 (t 3 )

1+µ 1+2µ

≡

τ2 (x(s)) 1+µ

τ3 (x(s)) 1+2µ

is expanded. (iii) One of the following cases may occur  0,    τ2 ∞, for s → 0 → 1+µ ζ  0   τ31+2µ no limit, where ζ0 is a nonzero complex number. If the limit does not exist, the problem becomes complicated. This may actually occur for particular values of the monodromy data (we’ll see later that this is the case of the quantum coho2 mology of CP2 , provided that we take et (t 3 )3 instead of t 2 (t 3 )−(1+µ)/(1+2µ)). In the other three cases the limit exists and we have a small quantity X = X(s) → 0 as s → 0:  τ 2  1+µ ,   τ 1+2µ     %3 τ &−1 2 , 1+µ X := 1+2µ  τ  3  τ2   1+µ − ζ0 .   1+2µ τ3

(iv) We invert the series X = X(s) and find a series s = s(X) for X → 0. Thus we can rewrite τ2 = τ2 (X), τ3 = τ3 (X), F = F (X). (v) We compute H as a series in X and as a function of t 3 : 1 3 1+2µ t 3 . H = H (X, t ) = τ3 (X) (vi) By substituting H (X, t 3 ) into F − F0 = F (X)H 3+2µ we obtain a series for F − F0 in the small variable X. In other words, we obtain ϕ(ζ ) as a series in ζ or ζ1 or ζ − ζ0 . (vii) Finally, we simply re-express X in term of the variables t 2 and t 3 and that’s all. We get the closed form F (t) as a series whose coefficients are classical functions of the monodromy data.

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DAVIDE GUZZETTI

7. Closed Form of F (t) from Algebraic Solutions of PVIµ We refer to [15] for manifold is 1 x∞ S= 0 1 0 0

the algebraic solutions of PVIµ . The Stokes’ matrix of the x0 x1 1

and in [15] branches of the algebraic solutions of PVIµ are reconstructed from the above monodromy data. The formulae (19) and (20) give the critical behavior of a branch of y(x) (branch cuts in the x-plane, like (−∞, 0) and (1, +∞), are understood). The analytic continuation of the branch has critical behavior still specified by the exponents (19) and the coefficients (20) computed on the entries of a new S obtained acting with the braid group. This action was described in the Introduction, and it is generated by the two elementary braids (x0 , x1 , x∞ ) → (−x0 , x∞ − x0 x1 , x1 ), (x0 , x1 , x∞ ) → (x∞ , −x1 , x0 − x1 x∞ ). A triple (x0 , x1 , x∞ ) specifies an algebraic solution if and only if its orbit under the action of the braid group is finite. There are only five finite orbits, all classified in [15]. For them, the entries are xi = −2 cos π ri , 0 ri 1 rational, i = 0, 1, ∞. Moreover, µ must be real. In [15] it is proved that the Stokes matrices coincide with the Stokes matrices of the Coxeter groups A3 , B3 , H3 . Namely, let us take one of the above Coxeter groups and choose a basis e1 , e2 , e3 of three vectors which generate the reflections in the planes normal to them w.r.t. an Euclidean metric ( , ). We compute the corresponding Stokes matrix with x1 := (e1 , e2 ), x0 := (e1 , e3 ), x∞ := (e2 , e3 ). If we choose another basis we find a new Stokes’ matrix. It turns out that for the group A3 all possible Stokes’ matrices constructed in this way belong to only one orbit w.r.t. the action of the braid group acting on one of the Stokes matrices. The same holds true for B3 . On the other hand, there are three orbits for H3 , therefore there are three inequivalent choices for the basis e1 , e2 , e3 . The five orbits correspond to the symmetries of five solids: tetrahedron, cube, icosahedron, great dodecahedron, great icosahedron (see [6]). The equation PVIµ admits a set of symmetries, which transform the equation for a given µ to another equation with −µ or µ + 1. They are discussed in [15] and they allows to reduce to the case 0 < µ < 1. With this restriction, there are five algebraic solutions, corresponding to the five orbits of S of A3 , B3 , H3 . We compute F (t) in closed form for these five algebraic solutions. ( I ) TETRAHEDRON

(A3 ), µ = − 41

There is only one orbit of S, completely given in [15]. From the entries of S we compute the critical behavior of y(x) through (19) and (20). This gives the leading terms of the Mj ’s (j = 1, 2, 3) according to the formulae of Section 4. They are

279


enough to start the expansion in the small parameter explained in the Appendix. This is a Puiseux expansion because the σ (i) ’s (i = 0, 1, ∞) are rational. Then we compute the Puiseux expansion of y(x) through (48) and we expand t = t (x, H ), F = F (x, H ) of Subsection 5.1. Finally, we apply the procedure explained in Section 6 to obtain the closed form F = F (t). This is the general procedure we’ll follow in all the cases below. (i) x → 0. We choose the triple (x0 , x1 , x∞ ) = (0, −1, −1). Through (19) and (20) we obtain y(x) = 12 x + O(x 2 ),

x ≡ s → 0.

Using the small parameter expansion explained in the Appendix, we compute the Puiseux expansion (a Taylor series in the example) of y up to order x m−1 , for a given large m. The small variable X is X=

t2 3

(t 3 ) 2

→ 0 for x → 0

and the final result obtained applying the procedure explained in Section 6 is F − F0 =

− k02 (t 3 )5 X 2 + O(X m )

4 4 3 5 2 2 2 3 2 = 15 k0 (t ) − k0 (t ) (t ) + O 4 4 3 5 k (t ) 15 0

X→0 m t2 , 3

(t 3 ) 2

k0 is the arbitrary integration constant in k(x, H ). Note that different solutions F (t) corresponding to different values of k0 are connected by symmetries of the WDVV equations [11]. (ii) x → 1. Let us choose (x0 , x1 , x∞ ) = (−1, 0, −1). Hence: y(x(s)) = 1 − 12 s + O(s 2 ),

s = 1 − x → 0.

X is like in (i) and F − F0 =

+ k02 (t 3 )5 X 2 + O(X m )

4 4 3 5 2 2 2 3 2 = 15 k0 (t ) + k0 (t ) (t ) + O 4 4 3 5 k (t ) 15 0

X→0 m t2 . 3

(t 3 ) 2

Here k02 has the opposite sign w.r.t. the previous case. (iii) x → ∞. We choose (x0 , x1 , x∞ ) = (−1, −1, 0). Hence, y(x(s)) = 12 [1 + O(s)],

s=

1 x

→ 0.

Again, X is as in (i) and (ii), and the result is precisely as in (i).

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DAVIDE GUZZETTI

(iv) We consider another example for x → 0: we choose (x0 , x1 , x∞ ) = (1, 1, 1). Hence, 2

43 2 y(x) = x 3 (1 + O(x δ )), 50

0 < δ < 1, x = s → 0

as determined by the formulae (19), (20). This time the computation of the expansion of y(x) is harder than before, because of the fractional exponent. The final result of the procedure of Section 6 is t2 3

(t 3 ) 2 X=

72 √ 2k0 , x → 0, 25 − ζ0 → 0, x → 0,

→ ζ0 = −

t2 3

(t 3 ) 2 √ 119751372 4 3 5 314928 2 3 3 5 2 2187 2 3 5 4 k (t ) − k0 (t ) X + k (t ) X + F − F0 = 1953125 0 15625 625 0 m + O(X ), X → 0.

Substituting X as a function of t 2 and t 3 we obtain F − F0 =

4 2 3 5 α (t ) − α(t 2 )2 (t 3 )2 + O(X m ), 15

α=−

2187 2 k . 625 0

We applied the procedure for all the triples in the orbit of S of A3 and we considered x → 0, 1, ∞ obtaining the same results of the examples above. ( II ) CUBE

(B3 ), µ = − 13

The computations are similar to those for the case A3 . We√just give one example, namely only the case x → 0 and (x0 , x1 , x∞ ) = (0, −1, − 2), which implies: y(x) = 23 x + O(x 2 ),

x → 0.

We repeated the computation for all the monodromy data x0 , x1 , x∞ of [15] obtaining the same result of the example explained here. For it, the small quantity X is X=

t2 →0 (t 3 )2

and the procedure of Section 6 gives √ 512 6 3 7 16 3 2 2 3 3 2i 2 23 2 3 3 k (t ) − k0 (t ) (t ) − k (t ) t + O(X m). F − F0 = 8505 0 27 9 0 For the case H3 we need to distinguish three sub-cases, corresponding to three inequivalent orbits of S. Again, all these orbits are explicitly listed in [15]. From

281


them we compute the leading terms of y(x) from (19) and (20) and then the Puiseux expansion up to any desired order by the small parameter expansion explained in the Appendix. ( III ) ICOSAHEDRON

(H3 ), µ = − 25

We choose for example √

1+ 5 . (x0 , x1 , x∞ ) = 0, 1, 2 Therefore, √ 3+ 5 √ x + O(x 2 ), x → 0. y(x) = 5+ 5 Now t2 X = 3 3 → 0. (t ) The final result is: 18 9 F − F0 = α 4 (t 3 )11 + α 2 (t 2 )2 (t 3 )5 + α(t 2 )3 (t 3 )2 + O(X m ), 55 5 5 √ √ 5 √ 5 where α = −512 5/[3( 5 − 5) 2 ( 5 + 5) 2 ]k02 . Remark. of [11]

In (I), (II), (III) we have recovered the famous polynomial solutions

F (t) = F0 (t) + a(t 2 )2 (t 3 )2 +

4 2 3 5 a (t ) , 15 2 2 2 3 3

F (t) = F0 (t) + a(t 2 )3 t 3 + 6a (t ) (t ) + F = F0 (t) + a(t 2 )3 (t 3 )2 + 95 a 2 (t 2 )2 (t 3 )5 +

a ∈ C, d = 12 , 216 4 3 7 a (t ) , 35 18 4 3 11 a (t ) , 55

a ∈ C, d = 23 , a ∈ C, d = 45 ,

which correspond to the Frobenius structures on the space of orbits of the Coxeter groups A3 , B3 , H3 [11, 30, 31]. ( IV ) GREAT DODECAHEDRON

(H3 ), µ = − 13

We give the final result, which is a series:

2 k ∞ t 3 7 Ak F − F0 = (t ) A0 + 3 )2 (t k=2

√ 512 6 3 7 16 3 2 2 3 3 4i 5 32 3 2 3 k (t ) − k0 (t ) (t ) + k t (t ) + = 8505 0 27 9 0 √ i 5 (t 2 )5 1 (t 2 )6 1 (t 2 )4 + − + ···. + 3 3 3 8 t3 64k03 (t 3 )5 80k 2 (t ) 0

The series can be computed up to any order in X = t 2 /(t 3 )2 → 0.

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DAVIDE GUZZETTI

( V ) GREAT ICOSAHEDRON

(H3 ), µ = − 15

The final result is a Puisex series:

2 k ∞ t 3 13 3 Ak F − F0 = (t ) A0 + 4 (t 3 ) 3 k=2 3 2 2 2 3 5 54 13 α 4 (t 3 ) 3 − α (t ) (t ) 3 + = 284375 125 1 (t 2 )4 i (t 2 )5 i 1 + + ··· + α(t 2 )3 (t 3 ) 3 + 15 72 t 3 108α (t 3 ) 73 4

1

6

for X = t 2 /(t 3 ) 3 → 0. Here k0 = 270 5 /30α 5 . 8. Closed form for QH ∗ (CP 2 ) In this case the factorization is t 3 = τ3 (x)H −1 and F − F0 = F (x)H , but x 3 = ln(H ) + (...). t 2 = h(1) 32 2

This implies that et (t 3 )3 is independent of H and that F (t) = F0 (t) +

1 t 2 3 3 ϕ e (t ) t3

or 2 t2 F (t) = F0 (t) + e 3 ϕ1 et (t 3 )3 . The situation is more complicated now, because the behavior of y(x) for x → 0 is not like ax 1−σ (1+ higher-order terms) as in the previous section. The same holds for x → 1 and x → ∞. The reason for this is that the monodromy data are in the orbit w.r.t the action of the braid group of the triple (x0 , x1 , x∞ ) = (3, 3, 3). Hence, the entries of S are real and their absolute value is greater than 2, so σ (i) = 1. For the data (3, 3, 3) we have √

3+ 5 2 (i) . σ = 1 − iν, ν = − ln π 2 This case corresponds to an oscillatory behaviour (21) if x converges to the critical points along radial directions, as we explained in the Introduction. We recall that the effective parameter s → 0 introduced in Section 6, point (i), is s = x or 1 − x 2 or 1/x. It turns out that et (t 3 )3 has no limit as s → 0. To overcome this difficulty, we compute F (t) in closed form starting from the expansion of y(x) and of M1 (x), M2 (x), M3 (x) close to the nonsingular point xc =

283


exp{−i(π/3)} and we prove that the reduction of (52)–(54) and (55) to closed form is precisely the expansion of F (t) due to Kontsevich [24], namely: ∞ 1 Nk 3 3 t 2 k (t ) e , F (t) = F0 (t) + 3 t k=1 (3k − 1)!

(56)

where Nk is the number of rational curves from CP1 → CP2 of degree k through 3k − 1 generic points. Our result is interesting because it allows us to obtain such a relevant expression starting from the isomonodromy deformation theory. On the other hand, it is not completely satisfactory. Since the Frobenius manifold can in principle be reconstructed from its monodromy data, we would like to express the coefficients Nk as functions of the monodromy data. The critical behavior of y(x) close to a critical point depends upon two parameters a (i) , σ (i) which are classical functions of the monodromy data (rational, trigonometric and D functions). Thus, our aim is to compute the closed form F (t) from the critical behavior of y(x) close to a critical point, in order to obtain the Nk ’s as classical functions of the monodromy data. The choice of a nonsingular point xc is not satisfactory because the expansion of y(x) close to xc depends on two parameters (initial data y(xc ), y (xc )), which in general are not classical functions of a (i) and σ (i) , i = 0, 1, ∞ (generically, the Painlevé transcendents cannot be expressed via classical functions). Thus, they are not classical functions of the monodromy data. π We now compute F (t) in closed form. We expand y(x) close to xc = e−i 3 . This choice comes from the knowledge of the structure of QH ∗ (CP 2 ) at the point t 1 = t 3 = 0 [12, 17]. Namely, we know that 0 0 3q 2 U = 3 0 0 , q := et 0 3 0 with eigenvalues 1

u1 = 3q 3 ,

1

The matrix φ0 is  1 q− 3 1 π φ0 =  q − 3 e−i 3 1 π q − 3 ei 3

 1 1 q3 1 π −1 q 3 ei 3  . 1 π −1 q 3 e−i 3

Thus xc =

2π

u2 = 3q 3 e−i 3 ,

u3 − u1 π = e−i 3 . u2 − u1

1

2π

u3 = 3q 3 ei 3 .

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DAVIDE GUZZETTI

Remark. We understand that the choice of xc is not satisfactory for another reason, namely because we have to rely on the knowledge of QH ∗ (CP2) at the point t 1 = t 3 = 0, and not only on the monodromy data. In order to compute y(x) we start from M1 (x), M2 (x), M3 (x). We look for a regular expansion Mi (x) =

∞

k M(k) i (x − xc ) ,

i = 1, 2, 3.

k=0

We need the initial conditions M(0) i . We can compute them using Mi = iµφi2,0 : i M(0) 1 = −√ , 3

i M(0) 2 = √ , 3

i M(0) 3 = √ . 3

Then we plug the expansion into (38) and we recursively compute the coefficients at any desired order. Finally, we obtain y(x) from (48). We give only the first terms of the expansions, the effective small variable being s := x − xc → 0: √

1 1 √ 1 √ i 3 1 √ 2 1 − + i 3 s + i 3s + − i 3 s3− M1 = − 3 6 6 9 18 18

5 √ 5 − i 3 s4 + · · · , − 36 108 √

1 1 √ 1 √ i 3 1 √ 2 1 + − i 3 s − i 3s − + i 3 s3− M2 = 3 6 6 9 18 18

5 √ 5 + i 3 s4 + · · · , − 36 108 √ 2 √ i 3 1 4 13 √ − s + i 3s 2 + s 3 − i 3s 4 + · · · , M3 = 3 3 9 9 54 1 √ 2 1 3 1 √ 4 13 5 1 1 1 √ y(x) = − i 3 + s − i 3s − s + i 3s + s − 2 6 3 3 3 9 45 37 √ 6 17 7 i 3s − s + · · · . − 135 27 Once we have M1 , M2 , M3 , we can compute the Eij ’s, the φp ’s (p = 0, 1, 2, 3) and finally the flat coordinates t (x, H ) and F (x, H ) through (52)–(55). At low orders: 1 1 √ 1 1 2 − i 3 + s + O(s ) H, t = u1 + 2 6 3 3 2 −1 t = [−9s + O(s )]H , √ 1 √ i 3q0 1 + i 3s + O(s 2 ) H 3 , q = exp(t 2 ) = 143

285


where q0 is an arbitrary integration constant (recall that t 2 is obtained by integration). As for F , we have 1 √ 1 1 √ F = i 3s 2 + s 3 − i 3s 4 + O(s 5 ). 6 6 18 1

We introduce the small quantity X, independent of H : X := t 3 q 3 . For example, if 1 √ √ 1 1 i 3)q03 of 143 i 3q0 we compute we take the cubic root (− 16 + 18

1 1 1 √ 3 1 √ 2 3 3 − i 3 s− + i 3 s + O(s ) . X = q0 2 2 2 2 Another choice of the branch of the cubic root does not affect the final result (actually, we will see that F − F0 is a series in X 3 ). We invert the series and find s = s(X), and then we find H = τ3 (X)/t 3 as a series in X → 0. Finally, F is computed through (55): 1 1 3 1 1 31 X + X6 + X9 + X 12 + F − F0 = 3 2 3 4 t 2q0 120q0 3360q0 1995840q0 1559 X 15 + O(X 17 ) . + 1556755200q05 We obtained this expansion through the expansions of the Mi ’s and of y(x) at order 16. If we put q0 = 1, this is exactly Kontsevich’s solution (56), with N1 = 1,

N2 = 1,

N3 = 12,

N4 = 620,

N5 = 87304.

Though not completely satisfactory to our theoretical purposes, the above is an explicit procedure to compute the Gromov–Witten invariants Nk , which is alternative to the usual procedure consisting in the direct substitution of the expansion (56) in the WDVV equations.1 The formulae (52)–(54) and (55) are completely explicit as rational functions of dy/dx, y(x) and its quadratures. Still, the problem of inversion to obtain a closed form F (t) close to a critical point is very hard because the behavior of y(x) is not 1 We observe that for the very special case of QH ∗ (CP 2 )

∂2 (F − F0 ) = u1 + u2 + u3 − 3t 1 . ∂(t 2 )2 This follows from the computation of the intersection form of the Frobenius manifold QH ∗ (CP 2 ) in terms of F (see [11]). Therefore, ∂2 (F − F0 ) = u1 + u2 + u3 − 3u1 − 3a(x)H ∂(t 2 )2 = H (1 + x − 3a(x)). The above formula allows to compute F − F0 faster than (55).

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given by a Puiseux series, but it is oscillatory. We plan to continue the project of this paper of understanding the global analytic structure of QH ∗ (CP2 ) in the near future. 9. Appendix We present a procedure to compute the expansion of the Painlevé transcendents of PVIµ and of the solutions of dM1 1 = M2 M3 , dx x

1 dM2 = M1 M3 , dx 1−x

1 dM3 = M1 M2 (57) dx x(x − 1)

close to the critical point x = 0. The corresponding solution of the PVIµ equation is −xR(x) M1 M2 + µM3 2 . (58) y(x) = , R(x) := 1 − x(1 + R(x)) µ2 + M22 9.1. EXPANSION WITH RESPECT TO A SMALL PARAMETER We want to study the behavior of the solution of (57) for x → 0. Let x := Iz, where I is the small parameter. The system (57) becomes: 1 dM1 = M2 M3 , dz z

I dM2 = M1 M3 , dz 1 − Iz

1 dM3 = M1 M2 . dz z(Iz − 1)

(59)

The coefficient of the new system are holomorphic for I ∈ E := {I ∈ C | |I| I0 } and for 0 < |z| < |I10 | , in particular for z ∈ D := {z ∈ C|R1 |z| R2 }, where R1 and R2 are independent of I and satisfy 0 < R1 < R2 < I10 . We will use the small parameter expansion as a formal way to compute the expansions of the Mj s for x → 0, the only justification being that in the cases we apply it we find expansions in x which we already know they are convergent. To our knowledge, there is no rigorous justification of the (uniform) convergence of the expansions for the Mj ’s in terms of the variable x restored after the small parameter expansion in powers of I. For I ∈ E and z ∈ D we can expand the fractions as follows: 1 dM1 = M2 M3 , dz z

∞ dM2 =I z n I n M1 M3 , dz n=0

∞

1 n n dM3 =− z I M1 M2 , (60) dz z n=0

and we look for a solution expanded in powers of I: Mj (z, I) =

∞ n=0

n M(n) j (z)I ,

j = 1, 2, 3.

(61)

287

INVERSE PROBLEM FOR FROBENIUS MANIFOLDS 0 We compute the M(n) j ’s substituting (61) into (60). To order I we find

= 0 ⇒ M(0) M(0) 2 2 =

iσ , 2

1 M(0) = M(0) M(0), 1 z 2 3

1 M(0) = − M(0) M(0) , 3 z 2 1

where σ is so far an arbitrary constant, and the prime denotes the derivative w.r.t. (0) z. Then we solve the linear system for M(0) 1 and M3 and find σ ˜ − σ2 + az ˜ 2, M(0) 1 = bz

σ ˜ − σ2 − i az M(0) ˜ 2, 3 = i bz

where a˜ and b˜ are integration constants. The higher orders are M(n) 2 (z)

=

z

dζ

n−1

ζk

k=0

n−1−k

(n−1−k−l) M(l) (ζ ), 1 (ζ )M3

l=0

1 = M(0) M(n) + A(n) M(n) 1 1 (z), z 2 3 1 = − M(0) M(n) + A(n) M(n) 3 3 (z), z 2 1 where 1 (k) M (z)M(n−k) (z), 3 z k=1 2 n n−k n 1 (k) (n−k−l) M (z)M(n−k) (z) + zk M(l) (z) . A3 (z) = − 1 1 (z)M2 z k=1 2 k=1 l=0 n

A(n) 1 (z) =

(n) The system for M(n) 1 , M3 is closed and nonhomogeneous. By variation of parameters we obtain the particular solution zσ/2 z z−σ/2 z 1+ σ (n) (n) 1− σ2 (n) dζ ζ R1 (ζ ) − ζ 2 R1 (ζ ), M1 (z) = σ σ z (n) M1 (z) − A(n) M(n) 3 (z) = 1 (z) , iσ/2

where 1 iσ (n) A3 (z) + A(n) R1(1)(z) = A(n) 1 (z) + 1 (z) . z 2z Restoring x we have: Mj (x) = x − 2 σ

∞

σ

(j )

bkq x k+(1−σ )q + x 2

k,q=0

M2 (x) =

∞ k,q=0

(2) k+(1−σ )q bkq x +

∞

(j )

akq x k+(1+σ )q ,

j = 1, 3,

k,q=0 ∞ k,q=0

(2) k+(1+σ )q akq x .

(62)

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The coefficients akq and bkq contain I. In fact, they are functions of a := aI ˜ −2 , ˜ σ2 . b := bI (j )

σ

(j )

9.2. SOLUTION BY FORMAL COMPUTATION Consider the system (57) and expand the fractions as x → 0. We find 1 dM1 = M2 M3 , dx x

∞

dM2 n = x M1 M3 , dx n=0

∞ 1 n dM3 =− x M1 M2 . dx x n=0

(63)

We can look for a solution written in formal series Mj (x) = x

∞

− σ2

(j ) bkq x k+(1−σ )q

+x

k,q=0

M2 (x) =

∞

(2) k+(1−σ )q bkq x +

k,q=0

σ 2

∞

(j )

akq x k+(1+σ )q ,

j = 1, 3,

k,q=0 ∞

(2) k+(1+σ )q akq x .

k,q=0

Plugging the series into the equation we find solvable relations between the coefficients and we can determine them. For example, the first relations give iσ + M2 = 2

(1) 2 (1) 2 i[b00 i[a00 ] 1+σ ] 1−σ x x + ··· − + ··· , 1−σ 1+σ

(1) − 2 (1) 2 x + · · ·) + (a00 x + · · ·), M1 = (b00 σ

σ

(1) − 2 (1) 2 x + · · ·) + (−ia00 x + · · ·). M3 = (ib00 σ

σ

(1) (1) , a00 . All the coefficients determined by successive relations are functions of σ , b00 These are the three integration constants on which the solution of (57) must depend. (1) (1) with b and a00 with a. We can identify b00

9.3. THE RANGE OF σ IN THE SMALL PARAMETER EXPANSION The above computations make sense if σ is not an odd integer, otherwise some coefficients of the expansions for the Mj ’s diverge (see for example the first terms of M2 in the preceding section). Moreover, the expansion in the small parameter yields the following approximation at order 0 for M2 : M2 ≈ iσ/2 ≡ constant. The approximation at order 1 contains powers z1−σ , z1+σ . If we assume that the approximation at order 0 in I is actually the limit of M2 as x = Iz → 0, than we need −1 < σ < 1. Of course, this makes sense if x → 0 along a radial path (i.e. within a sector of amplitude less than 2π ).


289

The ordering of the expansion (62) is somehow conventional: namely, we could σ σ transfer some terms multiplied by x 2 in the series multiplied by x − 2 , and conversely. I report the first terms:

a2 σ2 σ b2 1−σ 1+σ x + x + x + · · · + M1 (x) = bx − 2 1 − (1 − σ )2 4(1 − σ ) (1 + σ )2

a2 σ2 σ b2 1−σ 1+σ x− x + x + ··· , + ax 2 1 + (1 − σ )2 4(1 + σ ) (1 + σ )2

a2 σ (σ − 2) b2 − σ2 1−σ 1+σ x+ x + x + ··· − 1− M3 (x) = ibx (1 − σ )2 4(1 − σ ) (1 + σ )2

σ (σ + 2) a2 b2 σ 1−σ 1+σ 2 x + x + ··· , x− − iax 1 + (1 − σ )2 4(1 + σ ) (1 + σ )2 b2 σ a 2 1+σ 1−σ x M2 (x) = i + i x − i + ···. 2 (1 − σ )2 1+σ Note that the dots do not mean higher-order terms. There may be terms bigger than those written above (which are computed through the expansion in the small parameter up to order I) depending on the value of σ in (−1, 1). Finally, we note that we can always assume that 0 σ < 1, because that would not affect the expansion of the solutions but for the change of two signs. With this in mind, the expansions above are: M1 = bx − 2 (1 + O(x 1−σ )) + ax 2 (1 + O(x)), σ σ M3 = ibx − 2 (1 + O(x 1−σ )) − iax 2 (1 + O(x)), iσ (1 + O(x 1−σ )). M2 = 2 σ

σ

If we substitute them in the formula (58) we find a Painlevé transcendent with the behavior y(x) ∼ ax 1−σ for x → 0, as we already explained in the Introduction. Acknowledgements I am grateful to B. Dubrovin for introducing me to the theory of Frobenius manifolds and for many discussions and advice. The author is supported by a fellowship of the Japan Society for the Promotion of Science (JSPS). References 1. 2.

Anosov, D. V. and Bolibruch, A. A.: The Riemann–Hilbert Problem, Publ. Steklov Institute Math., 1994. Balser, W., Jurkat, W. B. and Lutz, D. A.: Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations, J. Math. Anal. Appl. 71 (1979), 48–94.

290 3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

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Balser, W., Jurkat, W. B. and Lutz, D. A.: On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities, SIAM J. Math. Anal. 12 (1981), 691–721. Bertola, M.: Jacobi groups, Hurwitz spaces and Frobenius structures, Preprint SISSA 74/98/FM, 1998, to appear in Differential Geom. Appl. Birman, J. S.: Braids, Links, and Mapping Class Groups, Ann. Math. Stud. 82, Princeton Univ. Press, 1975. Coxeter, H. S. M.: Regular Polytopes, Dover, New York, 1963. Di Francesco, P. and Itzykson, C.: Quantum intersection rings, In: R. Dijkgraaf, C. Faber and G. B. M. van der Geer (eds), The Moduli Space of Curves, 1995. Dijkgraaf, R., Verlinde, E. and Verlinde, H.: Topological strings in d < 1, Nuclear Phys. B 352 (1991), 59–86. Dubrovin, B.: Integrable systems in topological field theory, Nuclear Phys. B 379 (1992), 627– 689. Dubrovin, B.: Geometry and itegrability of topological-antitopological fusion, Comm. Math. Phys. 152 (1993), 539–564. Dubrovin, B.: Geometry of 2D topological field theories, In: Lecture Notes in Math. 1620, Springer, New York, 1996, pp. 120–348. Dubrovin, B.: Painlevé trascendents in two-dimensional topological field theory, In: R. Conte (ed.), The Painlevé Property, One Century Later, Springer, New York, 1999. Dubrovin, B.: Geometry and analytic theory of Frobenius manifolds, math.AG/9807034, 1998. Dubrovin, B.: Differential geometry on the space of orbits of a Coxeter group, math.AG/9807034, 1998. Dubrovin, B. and Mazzocco, M.: Monodromy of certain Painlevé-VI trascendents and reflection groups, Invent. Math. 141 (2000), 55–147. Gambier, B.: Sur des équations differentielles du second ordre et du premier degré dont l’intégrale est à points critiques fixes, Acta Math. 33 (1910), 1–55. Guzzetti, D: Stokes matrices and monodromy for the quantum cohomology of projective spaces, Comm. Math. Phys. 207 (1999), 341–383. Also see the preprint math/9904099. Its, A. R. and Novokshenov, V. Y.: The Isomonodromic Deformation Method in the Theory of Painleve Equations, Lecture Notes in Math. 1191, Springer, New York, 1986. Iwasaki, K., Kimura, H., Shimomura, S. and Yoshida, M.: From Gauss to Painlevé, Aspects Math. 16, Vieweg, Braunschweig, 1991. Jimbo, M.: Monodromy problem and the boundary condition for some Painlevé transcendents, Publ. RIMS, Kyoto Univ. 18 (1982), 1137–1161. Jimbo, M., Miwa, T. and Ueno, K.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients (I), Physica D 2 (1981), 306. Jimbo, M. and Miwa, T.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients (II), Physica D 2 (1981), 407–448. Jimbo, M. and Miwa, T.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients (III), Physica D 4 (1981), 26. Kontsevich, M. and Manin, Y. I.: Gromov–Witten classes, quantum cohomology and enumerative geometry, Comm. Math. Phys 164 (1994), 525–562. Malgrange, B.: Équations différentielles à coefficientes polynomiaux, Birkhauser, Basel, 1991. Manin, V. I.: Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, Max Planck Institut fur Mathematik, Bonn, Germany, 1998. Miwa, T.: Painlevé property of monodromy preserving equations and the analyticity of τ -functions, Publ. RIMS 17 (1981), 703–721. Painlevé, P.: Sur les équations differentielles du second ordre et d’ordre supérieur, dont l’intégrale générale est uniforme, Acta Math. 25 (1900), 1–86.


29. 30. 31. 32. 33.

34.

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Ruan, Y. and Tian, G.: A mathematical theory of quantum cohomology, Math. Res. Lett. 1 (1994), 269–278. Saito, K.: Preprint RIMS-288, 1979 and Publ. RIMS 19 (1983), 1231–1264. Saito, K., Yano, T. and Sekeguchi, J.: Comm. Algebra 8(4) (1980), 373–408. Sato, M., Miwa, T. and Jimbo, M.: Holonomic quantum fields. II – The Riemann–Hilbert problem, Publ. RIMS Kyoto. Univ. 15 (1979), 201–278. Shimomura, S.: Painlevé trascendents in the neighbourhood of fixed singular points, Funkcial. Ekvac. 25 (1982), 163–184. Series expansions of Painlevé trascendents in the neighbourhood of a fixed singular point, Funkcial. Ekvac. 25 (1982), 185–197. Supplement to ‘Series expansions of Painlevé trascendents in the neighbourhood of a fixed singular point’, Funkcial. Ekvac. 25 (1982), 363–371. A family of solutions of a nonlinear ordinary differntial equation and its application to Painlevé equations (III), (V), (VI), J. Math. Soc. Japan 39 (1987), 649–662. Witten, E.: Nuclear Phys. B 340 (1990), 281–332.


293

On the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painlevé VI Equation DAVIDE GUZZETTI Research Institute for Mathematical Sciences (RIMS), Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan. e-mail: [email protected] (Received: 4 May 2001, in final form: 20 November 2001) Abstract. In this paper we find a class of solutions of the sixth Painlevé equation appearing in the theory of WDVV equations. This class covers almost all the monodromy data associated to the equation, except one point in the space of the data. We describe the critical behavior close to the critical points in terms of two parameters and we find the relation among the parameters at the different critical points (connection problem). We also study the critical behavior of Painlevé transcendents in the elliptic representation. Mathematics Subject Classification (2000): 34M55. Key words: Painlevé equation, elliptic function, isomonodromic deformation, Fuchsian system, connection problem, monodromy.

1. Introduction This work is devoted to the study of the critical behavior of the solutions of a Painlevé VI equation given by a particular choice of the four parameters α, β, γ , δ of the equation (the notations are the standard ones, see [18]): 1 (2µ − 1)2 , β = γ = 0, δ= , µ ∈ C. 2 2 The equation is 2 1 1 dy 1 1 dy 1 1 1 d2 y + + + + + = − 2 dx 2 y y−1 y−x dx x x − 1 y − x dx x(x − 1) 1 y(y − 1)(y − x) 2 + (2µ − 1) , µ ∈ C. (1) + 2 x 2 (x − 1)2 (y − x)2 α=

Such an equation will be denoted PVIµ in the paper. The motivation of our work is that (1) is equivalent to the WDVV equations of associativity in two-dimensional topological field theory introduced by Witten [39], Dijkgraaf, E. Verlinde and H. Verlinde [6]. Such an equivalence is discussed in [9] and it is a consequence

294

DAVIDE GUZZETTI

of the theory of Frobenius manifolds. Frobenius manifolds are the geometrical setting for the WDVV equations and were introduced by Dubrovin in [7]. They are an important object in many branches of mathematics like singularity theory and reflection groups [9, 12, 34, 35], algebraic and enumerative geometry [24, 26]. The six classical Painlevé equations were discovered by Painlevé [31] and Gambier [15], who classified all the second-order ordinary differential equations of the type d2 y dy , = R x, y, dx 2 dx where R is rational in dy/dx, x and y. The Painlevé equations satisfy the Painlevé property of absence of movable branch points and essential singularities. These singularities will be called critical points; for PVIµ they are 0, 1, ∞. The behavior of a solution close to a critical point is called critical behavior. The general solution of the sixth Painlevé equation can be analytically continued to a meromorphic function on the universal covering of P1 \{0, 1, ∞}. For generic values of the integration constants and of the parameters in the equation, it cannot be expressed via elementary or classical transcendental functions. For this reason, it is called a Painlevé transcendent. The critical behavior for a class of solutions to the Painlevé VI equation was found by Jimbo in [20] for the general Painlevé equation with generic values of α, β, γ δ (we refer to [20] for a precise definition of generic). A transcendent in this class has the behavior (0)

y(x) = a (0) x 1−σ (1 + O(|x|δ )), y(x) = 1 − a (1) (1 − x) y(x) = a

(∞)

x

−σ (∞)

1−σ (1)

x → 0,

(1 + O(|1 − x|δ )), −δ

(1 + O(|x| )),

x → ∞,

(2) x → 1,

(3) (4)

where δ is a small positive number, a (i) and σ (i) are complex numbers such that a (i) = 0 and 0 σ (i) < 1.

(5)

We remark that x converges to the critical points inside a sector with vertex on the corresponding critical point. The connection problem, i.e. the problem of finding the relation among the three pairs (σ (i) , a (i) ), i = 0, 1, ∞, was solved by Jimbo in [20] for the above class of transcendents using the isomonodromy deformations theory. He considered a Fuchsian system A0 (x) Ax (x) A1 (x) dY = + + Y dz z z−x z−1 such that the 2 × 2 matrices Ai (x) (i = 0, x, 1 are labels) satisfy Schlesinger equations. This ensures that the dependence on x is isomonodromic, according to

CRITICAL BEHAVIOR OF PAINLEVE´ VI

295

the isomonodromic deformation theory developed in [21]. Moreover, for a special choice of the matrices, the Schlesinger equations are equivalent to the sixth Painlevé equation, as it is explained in [22]. In particular, the local behaviors (2), (3), (4) were obtained using a result on the asymptotic behavior of a class of solutions of Schlesinger equations proved by Sato, Miwa and Jimbo in [33]. The connection problem was solved because the parameters σ (i) , a (i) were expressed as functions of the monodromy data of the Fuchsian system. For studies on the asymptotic behavior of the coefficients of Fuchsian systems and Schlesinger equations see also [5]. Later, Dubrovin and Mazzocco [13] applied Jimbo’s procedure to PVIµ , with the restriction that 2µ ∈ / Z. We remark that this case was not studied by Jimbo, being a nongeneric case. Dubrovin and Mazzocco obtained a class of transcendents with behaviors (2), (3), (4) (again, x converges to a critical point inside a sector) and restriction (5). They also solved the connection problem. In the case of PVIµ , the monodromy data of the Fuchsian system, to be introduced later, turn out to be expressed in terms of a triple of complex numbers (x0 , x1 , x∞ ). The two integration constants in y(x) and the parameter µ are contained in the triple. The following relation holds: 2 − x0 x1 x∞ = 4 sin2 (π µ). x02 + x12 + x∞

(6)

There exists a one-to-one correspondence between triples (define up to the change of two signs) and branches of the Painlevé transcendents. In other words, any branch y(x) is parameterized by a triple y(x) = y(x; x0 , x1 , x∞ ). As is proved in [13], the transcendents (2), (3), (4) are parameterized by a triple according to the formulae 2 2 π (i) σ , i = 0, 1, ∞, 0 σ (i) < 1. xi = 4 sin 2 A more complicated expression gives a (i) = a (i) (x0 , x1 , x∞ ) in [13]. We recall that a branch is defined by the choice of branch cuts, like |arg(x)| < π , |arg(1−x)| < π . The analytic continuation of a branch when x crosses the cuts is obtained by an action of the braid group on the triple. This is explained in [13] and in Section 6. As we mentioned above, it is very important to concentrate on PVIµ due to its equivalence to WDVV equations in 2-D topological field theory, and due to its central role in the construction of three-dimensional Frobenius manifolds. It is known [9] that the structure of a local chart of a Frobenius manifold can in principle be constructed from a set of monodromy data. To any manifold a PVIµ equation is associated and the monodromy data of the local chart are contained in µ and in the There are only some exceptions to the one-to-one correspondence above, which are already

treated in [28]. In order to rule them out, we require that at most one of the entries xi of the triple / {(2, 2, 2) (−2, −2, 2), (2, −2, −2), (−2, 2, −2)}. Two triples may be zero and that (x0 , x1 , x∞ ) ∈ which differ by the change of two signs identify the same transcendent. They are called equivalent triples. The one-to-one correspondence is between transcendents and classes of equivalence.

296

DAVIDE GUZZETTI

triple (x0 , x1 , x∞ ) of a Painlevé transcendent. The mentioned action of the braid group, which gives the analytic continuation of the transcendent, allows us to pass from one local chart to another. The local structure of a Frobenius manifold is explicitly constructed in [17] starting from the Painlevé transcendents. In [17] it is shown that in order to obtain a local chart from its monodromy data we need to know the critical behavior of the corresponding transcendent in terms of the triple (x0 , x1 , x∞ ) (note that this is equivalent to solving the connection problem). Recently, Frobenius manifolds have become important in enumerative geometry and quantum cohomology [24, 26]. As is shown in [17], it is possible to compute Gromov–Witten invariants for the quantum cohomology of the two-dimensional projective space starting from a special PVIµ , with µ = −1. In this case the triple is (x0 , x1 , x∞ ) = (3, 3, 3), as it is proved in [10] and [16]. Due to the restriction 0 σ (i) < 1, the formulae for the critical behavior and the connection problem obtained by Dubrovin–Mazzocco do not apply if at least one xi (i = 0, 1, ∞) is real and |xi | 2. Thus, they do not apply in the case of quantum cohomology, because xi = 3 and σ (i) = 1. Therefore, the motivation of our paper becomes clear: in the attempt to extend the results of [13] to the case of quantum cohomology, we actually extended them to almost all monodromy data, namely we found the critical behavior and we solved the connection problem for all the triples satisfying xi = ±2 ⇒ σ (i) = 1,

i = 0, 1, ∞.

In order to do this, we extended the Jimbo and Dubrovin and Mazzocco methods and we analyzed the elliptic representation of the Painlevé VI equation. 1.1. OUR RESULTS We observe that the branch y(x; x0 , x1 , x∞ ) has analytic continuation on the universal covering of P1 \{0, 1, ∞}. We still denote this continuation by y(x; x0 , x1 , x∞ ), where x is now a point in the universal covering. Therefore: There is a one-to-one correspondence between triples of monodromy data (x0 , x1 , x∞ ) (defined up to the change of two signs) and Painlevé transcendents, 1, ∞}. namely y(x) = y(x; x0 , x1 , x∞ ), x ∈ P1 \{0, We mentioned that if we fix a branch, namely if we choose branch cuts like |arg x| < π , |arg(1 − x)| < π , then the branch of y(x; x0 , x1 , x∞ ) has analytic ) in the cut plane, where (x0 , x1 , x∞ ) is obtained from continuation y(x; x0 , x1 , x∞ (x0 , x1 , x∞ ) by an action of the braid group (see Section 6 for details). We obtained the following results: (1) A transcendent y(x; x0 , x1 , x∞ ) such that |xi | = 2 has behaviors (2), (3), (4) 1 \{∞} C\{1}, P in suitable domains, to be defined below, contained in C\{0},

297


/ (−∞, 0) ∪ respectively. The exponent are restricted by the condition σ (i) ∈ [1, +∞), which extends (5). (2) The parameters σ (i) , a (i) are computed as functions of (x0 , x1 , x∞ ), and vice versa, by explicit formulae which extend those of [13]. (3) If we enlarge the domains where (2), (3), (4) hold, the behavior of y(x; x0 , x1 , x∞ ) becomes oscillatory. The movable poles of the transcendent lie outside the enlarged domains. In proving this, we investigated the elliptic representation of the transcendent, providing a general result stated in Theorem 3 below. / We state result (1) in more detail. Let σ (0) be a complex number such that σ (0) ∈ (−∞, 0) ∪ [1, +∞). We introduce additional parameters θ1 , θ2 ∈ R, 0 < σ˜ < 1 to define a domain D ; σ (0) ; θ1 , θ2 , σ˜ 0 s.t. |x| < , e−θ1 σ (0) |x|σ˜ |x σ (0) | e−θ2 σ (0) , 0 < σ˜ < 1 , := x ∈ C which can can be rewritten as |x| < , σ (0) log |x| + θ2 σ (0) σ (0) arg(x) ( σ (0) − σ˜ ) log |x| + θ1 σ (0). For real σ (0), the domain is more simply defined as 0 s.t. |x| < }, for 0 σ (0) < 1. D σ (0); := {x ∈ C

(7)

For simplicity, we study the critical behavior of the transcendent for x → 0 along the family of paths defined below. Such paths start at some point x0 belonging to the domain. If σ = 0, any regular path will be allowed. If σ = 0, we considered the family |x| |x0 | < , arg x = arg x0 +

σ (0) − |x| , ln (0) σ |x0 |

0 σ˜ .

(8)

The condition 0 σ˜ ensures that the paths remain in the domain as x → 0. In general, these paths are spirals. / (−∞, 0) ∪ [1, +∞), for any a (0) ∈ C, THEOREM 1. Let µ = 0. For any σ (0) ∈ (0) a = 0, for any θ1 , θ2 ∈ R and for any 0 < σ˜ < 1, there exists a sufficiently small positive and a small positive number δ such that Equation (1) has a solution (0) (9) y x; σ (0) , a (0) = a(x)x 1−σ (1 + O(|x|δ )), 0 < δ < 1, as x → 0 along (8) in the domain D(; σ (0); θ1 , θ2 , σ˜ ) defined for nonreal σ (0), or along any regular path in D(; σ (0)) defined for real 0 σ (0) < 1. The amplitude a(x) is

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DAVIDE GUZZETTI

a(x) := a (0) , for 0 < σ˜ , or for real σ (0)

σ (0) 2 2iα(x) 1 1

σ (0)

iα(x) (0)

x e a(x) := a + 1 + (0) x0 e = O(1), 2a 16[a (0) ]2 0 for = 0, (10) where we have used the notation α(x) to denote the real phase of x σ (0) ≡ |x0σ |eiα(x) , when = 0.

(0)

(0)

= |x σ |eiα(x)

Note that in the case (10), we can rewrite as (0) π i iσ ln x − ln 4a (0) − x(1 + O(|x|δ )). (11) y x; σ (0) , a (0) = sin2 2 2 2 For brevity, we will sometimes denote the domain by D(σ (0)). The condition µ = 0 is not restrictive because PVIµ=0 coincides with PVIµ=1 . From Theorem 1 and the symmetries of (1), we prove the existence of solutions with the following local behaviors: (1) y x, σ (1) , a (1) = 1 − a (1) (1 − x)1−σ (1 + O(|1 − x|δ )), a (1) = 0,

x → 1,

σ (1) ∈ / (−∞, 0) ∪ [1, +∞)

and

y x; σ

(∞)

,a

a (∞) = 0,

(∞)

=a

(∞) σ (∞)

x

1 1+O , |x|δ

x → ∞,

σ (∞) ∈ / (−∞, 0) ∪ [1, +∞)

in domains D(σ (1)), D(σ (∞)) given by (51), (48), respectively. The critical behaviors above coincide with (2), (3) and (4) for 0 σ (i) < 1, i = 0, 1, ∞. But our result is more general because it extends the range of σ (i) to σ (i) < 0 and σ (i) 1. For this larger range, x may tend to x = i (i = 0, 1, ∞) along a spiral, according to the shape of D(σ (i)). For more comments, see Sections 3 and 7. Result (2) is stated in the theorem below – where we write σ, a instead of (0) σ , a (0) – and in its comment. THEOREM 2. Let µ be any non zero complex number. The transcendent y(x; σ, a) of Theorem 1, defined for σ ∈ / (−∞, 0) ∪ [1, +∞) and a = 0, is the representation of a transcendent y(x; x0 , x1 , x∞ ) in D(σ ). The triple (x0 , x1 , x∞ ) is uniquely determined (up to the change of two signs) by the following formulae: (i) σ = 0, ±2µ + 2m for any m ∈ Z.

299


π σ , x0 = 2 sin 2 √ 1 1 a − G(σ, µ) √ , x1 = i f (σ, µ)G(σ, µ) a √ 1 πσ 1 a + G(σ, µ)e−i 2 √ , x∞ = − iπ2σ a f (σ, µ)G(σ, µ)e where

2 cos2 π2 σ , f (σ, µ) = cos(π σ ) − cos(2π µ) 2 4σ σ +1 1 2 . G(σ, µ) = 2 1 − µ + σ2 µ + σ2 √ √ Any sign of a is good (changing the sign of a is equivalent to changing the sign of both x1 , x∞ ).

(ii) σ = 0 x0 = 0,

√ x12 = 2 sin(π µ) 1 − a,

√ 2 x∞ = 2 sin(π µ) a.

We can take any sign of the square roots (iii) σ = ±2µ + 2m. (iii1 ) σ = 2µ + 2m, m = 0, 1, 2, . . . x0 = 2 sin(π µ), x1 = −

i 16µ+m (µ + m + 12 )2 1 √ , 2 (m + 1)(2µ + m) a

x∞ = ix1 e−iπµ (iii2 ) σ = 2µ + 2m, m = −1, −2, −3, . . . x0 = 2 sin(π µ), √ 1 π2 a, x1 = 2i cos2 (π µ) 16µ+m (µ + m + 12 )2 (−2µ − m + 1)(−m) x∞ = −ix1 eiπµ , (iii3 ) σ = −2µ + 2m, m = 1, 2, 3, . . . x0 = −2 sin(π µ), x1 = −

i 16−µ+m (−µ + m + 12 )2 1 √ , 2 (m − 2µ + 1)(m) a

x∞ = ix1 eiπµ ,

300

DAVIDE GUZZETTI

(iii4 ) σ = −2µ + 2m, m = 0, −1, −2, −3, . . . x0 = −2 sin(π µ), x1 = 2i

√ π2 1 a, cos2 (π µ) 16−µ+m (−µ + m + 12 )2 (2µ − m)(1 − m)

x∞ = −ix1 e−iπµ . 2 − x0 x1 x∞ = 4 sin2 (π µ) is In all the above formulae, the relation x02 + x12 + x∞ automatically satisfied. Note that σ = 1 implies x0 = ±2. Changes of two signs in the triple of the formulae above are allowed. 2 −x0 x1 x∞ = Conversely, a transcendent y(x; x0 , x1 , x∞ ), such that x02 +x12 +x∞ 2 4 sin (π µ), xi = ±2, has representation y(x; σ, a) in D(σ ) of Theorem 1 with parameters σ and a obtained as follows:

(I) Generic case cos(π σ ) = 1 − a=

x02 , 2

iG(σ, µ)2 2 2(1 + e−iπσ ) − f (x0 , x1 , x∞ )(x∞ + e−iπσ x12 ) × 2 sin(π σ ) × f (x0 , x1 , x∞ ),

where f (x0 , x1 , x∞ ) := f (σ (x0 ), µ) = =

4 − x02 2 − x02 − 2 cos(2π µ)

4 − x02 . 2 −x x x x12 + x∞ 0 1 ∞

σ is determined up to the ambiguity σ → ±σ + 2n, n ∈ Z [see remark below]. If σ is real we can only choose the solution satisfying 0 σ < 1. Any solution σ of the first equation must satisfy the additional restriction σ = ±2µ + 2m for any m ∈ Z, otherwise we encounter the singularities in G(σ, µ) and in f (σ, µ). (II) x0 = 0. σ = 0,

a=

2 x∞ 2 x12 + x∞

provided that x1 = 0 and x∞ = 0, namely µ ∈ / Z. 2 = −x12 exp(±2π iµ). Four cases (III) x02 = 4 sin2 (π µ). Then (6) implies x∞ which yield the values of σ not included in (I) and (II) must be considered


301

2 = −x12 e−2πiµ , then (III1 ) If x∞

σ = 2µ + 2m, m = 0, 1, 2, . . . , 1 162µ+2m (µ + m + 12 )4 a=− 2 . 4x1 (m + 1)2 (2µ + m)2 2 = −x12 e2πiµ then (III2 ) If x∞

σ = 2µ + 2m, m = −1, −2, −3, . . . , cos4 (π µ) 2µ+2m 16 × a=− 4 4π 4 × µ + m + 12 (−2µ − m + 1)2 (−m)2 x12 . 2 = −x12 e2πiµ then (III3 ) If x∞

σ = −2µ + 2m, m = 1, 2, 3, . . . , 1 16−2µ+2m (−µ + m + 12 )4 . a=− 2 4x1 (m − 2µ + 1)2 (m)2 2 (III4 ) If x∞ = −x12 e−2πiµ then

σ = −2µ + 2m, m = 0, −1, −2, −3, . . . , 4 cos4 (π µ) −2µ+2m 16 −µ + m + 12 × a=− 4 4π × (2µ − m)2 (1 − m)2 x12 . Let us restore the notation σ (0), a (0) . At x = 1, ∞ the exponents σ (i) , i = 1, ∞ are given by cos(π σ (i) ) = 1 − (xi2 /2) and the coefficients a (1) , a (∞) are obtained from the formula of a = a (0) of Theorem 2, provided that we do the substitutions and (x0 , x1 , x∞ ) → (x1 , x0 , x0 x1 − x∞ ), σ (0) → σ (1) (0) (∞) (x0 , x1 , x∞ ) → (x∞ , −x1 , x0 − x1 x∞ ), σ → σ , respectively. This also solves the connection problem for the transcendents y(x; σ (i) , a (i) ), because we are able to compute (σ (i) , a (i) ) for i = 0, 1, ∞ in terms of a fixed triple (x0 , x1 , x∞ ). Remark. Let (x0 , x1 , x∞ ) be given and let us compute σ and a by the formulae of Theorem 2. The equation cos(π σ ) = 1 −

x02 2

(12)

does not determine σ uniquely. We can choose σ such that 0 σ 1. This convention will be assumed in the paper. Therefore, all the solutions of (12) are ±σ + 2n, n ∈ Z. If σ is real, we can only choose 0 σ < 1. With this convention,

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DAVIDE GUZZETTI

there is a one-to-one correspondence between (σ, a) and (a class of equivalence, defined by the change of two signs, of) an admissible triple (x0 , x1 , x∞ ). We observe that σ = σ (x0 ) and a = a(σ ; x0 , x1 , x∞ ); namely, the transformation σ → ±σ + 2n affects a. The transcendent y(x; x0 , x1 , x∞ ) has representation y(x; σ (x0 ), a(σ ; x0 , x1 , x∞ )) in D(σ ). If we choose another solution ±σ + 2n we again have y(x; x0 , x1 , x∞ ) = y(x; ±σ (x0 ) + 2n, a(±σ (x0 ) + 2n; x0 , x1 , x∞ )) in the new domain D(±σ + 2n). Hence – and this is very important! the transcendent y(x; x0 , x1 , x∞ ) has different representations and different critical behaviors in different domains. Outside the union of these domains we are not able to describe the transcendents and we believe that the movable poles lie there (we show this in one example in the paper). According to the above remark, we restrict to the case 0 σ (i) 1, σ (i) = 1. So the critical behaviors of y(x; σ (i) , a (i) ) coincide with (2), (3), (4) when 0 σ (i) < 1. But for σ (i) = 1 the critical behaviors (2), (3), (4) hold true only if x converges to a critical point along spirals. We finally describe the third result. In the case σ (i) = 1, we obtained the critical behaviors along radial paths using the elliptic representation of Painlevé transcendents. We only consider now the critical point x = 0, because the symmetries of (1), to be discussed in Section 7, yield the behavior close to the other critical points. The elliptic representation was introduced by Fuchs in [14]: 1+x u(x) ; ω1 (x), ω2 (x) + . y(x) = ℘ 2 3 Here u(x) solves a nonlinear second-order differential equation to be studied later and ω1 (x), ω2 (x) are two elliptic integrals, expanded for |x| < 1 in terms of hypergeometric functions: 1 2 ∞ π 2 n n x , ω1 (x) = 2 n=0 (n!)2 ∞ 1 2 ∞ 1 2

n i 2 n n 2 n 1 x ln(x) + 2 ψ n + 2 − ψ(n + 1) x , ω2 (x) = − 2 n=0 (n!)2 (n!)2 n=0 where ψ(z) := d/dz ln (z). We introduce a new domain, depending on two complex numbers ν1 , ν2 and on the small real number r:

−iπν

iπν

1

e

e 1 ν

2−ν2

2

˜ D(r; ν1 , ν2 ) := x ∈ C0 such that |x| < r, 2−ν x

< r, 16ν2 x < r . 16 2 The domain can be also written as follows: |x| < r, ν2 ln |x| + C1 − ln r < ν2 arg x < ( ν2 − 2) ln |x| + C2 + ln r, C2 := C1 + 8 ln 2, C1 := −[4 ln 2 ν2 + π ν1 ], if ν2 = 0. If ν2 = 0, the domain is simply |x| < r.

303


/ (−∞, 0] ∪ [2, +∞), there THEOREM 3. For any complex ν1 , ν2 such that ν2 ∈ exists a sufficiently small r < 1 such that PVIµ has a solution of the form y(x) = ℘ (ν1 ω1 (x) + ν2 ω2 (x) + v(x); ω1 (x), ω2 (x)) +

1+x 3

in the domain D(r; ν1 , ν2 ) defined above. The function v(x) is holomorphic in D(r; ν1 , ν2 ) and has convergent expansion −iπν1 m

n n e 2−ν2 an x + bnm x x + v(x) = 162−ν2 n1 n0, m1 iπν1 m

n e cnm x x ν2 , (13) + 16ν2 n0, m1

where an , bnm , cnm are certain rational functions of ν2 . Moreover, there exists a constant M(ν2 ) depending on ν2 such that

−iπν

iπν

1

e

e 1 ν

2−ν2

2

v(x) M(ν2 ) |x| + 2−ν x

+ 16ν2 x

16 2 in D(r; ν1 , ν2 ). Theorem 3 allows us to compute the critical behavior. We consider a family of paths along which x may tend to zero, contained in the domain of the theorem. If 0 < ν2 < 2, any regular path is allowed. If ν2 is any nonreal number, we consider the following family, starting at x0 ∈ D(r; ν1 , ν2 ): |x| |x0 | < r,

arg(x) = arg(x0 ) +

ν2 − V |x| , ln ν2 |x0 |

0 V 2. (14)

The restriction 0 V 2 ensures that the paths remain in the domain as x → 0. COROLLARY. Consider a transcendent y(x) = ℘ (ν1 ω1 (x) + ν2 ω2 (x) + v(x); ω1 (x), ω2 (x)) +

1+x 3

of Theorem 3. Its critical behavior for x → 0 in D(r; ν1 , ν2 ) along (14) if ν2 = 0 and 0 < V < 2, or along any regular path if 0 < ν2 < 2 is 1 eiπν1 1 eiπν1 −1 2−ν2 1 ν2 x− x (15) x − (1 + O(x δ )), y(x) = 2 4 16ν2 −1 4 16ν2 −1 for some 0 < δ < 1. If ν2 = 0 and V = 0 the behavior along (14) is y(x) =

2

sin −i ν22 ln x + i ν22 ln 16 + × (1 + O(x)).

1 πν1 2

+

∞

m=1 c0m (ν2 )

eiπ ν1 16ν2

x ν2

m ×

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DAVIDE GUZZETTI

If ν2 = 0 and V = 2, the behavior along (14) is y(x) =

sin ln x + × (1 + O(x)). 2

2 i 2−ν 2

1 i ν22−2

ln 16 +

πν1 2

+

∞ m=1

b0m (ν2 )

e−iπ ν1 162−ν2

x 2−ν2

m ×

Note that (15) is 1 eiπν1 (16) x ν2 (1 + O(x δ )), if 0 < V < 1, or 0 < ν2 < 1, y(x) = − 4 16ν2 −1 1 eiπν1 −1 2−ν2 y(x) = − x (1 + O(x δ )), if 1 < V < 2, or 1 < ν2 < 2 (17) 4 16ν2 −1 and

1 − ν2 |x| π ν1 ln + x(1 + O(x)), y(x) = sin i 2 16 2 2

if V = 1, or ν2 = 1.

(18)

The elliptic representation has been studied from the point of view of algebraic geometry in [27] but, to our knowledge, Theorem 3 and its corollary are the first general results on its critical behavior. We note, however, that for the very special value µ = 12 the function v(x) vanishes; the transcendents are called Picard solutions in [28] because they were known to Picard [32]. Their critical behavior is studied in [28] and agrees with the corollary. Comparing (9) with (16), we prove in Section 5.1 that the transcendent of Theorem 3 coincides with y(x; σ (0) , a (0) ) of Theorem 1 on the domain D(, σ (0)) ∩ D(r; ν1 , ν2 ) with the identification 1 eiπν1 σ (0) = 1 − ν2 and a (0) = − 4 16ν2 −1 (note also that (11) is (18)). The identification of a (0) and σ (0) makes it possible to connect ν1 and ν2 to the monodromy data (x0 , x1 , x∞ ) according to Theorem 2 and to solve the connection problem for the elliptic representation. The corollary provides the behavior of the transcendents when σ (0) = 1 (σ (0) = 1) and x → 0 along a radial path. This corresponds to the case ν2 = 0 (ν2 = 0), with the identification 1 eiπν1 (0) (0) . σ = 1 − ν2 and a = − 4 16ν2 −1 The critical behavior along a radial path is then y(x) =

2 ν

sin 2 ln x − ν ln 16 + x → 0.

1 πν1 2

+

∞

m=1 c0m (ν)

eiπ ν1 16iν

x iν

m (1 + O(x)), (19)

305


∞

The number ν is real, ν = 0 and σ (0) = 1 − iν. The series m=1 c0m (ν)[(eiπν1 / eiν )x iν ]m converges and defines a holomorphic and bounded function in the domain D(r; ν1 , iν) |x| < r,

C1 − ln r < ν arg x < −2 ln |x| + C2 + ln r.

Note that not all the values of arg x are allowed, namely C1 − ln r < ν arg(x). Our belief is that y(x) may have movable poles if we extend the range of arg x. We are not able to prove it in general, but we will give an example in Section 5. We finally remark that the critical behavior of Painlevé transcendents can also be investigated using a representation due to Shimomura [19, 37]. We will review this representation in the paper. However, the connection problem in this representation was not solved. To summarize, in this paper we give an extended and unified picture of both elliptic and Shimomura’s representations and Dubrovin and Mazzocco’s works, showing that the transcendents obtained in these three different ways all are included in the wider class of Theorem 1. In this way we solve the connection problem for elliptic and Shimomura’s representations by virtue of Theorem 2. Finally, Theorem 3 provides the oscillatory behavior along radial paths when σ (0) = 1. 2. Monodromy Data and Review of Previous Results Before giving further details about the result stated above, we review the connection between PVIµ and the theory of isomonodromic deformations. We also give a detailed expositions of the results of [13, 28]. The equation PVIµ is equivalent to the equations of isomonodromy deformation (Schlesinger equations) of the Fuchsian system A1 (u) A2 (u) A3 (u) dY = A(z; u)Y, A(z; u) := + + , dz z − u1 z − u2 z − u3 u := (u1 , u2 , u3 ),

tr(Ai ) = det Ai = 0,

3

Ai = −diag(µ, −µ). (20)

i=1

The dependence of the system (20) on u is isomonodromic, as it is explained below. From the system we obtain a transcendent y(x) of PVIµ as follows: x=

u2 − u1 , u3 − u1

y(x) =

q(u) − u1 , u3 − u1

where q(u1 , u2 , u3 ) is the root of [A(q; u1 , u2 , u3 )]12 = 0,

if µ = 0.

The case µ = 0 is disregarded, because PVIµ=0 ≡ PVIµ=1 .

306

DAVIDE GUZZETTI

Conversely, given a transcendent y(x) the system (20) associated to it is obtained as follows. Let’s define x )−ζ k0 exp (2µ − 1) dζ y(ζ ζ(ζ −1) k = k(x, u3 − u1 ) := , k0 ∈ C\{0}. (u3 − u1 )2µ−1 We have Ai = −µ

φi1 φi3 2 φi1

2 −φi3 φi1 φi3

,

i = 1, 2, 3,

(21)

where

√ √ k y φ13 = i √ √ , u3 − u1 x √ √ k y−x φ23 = − √ , √ √ u3 − u1 x 1 − x √ √ k y−1 , √ φ33 = i √ u3 − u1 1 − x √ √ u3 − u1 y 2µ i 2 A B+ + µ (y − 1 − x) , √ φ11 = √ 2µ2 y k(x) x √ √ 2µ 1 u3 − u1 y − x 2 A B+ + µ (y − 1 + x) , φ21 = − 2 √ √ √ 2µ k(x) x 1 − x y−x √ √ 2µ i u3 − u1 y − 1 2 A B + + µ (y + 1 − x) , φ31 = √ √ 2µ2 y−1 k(x) 1 − x 1 dy x(x − 1) − y(y − 1) , A = A(x) := 2 dx A . B = B(x) := y(y − 1)(y − x)

Any branch of the square roots can be chosen. For a derivation of the above formulae, see [9, 17, 22]. The system (20) has Fuchsian singularities at u1 , u2 , u3 . Let us fix a branch Y (z, u) of a fundamental matrix solution by choosing branch cuts in the z plane and a basis of loops in π(C\{u1, u2 , u3 }; z0 ), where z0 is a base-point. Let γi be a basis of loops encircling counter-clockwise the point ui , i = 1, 2, 3. See Figure 1. Then Y (z, u) → Y (z, u)Mi ,

i = 1, 2, 3,

det Mi = 0,

if z goes around a loop γi . Along the loop γ∞ := γ1 · γ2 · γ3 we have Y → Y M∞ , M∞ = M3 M2 M1 . The 2 × 2 matrices Mi are the monodromy matrices, and

307


Figure 1.

they give a representation of the fundamental group called monodromy representation. The transformations Y (z, u) = Y (z, u)B, det(B) = 0 yields all possible fundamental matrices, hence the monodromy matrices of (20) are defined up to conjugation Mi → Mi = B −1 Mi B. From the standard theory of Fuchsian systems, it follows that we can choose a fundamental solution behaving as follows I + O 1z z−µˆ zR C∞ , z → ∞, (22) Y (z; u) = Gi [I + O(z − ui )](z − ui )J Ci , z → ui , i = 1, 2, 3, where J = and

0 1 0 0

,

µˆ = diag(µ, −µ),

Gi J G−1 i = Ai

 0, if 2µ ∈ / Z,        0 R12  , µ>0  0 0 R=  if 2µ ∈ Z.   0 0     , µ0  if 2µ ∈ Z, 0 0 R=     0 0    , µ 0), (µ < 0), α = 0, C∞ C˜ ∞ 0 α β α    (iii) Any invertible matrix, if 2µ ∈ Z, R = 0.

if 2µ ∈ Z, R = 0, (27)

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DAVIDE GUZZETTI

Once the RH is solved, the sum of the matrix coefficients Ai (u) of the solution A(z; u1 , u2 , u3 ) =

3

Ai (u) z − ui i=1

must be diagonalized (to give − diag(µ, −µ)). After that, a branch y(x) of PVIµ can be computed from [A(q; u1 , u2 , u3 )]12 = 0. The fact that the RH has a unique solution for the given monodromy data (if 2µ ∈ / Z or 2µ ∈ Z and R = 0) means that there is a one-to-one correspondence between the triple M1 , M2 , M3 and the branch y(x). We review some known results [13, 28]: (1) One Mi = I if and only if q(u) ≡ ui . This does not correspond to a solution of PVIµ . (2) If the Mi ’s, i = 1, 2, 3, commute, then µ is integer (as follows from the fact that the 2 × 2 matrices with 1’s on the diagonals commute if and only if they can be Then (i) (ii)

z−µˆ C∞ = z−µˆ diag(α, β)C˜ ∞ = diag(α, β)zµˆ C˜ ∞ , 1 z−µˆ z−R C∞ = · · · = αI + |2µ| Q z−µˆ z−R C˜ ∞ , z

where

Q=

(iii)

z−µˆ C

0 β 0 0

or

Q=

0 β

0 0

.

Q1 |2µ| + Q0 + Q−1 z z−µˆ C˜ ∞ , ∞ = ··· = z|2µ|

where Q0 = diag(α, β), Q±1 are, respectively, upper and lower triangular (or lower and upper −1 = Q1 + Q0 + Q−1 . triangular, depending on the sign of µ), and C∞ C˜ ∞ ˜ This implies that the two solutions Y (z; u), Y (z; u) of the form (25) with Cν and C˜ ν , respectively (ν = 1, 2, 3, ∞), are such that Y (z; u)Y˜ (z; u)−1 is holomorphic at each ui , while at z = ∞ it is   (i) G∞ diag(α, β)G−1 ∞, Y (z; u)Y˜ (z; u)−1 → (ii) αI,  (iii) divergent. Thus, the two Fuchsian systems are conjugated only in the cases (i) and (ii), because in those cases Y Y˜ −1 is holomorphic everywhere on P1 , and then it is a constant. In other words, the RH has a unique solution, up to conjugation, for 2µ ∈ / Z or for 2µ ∈ Z and R = 0. Note that if G = I , then 3 A is already diagonal. Therefore, there is no loss of generality ∞ i i=1 if, for 2µ ∈ / Z, we solve the Riemann–Hilbert problem for given M1 , M2 , M3 with the choice of normalization Y (z; u)zµˆ → I if z → ∞. This uniquely determines A1 , A2 , A3 up to diagonal conjugation. Note that for any diagonal invertible matrix D, the sum of D −1 Ai D is still diagonal. On the other hand, if 2µ ∈ Z and R = 0, then Y (z; u)Y˜ (z; u)−1 = α, where α appears in (27), ˜ u) = case (ii). Therefore the two Fuchsian systems obtained from A(z; u) := dY/dzY −1 and A(z; dY˜ /dzY˜ −1 coincide. In both cases, A12 (z, u) changes at most for the multiplication by a constant, therefore the same y(x) is defined by A12 (z, u) = 0 and A˜ 12 (z, u) = 0.

311


simultaneously put in upper or lower triangular form). There are solutions of PVIµ only for 1 iπ a 1 iπ , M2 = , M1 = 0 1 0 1 1 iπ(1 − a) M3 = , a = 0, 1. 0 1 In this case R = 0 and M∞ = I . For µ = 1 the solution is y(x) = ax/(1 − (1 − a)x) and for other integers µ the solution is obtained from µ = 1 by a birational transformation [13, 28]. (3) Noncommuting Mi ’s. The parameters in the space of the monodromy representation, independent of conjugation, are 2 − x12 := tr(M1 M2 ),

2 − x22 := tr(M2 M3 ),

2 − x32 := tr(M1 M3 ).

The triple (x0 , x1 , x∞ ) in the Introduction is (x1 , x2 , x3 ). (3.1) If at least two of the xj ’s are zero, then one of the Mi ’s is I , or the matrices commute. We return to the case (1) or (2). Note that (x1 , x2 , x3 ) = (0, 0, 0) in case (2). (3.2) At most one of the xj ’s is zero. We say that the triple (x1 , x2 , x3 ) is admissible. In this case, it is possible to fully parameterize the monodromy using the triple (x1 , x2 , x3 ). Namely, there exists a fundamental matrix solution such that x2 1 + xx2 x1 3 − x21 1 −x1 1 0 , M2 = , M3 = , M1 = x32 0 1 x1 1 1 − x2 x3 x1

x1

if x1 = 0. If x1 = 0 we just choose a similar parameterization starting from x2 or x3 . The relation M3 M2 M1 similar to e−2πi µˆ e2πiR , implies x12 + x22 + x32 − x1 x2 x3 = 4 sin2 (π µ). The conjugation (26) changes the triple by two signs. Thus, the true parameters for the monodromy data are classes of equivalence of triples (x1 , x2 , x3 ) defined by the change of two signs. We have to distinguish three sub-cases of (3.2): (i) 2µ ∈ / Z. There is a one-to-one correspondence between (classes of equivalence of) monodromy data (x0 , x1 , x∞ ) ≡ (x1 , x2 , x3 ) and the branches of transcendents of PVIµ . The connection problem was solved in [13] for the class of transcendents with critical behavior (0)

y(x) = a (0) x 1−σ (1 + O(|x|δ )), y(x) = 1 − a (1 − x) (1)

y(x) = a (∞) x

−σ (∞)

1−σ (1)

x → 0,

(28)

(1 + O(|1 − x| )),

(1 + O(|x|−δ )),

δ

x → ∞,

x → 1,

(29) (30)

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DAVIDE GUZZETTI

where a (i) and σ (i) are complex numbers such that a (i) = 0 and 0 σ (i) < 1. δ is a small positive number. This behavior is true if x converges to the critical points inside a sector in the x-plane with vertex on the corresponding critical point and finite angular width. In [13], all the algebraic solutions are classified and related to the finite reflection groups A3 , B3 , H3 . (ii) The case µ half integer was studied in [28]. There is an infinite set of Picard type solutions in one-to-one correspondence to triples of monodromy data not in the equivalence class of (2, 2, 2). These solutions form a two parameter family, behave asymptotically as the solutions of the case (i), and comprise a denumerable subclass of algebraic solutions. In this case, R = 0. For any half integer µ = 12 , there is also a one-parameter family of Chazy solutions. In this case, R = 0 and the one-to-one correspondence with monodromy data is lost. In fact, they form an infinite family but any element of the family corresponds to the class of equivalence of the triple (2, 2, 2). The result of our paper applies to Picard’s solutions with xi = ±2. (iii) µ integer. In this case, R = 0. There is a one-to-one correspondence between monodromy data and branches. To our knowledge, this case has not been studied before. There are relevant examples of Frobenius manifolds included in this case, like the case of quantum cohomology of CP2 . For this manifold, µ = −1, the triple (x1 , x2 , x3 ) = (3, 3, 3) (see [10, 16]) and the real part of σ is equal to 1. In this paper, we find the critical behavior and we solve the connection problem for any µ = 0 and for all the triples (x1 , x2 , x3 ) except for the points xi = ±2 ⇒ σ (i) = 1, i = 0, 1, ∞.

3. Critical Behavior – Theorem 1 Theorem 1 has been stated in the Introduction and will be proved in Section 8. Here we give some comments about the domain D(σ ). The superscript of σ (i) , a (i) will be omitted in this section and we concentrate on a small punctured neighborhood of x = 0 (x = 1, ∞ will be treated in Section 7). The point x can be read as a point in the universal covering of C0 := C\{0} with 0 < |x| < ( < 1). / Namely, x = |x|ei arg(x), where −∞ < arg(x) < +∞. Let σ be such that σ ∈ (−∞, 0) ∪ [1, +∞). We defined the domains D(; σ ; θ1 , θ2 , σ˜ ), or D(σ ; ) for real σ in the Introduction. Theorem 1 holds in these domains; the small number depends on σ˜ , θ1 and a. In the following, we may sometimes omit , σ˜ , θi and write simply D(σ ). We observe that |x σ | = |x|σ (x), where σ (x) := σ −

σ arg(x) . log |x|

(R = 0 only in the case (2) of commuting monodromy matrices and µ integer.)

313


In particular, we have σ (x) = σ for real σ . The exponent σ (x) satisfies the restriction 0 σ (x) < 1 for x → 0, if x lies in the domain, because −

θ1 σ θ2 σ σ (x) σ˜ − , ln |x| ln |x|

and θ2 σ → 0, − ln |x|

θ1 σ σ˜ − → σ˜ < 1 ln |x|

for x → 0. Figure 2 shows the domains in the (ln |x|, σ arg x)-plane (in the (ln |x|, arg x)-plane if σ = 0). In Figure 2, we draw the paths along which x → 0. Any regular path is allowed if σ = 0. If σ = 0, we considered the family of paths (8) connecting a point x0 ∈ D(σ ) to x = 0. In general, these paths are spirals, represented in Figure 2 both in the plane (ln |x|, σ arg x) and in the x-plane. They are radial paths if 0 σ < 1 and = σ , because in this case arg(x) = constant. But there are only spiral paths whenever σ < 0 and σ 1. In particular, the paths σ arg(x) = σ arg(x0 ) + σ log

|x| |x0 |

are parallel to one of the boundary lines of D(σ ) in the plane (ln |x|, σ arg(x)) and the critical behavior is (11). The boundary line is σ arg(x) = σ ln |x| + σ θ2 and it is shared by D(σ ) and D(−σ ) (with the same θ2 – see also Remark 2 of Section 4). • Important Remark on the Domain: Consider the domain D(; σ ; θ1 , θ2 , σ˜ ) for σ = 0. In Theorem 1 we can arbitrarily choose θ1 . Apparently, if we increase θ1 σ the domain D(; σ ; θ1 , θ2 ) becomes larger. But itself depends on θ1 . In the proof of Theorem 1 (Section 8), we will show that 1−σ˜ ce−θ1 σ , where c is a constant, depending on a. Equivalently, θ1 σ (σ˜ −1) ln +ln c. This means that if we increase σ θ1 , we have to decrease . Therefore, for x ∈ D(; σ ; θ1 , θ2 , σ˜ ), we have σ arg(x) ( σ − σ˜ ) ln |x| + θ1 σ ( σ − σ˜ ) ln |x| + (σ˜ − 1) ln + ln c. We advise the reader to visualize a point x in the plane (ln |x|, σ arg(x)). With this visualization in mind, let x be the point {σ arg(x) = ( σ − σ˜ ) ln |x| + (σ˜ − 1) ln + ln c} ∩ {|x| = } (see Figure 3). Namely, σ arg(x ) = ( σ − 1) ln + ln c. This has the following implication: Let σ , a, σ˜ , θ2 be fixed. The union of the domains D( = (θ1 ); σ ; θ1 , θ2 , σ˜ ) obtained by letting θ1 vary is D (θ1 ); σ ; θ1 , θ2 , σ˜ ⊆ B(σ, a; θ2 , σ˜ ), θ1

314

DAVIDE GUZZETTI

Figure 2.


315

Figure 3.

where B(σ, a; θ2 , σ˜ ) := {|x| < 1 such that σ ln |x| + θ2 σ σ arg(x) < ( σ − 1) ln |x| + ln c}.

(31)

The dependence on a of the domain B is motivated by the fact that c depends on a (but not on θ1 , θ2 ). If 0 σ < 1, the above result is not a limitation on the values of arg(x) in D(; σ ; θ1 , θ2 , σ˜ ), provided that |x| is sufficiently small. Also in the case σ < 0, there is no limitation, because any point x, such that |x| < , can be included in D(; σ ; θ1 , θ2 , σ˜ ) for a suitable θ2 . In fact, we can always decrease σ θ2 without affecting . But if σ 1, the situation is different. Actually, all the points x which lie above the set B(σ, a; θ2 , σ˜ ) in the (ln |x|, σ arg(x))-plane can never be included in any D(; σ ; θ1 , θ2 , σ˜ ). See Figure 4. This is an important restriction on the domains of Theorem 1.

4. Parameterization of a Branch Through Monodromy Data – Theorem 2 The second step in our discussion is to compute the relation between the parameters σ , a of Theorem 1, stated for x = 0, and the monodromy data (x0 , x1 , x∞ ), to which a unique transcendent y(x; x0 , x1 , x∞ ) is associated. Also in this section, σ (0), a (0) are denoted σ, a. The points x = 1, ∞ are studied in Section 7.

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DAVIDE GUZZETTI

Figure 4.

We consider the Fuchsian system (20) for the special choice u1 = 0, u2 = x, u3 = 1. The labels i = 1, 2, 3 will be substituted by the labels i = 0, x, 1, and the system becomes A0 (x) Ax (x) A1 (x) dY = + + Y. (32) dz z z−x z−1 Also, the triple (x1 , x2 , x3 ) will be denoted by (x0 , x1 , x∞ ), as in [13] and in the Introduction. We consider only admissible triples and xi = ±2, i = 0, 1, ∞. We recall that an admissible triple is defined in [13] by the condition that only one xi , i = 0, 1, ∞ may be zero. Two admissible triples are equivalent if their elements differ just by the change of two signs and 2 − x0 x1 x∞ = 4 sin2 (π µ). x02 + x12 + x∞

(33)

In the Introduction, we called y(x; x0 , x1 , x∞ ) the branch in one-to-one correspondence with an equivalence class of (x0 , x1 , x∞ ). The branch has analytic continuation on the universal covering of P1 \{0, 1, ∞}. We also denote this continuation by y(x; x0 , x1 , x∞ ), where x is now a point in the universal covering. Theorem 2 has been stated in full generality in the Introduction and will be proved in Section 9. The result is a generalization of the formulae of [13] to any µ = 0 (including half-integer µ) and to all xi = ±2, i = 0, 1, ∞. The proof of the theorem is also valid for the resonant case 2µ ∈ Z\{0}. To read the formulae in this case, it is enough to just substitute an integer for 2µ in the formulae (i) or (I) of the theorem. The cases (ii), (iii); (II), (III) do not occur when 2µ ∈ Z\{0}.

317


Note that for µ integer, the case (ii), (II) degenerates to (x0 , x1 , x∞ ) = (0, 0, 0) and a arbitrary. This is the case in which the triple is not a good parameterization for the monodromy (not admissible triple). Anyway, we know that in this case there is a one-parameter family of rational solutions [28], which are all obtained by a birational transformation from the family ax y(x) = , µ = 1. 1 − (1 − a)x At x = 0, the behavior is y(x) = ax(1 + O(x)), and then the limit of Theorem 2 for µ → n ∈ Z\{0} and σ = 0 yields the above one-parameter family. Recall that R = 0 in this case. Remark 1. We repeat the remark to Theorem 2 we made in the Introduction; namely, Equation (12) does not uniquely determine σ . We decided to choose σ such that 0 σ 1, so that all the solutions of (12) are ±σ + 2n, n ∈ Z. If σ is real, we can only choose 0 σ < 1. With this convention, there is a one-to-one correspondence between (σ, a) and (a class of equivalence of) an admissible triple (x0 , x1 , x∞ ). We observed that a = a(σ ; x0 , x1 , x∞ ) is affected by ±σ + 2n, n ∈ Z. Hence, y(x; x0 , x1 , x∞ ) has different critical behaviors in different domains D(±σ + 2n). Outside their union, we expect movable poles. Remark 2. The domains D(σ ) and D(−σ ), with the same θ2 , intersect along the common boundary σ arg(x) = σ log |x| + θ2 σ (see Figure 2). The critical behavior of y(x; x0 , x1 , x∞ ) along the common boundary is given in terms of (σ (x0 ), a(σ ; x0 , x1 , x∞ )) and (−σ (x0 ), a(−σ ; x0 , x1 , x∞ )), respectively. According to Theorem 1, the critical behaviors in D(σ ) and D(−σ ) are different, but they become equal on the common boundary. Actually, along the boundary of D(σ ) the behavior is given by (11), which we rewrite as y(x) = A(x; σ, a(σ ))x(1 + O(|x|δ )), where δ is a small number between 0 and 1 and A(x; σ, a(σ )) = a(Ceiα(x;σ ))−1 +

1 1 + Ceiα(x;σ ), 2 16a

C = e−θ2 σ , x σ = Ceiα(x;σ ),

α(x; σ ) = σ arg(x) + σ ln |x| σ arg(x)= σ log |x|+θ

2 σ

.

We observe that α(x; −σ ) = −α(x; σ ). At the end of Section 9, we prove that a(σ ) = 1/(16a(−σ )). This implies that A(x; −σ, a(−σ )) = A(x; σ, a(σ )). Therefore, the critical behavior, as prescribed by Theorem 1 in D(σ ) and D(−σ ), is the same along the common boundary of the two domains.

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DAVIDE GUZZETTI

We end the section with the following proposition: PROPOSITION [unicity]. Let σ ∈ / (−∞, 0) ∪ [1, +∞) and a = 0. Let y(x) be a solution of PVIµ such that y(x) = ax 1−σ (1+ higher-order terms) as x → 0 in the domain D(; σ ). Suppose that the triple (x0 , x1 , x∞ ) computed by the formulae of Theorem 2 in terms of σ and a is admissible. Then, y(x) coincides with y(x; σ, a) of Theorem 1. Proof. See Section 9. ✷

5. Other Representations of the Transcendents – Theorem 3 We need to further investigate the critical behavior close to x = 0, in order to extend the results of Theorem 1 for x → 0 along paths not allowed by the theorem. In this section we discuss the critical behavior of the elliptic representation of Painlevé transcendents. According to Remark 1 of Section 4, we restrict to 0 σ 1 for σ = 0, or 0 σ < 1 for σ real. In Figure 5 (left) we draw domains D(σ ), D(−σ ), D(−σ + 2), D(2 − σ ), etc., where y(x; x0 , x1 , x∞ ) has different critical behaviors. Some small sectors remain uncovered by the union of the domains (Figure 5 (right)). If x → 0 inside these sectors, we do not know the behavior of the transcendent. For example, if σ = 1,

Figure 5.


319

Figure 6. The figure represents a possible configuration of the strips where Theorem 1 does not give answers. It is in these strips that we might expect movable poles.

a radial path converging to x = 0 will end up in a forbidden small sector (see also Figure 7 for the case σ = 1). If we draw, for the same θ2 , the domains B(σ ), B(−σ ), B(−σ + 2), etc., defined in (31) we obtain strips in the (ln |x|, σ arg(x))-plane which are certainly forbidden to Theorem 1 (see Figure 6). In the strips we know nothing about the transcendent. We guess that there might be poles there, as we verify in one example later. What is the behavior along the directions not allowed by Theorem 1? In the very particular case (x0 , x1 , x∞ ) ∈ {(2, 2, 2), (2, −2, −2), (−2, −2, 2), (−2, 2, −2)}, it is known that PVIµ=−1/2 has a 1-parameter family of classical solutions. The critical behavior of a branch for radial convergence to the critical points 0, 1, ∞ was computed in [28]:   − ln(x)−2 (1 + O(ln(x)−1 )), x → 0, y(x) = 1 + ln(1 − x)−2 (1 + O(ln(1 − x)−1 )), x → 1,  −x ln(1/x)−2 (1 + O(ln(1/x)−1 ), x → ∞.

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DAVIDE GUZZETTI

The branch is specified by |arg(x)| < π , |arg(1 − x)| < π . This behavior is completely different from ∼a(x)x 1−σ as x → 0. Intuitively, as x0 approaches the value 2, 1 − σ approaches 0 and the decay of y(x) ∼ ax 1−σ becomes logarithmic. These solutions were called Chazy solutions in [28], because they can be computed as functions of solutions of the Chazy equation. This section is devoted to the investigation of the critical behavior at x = 0 in the regions not allowed in Theorem 1. 5.1. ELLIPTIC REPRESENTATION The transcendents of PVIµ can be represented in the elliptic form [14] 1+x u(x) ; ω1 (x), ω2 (x) + , y(x) = ℘ 2 3 where ℘ (z; ω1 , ω2 ) is the Weierstrass elliptic function of half-periods ω1 , ω2 . u(x) solves the nonlinear differential equation ∂ u (2µ − 1)2 α ℘ ; ω1 (x), ω2 (x) , , (34) α= L(u) = x(1 − x) ∂u 2 2 where the differential linear operator L applied to u is L(u) := x(1 − x)

du 1 d2 u − u. + (1 − 2x) 2 dx dx 4

The half-periods are two independent solutions of the hyper-geometric equation L(u) = 0: ω1 (x) :=

π F (x), 2

i ω2 (x) := − [F (x) ln x + F1 (x)], 2

where F (x) is the hyper-geometric function 1 2 ∞ 1 1 2 n , , 1; x = xn F (x) := F 2 2 2 (n!) n=0 and F1 (x) :=

∞

n=0

1 2 2 n (n!)2

d ln (z), ψ(z) = dz ψ(1) = −γ ,

2 ψ n + 12 − ψ(n + 1) x n , 1 ψ = −γ − 2 ln 2, 2

ψ(a + n) = ψ(a) +

n−1

l=0

1 . a+l

321


The solutions u of (34) were not studied in the literature, so we did that and we proved a general result in Theorem 3. But first, we give a special example, already known to Picard. EXAMPLE. The equation PVIµ=1/2 has a two-parameter family of solutions discovered by Picard [28, 30, 32]. It is easily obtained from (34). Since α = 0, u solves the hyper-geometric equation L(u) = 0 and has the general form u(x) := ν1 ω1 (x) + ν2 ω2 (x), νi ∈ C, 0 νi < 2, (ν1 , ν2 ) = (0, 0). 2 A branch of y(x) is specified by a branch of ln x in ω2 (x). The monodromy data computed in [28] are x1 = −2 cos π r2 , x∞ = −2 cos π r3 , x0 = −2 cos π r1 , ν2 ν1 ν1 − ν2 r2 = 1 − , r3 = , for ν1 > ν2 , r1 = , 2 2 2 ν2 ν1 ν2 − ν1 r2 = , r3 = , for ν1 < ν2 . r1 = 1 − , 2 2 2 The modular parameter is now a function of x: τ (x) =

1 4i ω2 (x) = (arg x − i ln |x|) + ln 2 + O(x), ω1 (x) π π

We see that τ > 0 as x → 0. Now, if

u(x)

4ω < τ, 1

x → 0.

(35)

we can expand the Weierstrass function in Fourier series. Condition (35) becomes

ν2 4 ln 2 ν2 1

arg(x) − ln |x| + ν ν + 1 2

2 π π π ln |x| 4 ln 2 + + O(x), as x → 0. 0. (37)

This is the same critical behavior of Theorem 1. By virtue of the proposition of Section 4, the transcendent here coincides with y(x; σ, a) of Theorem 1 if we identify 1 − σ with ν2 for 0 < ν2 < 1, or with 2 − ν2 for 1 < ν2 < 2. In the case, ν2 = 1 the three terms x ν2 , x, x 2−ν2 have the same order and we find again the behavior (10) of Theorem 1 (oscillatory case): 1 2−ν2 x ν2 x (1 + O(x δ )) y(x) = ax + + 2 16a 1 −2iν2 1 −iν2 ν2 + x (1 + O(x δ )), = ax 1 + x 2a 16a 2 where a = − 14 [eiπν1 /16ν2 −1 ]. (b) ν2 = 0. Put ν2 = iν (namely, σ = 1 − iν). The domain (36) is now (for sufficiently small |x|): 2 ln |x| − π ν1 − 8 ln 2 < ν2 arg(x) < −2 ln |x| − π ν1 + 8 ln 2 or 2 ln |x| + π ν1 − 8 ln 2 < σ arg(x) < −2 ln |x| + π ν1 + 8 ln 2.

(38)

For radial convergence, we have y(x) =

sin2 ν2

1 + O(x) ln(x) +

ν F1 (x) 2 F (x)

+

πν1 2

+ O(x).

This is an oscillating function, and it may have poles. Suppose, for example, that ν1 is real. Since F1 (x)/F (x) is a convergent power series (|x| < 1) with real

323


coefficients and defines a bounded function, then y(x) has a sequence of poles on the positive real axis, converging to x = 0. In the domain (38), spiral convergence of x to zero is also allowed and the critical behavior is (37) because arg x is not constant. Finally, if ν = 0, namely ν2 = 0 (and then x0 = 2), we have y(x) =

1 (1 + O(|x|)). sin (π ν1 ) 2

The case (b) in the above example is helpful in understanding the limitation of Theorem 1 because it gives a complete description of the behavior of Painlevé transcendents. Actually, Theorem 1 yields the behavior (37) in the domain D(σ ) ∪ D(−σ ) ( σ = 1): (1 + σ˜ ) ln |x| + θ1 σ σ arg x (1 − σ˜ ) ln |x| + θ1 σ, where radial convergence to x = 0 is not allowed. On the other hand, the transformations σ → ±(σ − 2), gives a further domain D(σ − 2) ∪ D(−σ + 2): (−1 + σ˜ ) ln |x| + θ1 σ σ arg x −(1 + σ˜ ) ln |x| + θ1 σ, but again it is not possible for x to converge to x = 0 along a radial path. Figure 7 shows D(σ ) ∪ D(−σ ) ∪ D(2 − σ ) ∪ D(σ − 2). Note that a radial path would be allowed if it were possible to make σ˜ → 1. The interior of the set obtained as the limit for σ˜ → 1 of D(σ ) ∪ D(−σ ) ∪ D(2 − σ ) ∪ D(σ − 2) is like (38). Actually, the intersection of (38) and D(σ ) ∪ D(−σ ) ∪ D(2 − σ ) ∪ D(σ − 2) is never empty. In (38), the elliptic representation predicts an oscillating behavior and poles. So it is definitely clear that the ‘limit’ of theorem 1 for σ˜ → 1 is not trivial. Remark on the Example. For µ half integer all the possible values of (x0 , x1 , x∞ ) 2 − x0 x1 x∞ = 4 are covered by Chazy and Picard’s solutions, such that x02 + x12 + x∞ with the warning that for µ = 12 the image (through birational transformations) of Chazy solutions is y = ∞. See [28]. We turn to the general case. The elliptic representation has been studied from the point of view of algebraic geometry in [27], but to our knowledge Theorem 3 and its corollary, both stated in the Introduction, are the first general result about its critical behavior appearing in the literature. We prove Theorem 3 in Section 10. Here we prove the corollary. The critical behavior is obtained expanding y(x) in Fourier series: ∞ 2π 2 ne2πinτ π2 πu u ; ω1 , ω2 = − + 2 1 − cos n + ℘ 2 2ω1 12ω12 ω1 n=1 1 − e2πinτ +

1 π2 . 2 2 πu 4ω1 sin 4ω 1

(39)

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DAVIDE GUZZETTI

Figure 7. Domain D(σ ) ∪ D(−σ ) ∪ D(σ − 2) ∪ D(−σ + 2) for σ = 1 + iIm σ . Comparison with the domain where Piccard solution is expanded (top picture). We represent the domain D(r, ν1 , ν2 ) of Theorem 3 for immaginary ν2 , and we compare it to the domain D(σ ) with the identification ν2 = 1 − σ (and for suitable θ1 , θ 2 ). The numbers close to the boundary lines are their slopes (˜ε = 1 − σ˜ is arbitrarily small) (bottom picture).

The expansion can be performed if τ (x) > 0 and |( ωu(x) )| < τ ; these con1 (x) ditions are satisfied in D(r; ν1 , ν2 ). Let’s put F1 /F = −4 ln 2 + g(x), where g(x) = O(x). Taking into account (39) and Theorem 3, the expansion of y(x) for x → 0 in D(r; ν1 , ν2 ) is ∞ π2 n π2 1+x − × + y(x) = eg(x) 2n 2 2 2n 3 12ω1 (x) ω1 (x) n=1 1 − x iπν1 16 n g(x) 2n e inπ ωv(x) 2n n(ν2 +2)g(x) e 2+ν2 1 (x) − x −e x e × 2 2+ν2 16 16 e−iπν1 2−ν2 n −iπn ωv(x) 1 (x) x e + − en(2−ν2 )g(x) 162−ν2 1 π2 + . πv(x) 4ω1 (x)2 sin2 −i ν22 ln x + i ν22 ln 16 + πν2 1 − i ν22 g(x) + 2ω (x)

1

325


We observe that π 1 π ω1 (x) ≡ F (x) = (1 + x + O(x 2 )), 2 2 4 π2 1 1 1+x 1+x − − = x(1 + O(x)), ≡ 2 3 12ω1 (x) 3 3F (x) 2 g(x) e = 1 + O(x) and ±iπ ωv(x) (x)

e

1

iπν

−iπν 1

e 1 ν

e 2−ν2

2

= 1 + O |x| + 2−ν x

+ 16ν2 x . 16 2

In order to single out the leading terms, we observe that we are dealing with the powers x, x 2−ν2 , x ν2 in D(r; ν1 , ν2 ). If 0 < ν2 < 2 (the only allowed real values of ν2 ) |x ν2 | is leading if 0 < ν2 < 1 and |x 2−ν2 | is leading if 1 < ν2 < 2. We have eiπν1 ν2 1 ν2 2−ν2 1 + O(|x| + |x = −4 x | + |x |) . 2 16ν2 sin (. . .) Thus, there exists 0 < δ < 1 (explicitly computable in terms of ν2 ) such that 1 eiπν1 1 eiπν1 −1 2−ν2 1 ν2 x− x x − (1 + O(x δ )) y(x) = 2 4 16ν2 −1 4 16ν2 −1  2 πν1 if ν2 = 1, sin 2 x(1 + O(x δ )),    iπ ν 1 if 0 < ν2 < 1, = − 14 16e ν2−1 x ν2 (1 + O(x δ )),   iπ ν −1  1 e 1 − 4 16ν2−1 x 2−ν2 (1 + O(x δ )), if 1 < ν2 < 2. This behavior coincides with that of Theorem 1 for σ = 0 in the first case, σ = 1 − ν2 in the second, and σ = ν2 − 1 in the third. We turn to the case ν2 = 0. We consider a path contained in D(r; ν1 , ν2 ) of the equation ν2 arg(x) = ( ν2 − V) ln |x| + b,

0V2

(40)

with a suitable constant b (the path connects some x0 ∈ D(r; ν1 , ν2 ) to x = 0, therefore b = ν2 arg x0 − ( ν2 − V) ln |x0 |). We have|x 2−ν2 | = |x|2−V eb , |x ν2 | = |x|V e−b and so |x ν2 | is leading for 0 V < 1, |x ν2 |, |x|, |x 2−ν2 | have the same order for V = 1, |x 2−ν2 | is leading for 1 < V 2. If V = 0,

iπν

e 1 ν

2

16ν2 x < r,

but x ν2 → 0 as x → 0.

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DAVIDE GUZZETTI

If V = 2,

−iπν 1

e 2−ν2

< r,

162−ν2 x

but x 2−ν2 → 0 as x → 0.

This also implies that v(x) → 0 as x → 0 along the paths with V = 0 or V = 2, while v(x) → 0 for all other values 0 < V < 2. We conclude that (a) If x → 0 in D(r; ν1 , ν2 ) along (40) for V = 0, 2, then 1 eiπν1 1 eiπν1 −1 2−ν2 1 ν2 x− x x − (1 + O(x δ )), 0 < δ < 1. y(x) = 2 4 16ν2 −1 4 16ν2 −1 The three leading terms have the same order if the convergence is along a path asymptotic to (40) with V = 1. Namely π ν1 1 − ν2 2 ln x + + 2i(ν2 − 1) ln 2 (1 + O(x)), for V = 1. y(x) = x sin i 2 2 Otherwise

1 eiπν1 x ν2 (1 + O(x δ )), y(x) = − 4 16ν2 −1

or

for 0 < V < 1

1 eiπν1 −1 2−ν2 x (1 + O(x δ )), y(x) = − 4 16ν2 −1

for 1 < V < 2.

This is the behavior of Theorem 1 with 1 − σ = ν2 or 2 − ν2 . Important Observation. Let ν2 = 1−σ and consider the intersection D(r; ν1 , ν2 ) ∩ D(σ ) in the (ln |x|, ν2 arg(x))-plane. It is never empty (see Figure 7). We choose ν1 such that a = − 14 [eiπν1 /16ν2 −1 ]. According to the Proposition in Section 4, the transcendent of the elliptic representation and y(x; σ, a) of Theorem 1 coincide at the intersection. Equivalently, we can choose the identification 1 − σ = 2 − ν2 and repeat the argument. (b) If V = 0 the term sin

2

−i ν22

ln x +

1 i ν22

ln 16 +

πν1 2

− i ν22 g(x) +

πv(x) 2ω1 (x)

is oscillatory as x → 0 and does not vanish. Note that there are no poles because iπ ν the denominator does not vanish in D(r; ν1 , ν2 ) since | e16ν21 x ν2 | < r < 1. Then y(x) = O(x) + =

1 1 ν2 ν2 2 2 F (x) sin −i 2 ln x + i 2 ln 16 +

πν1 2

− i ν22 g(x) +

v(x) F (x)

1 + O(x) ν2 eiπ ν1 m + O(x). ν2 sin −i 2 ln x + i 2 ln 16 + πν2 1 + ∞ c (ν ) x ν2 0m 2 m=1 16ν2 2

327


The last step is obtained taking into account the nonvanishing term in (13) and v(x) π v(x) = = v(x)(1 + O(x)). 2ω1 (x) F (x) (c) If V = 2, the series −

∞

n=1

n

1−

eg(x) 2n 16

n(2−ν2 )g(x)

x 2n

e

e−iπν1 2−ν2 x 162−ν2

n

−iπn ωv(x) (x)

e

1

which appears in y(x) is oscillating. Simplifying we obtain −iπν1 n ∞

e −iπn ωv(x) 2−ν2 1 (x) n x e y(x) = O(x) − 4(1 + O(x)) 2−ν2 16 n=1 1 + O(x) = e−iπ ν1 m + O(x). ∞ ν2 −2 πν1 2 sin2 i 2−ν ln x + i ln 16 + b (ν ) x ν2 + 0m 2 m=1 2 2 2 162−ν2 The observation at the end of point (a) makes it possible to investigate the behavior of the transcendents of Theorem 1 along a path (8) with = 1. The path (8) coincides with (40) for V = 0 if we define 1 − σ := ν2 , for V = 2 if we define 1 − σ = 2 − ν2 . In particular, we can analyze the radial convergence when σ = 1. We identify ν2 = 1 − σ and choose ν2 = iν, ν = 0 real. Namely, σ = 1 − iν. Let x → 0 in D(r; ν1 , iν) along the line arg(x)= constant (it is the line with V = 0). We have 1 1 ν + O(x) πv(x) 2 2 F (x) sin 2 ln x − ν ln 16 + π2 ν1 + ν2 g(x) + 2ω 1 (x) 1 + O(x) = eiπ ν1 m + O(x) πν1 2 ν sin 2 ln x − ν ln 16 + 2 + ∞ x iν + O(x) m=1 c0m (ν) 16iν 1 + O(x) = eiπ ν1 m . πν1 2 ν sin 2 ln x − ν ln 16 + 2 + ∞ c (ν) x iν 0m m=1 16iν

y(x) =

The last step is possible because sin(f (x) + O(x)) = sin(f (x)) + O(x) = sin(f (x))(1 + O(x)) if f (x) → 0, as x → 0; this is our case for ∞

π ν1 ν + c0m (ν) f (x) = ln x − ν ln 16 + 2 2 m=1

eiπν1 iν m x 16iν

in D. We observe that for σ = 1 we have a limitation on arg(x) in D(r; ν1 , iν), namely (41) −π ν1 − ln r < ν arg(x). This is the analogous of the limitation imposed by B(σ, a; θ2 , σ˜ ) of (31).

328

DAVIDE GUZZETTI

Remark. If ν2 = 0, the freedom ν2 → ν2 +2N, N ∈ Z, is the analogous of the freedom σ → ±σ + 2n. Moreover, Theorem 3 yields different critical behaviors for the same transcendent on the different domains corresponding to ν2 + 2N. As a last remark, we observe that the coefficients in the expansion of v(x) can be computed by direct substitution of v into the elliptic form of PVIµ , the right-hand side being expanded in Fourier series. 5.2. SHIMOMURA’ S REPRESENTATION In [37] and [19] S. Shimomura proved the following statement for the Painlevé VI equation with any value of the parameters α, β, γ , δ. For any complex number k and for any σ ∈ / (−∞, 0] ∪ [1, +∞), there is a sufficiently small r such that the Painlevé VI equation for given α, β, γ , δ has a holomorphic solution in the domain Ds (r; σ, k) = {x ∈ C˜0 ||x| < r, |e−k x 1−σ | < r, |ek x σ | < r} with the following representation: y(x; σ, k) = where v(x) =

n1

+

1

cosh2 σ −1 2

an (σ )x n +

ln x +

k 2

+

v(x) 2

,

bnm (σ )x n (e−k x 1−σ )m +

n0,m1

cnm (σ )x n (ek x σ )m ,

n0, m1

an (σ ), bnm (σ ), cnm (σ ) are rational functions of σ and the series defining v(x) is convergent (and holomorphic) in D(r; σ, k). Moreover, there exists a constant M = M(σ ) such that |v(x)| M(σ )(|x| + |e−k x 1−σ | + |ek x σ |).

(42)

The domain D(r; σ, k) is specified by the conditions: |x| < r, σ ln |x| + [ k − ln r] < σ arg(x) < ( σ − 1) ln |x| + [ k + ln r]. (43) This is an open domain in the plane (ln |x|, arg(x)). It can be compared with the domain D(; σ, θ1 , θ2 ) of Theorem 1 (Figure 8). Note that (43) imposes a limitation on arg(x). For example, if σ = 1 we have σ arg(x) < [ k + ln r] (ln r < 0). This is similar to (41). We will show that Shimomura’s transcendents coincide with


329

Figure 8.

those of Theorem 1 (see point (a.1) below). So, the above limitation turns out to be the analogous of the limitation imposed to D(; σ ; θ1 , θ2 ) by B(σ, a; θ2 , σ˜ ) of (31). Like the elliptic representation, Shimomura’s allows us to investigate what happens when x → 0 along a path (8) with = 1, contained in Ds (r; σ, k). It is a radial path if σ = 1. Along (8), we have |x σ | = |x| e−b . We suppose σ = 0. (a) 0 < 1. We observe that |x 1−σ e−k | → 0 as x → 0 along the line. Then, 1 y(x; σ, k) = 2 σ −1 cosh 2 ln x + k2 + v(x) 2 4 = σ −1 k v(x) x ee + x 1−σ e−k e−v(x) + 2 1 = 4e−k e−v(x) x 1−σ −k −v(x) e x 1−σ )2 (1 + e−v(x) −k −v(x) 1−σ = 4e e x O(|e−k x 1−σ |) . 1+e Two sub-cases: (a.1) = 0. Then |x σ ek | → 0 and v(x) → 0 (see (42)). Thus y(x; σ, k) = 4e−k x 1−σ 1 + O(|x| + |ek x σ | + |e−k x 1−σ |) . By the proposition in Section 4, y(x; σ, k) and y(x; σ, a) coincide, for a = 4e−k , in Ds (r; σ, k) ∩ D(; σ ; θ1 , θ2 ). The intersection is not empty for any θ1 , θ2 . See Figure 8. (a.2) = 0. |x σ ek | → constant < r, so |v(x)| does not vanish. Then y(x) = a(x)x 1−σ 1 + O(|e−k x 1−σ |) , a(x) = 4e−k e−v(x), and a(x) must coincide with (10) of Theorem 1:

330

DAVIDE GUZZETTI

Figure 9.

(b) = 1. In this case Theorem 1 fails. Now |x 1−σ e−k | → (constant = 0) < r. Therefore, y(x) does not vanish as x → 0. We keep the representation y(x; σ, k) =

cosh2 σ −1 2

1 ln x +

k 2

+

v(x) 2

≡

1 σ −1 . sin i 2 ln x + i k2 + i v(x) − π2 2 2

v(x) does not vanish and y(x) is oscillating as x → 0, with no limit. We remark that like in the elliptic representation, cosh2 (. . .) does not vanish in Ds (r; σ, k), so we do not have poles. Figure 9 synthesizes points (a.1), (a.2), (b). As an application, we consider the case σ = 1, namely σ = 1−iν, ν ∈ R\{0}. Then, the path corresponding to = 1 is a radial path in the x-plane and y(x; 1 − iν, k) =

sin2 ν2

ln(x) +

ik 2

1 + O(x) . − + 2i m1 b0m (σ )(e−k x 1−σ )m π 2

6. Analytic Continuation of a Branch We describe the analytic continuation of the transcendent y(x; σ, a). We choose a basis γ0 , γ1 of two loops around 0 and 1, respectively, in the fundamental group π(P1\{0, 1, ∞}, b), where b is the base-point (Figure 10). The analytic continuation of a branch y(x; x0 , x1 , x∞ ) along paths encircling x = 0 and x = 1 (a loop around x = ∞ is homotopic to the product of γ0 , γ1 ) is given by the action of the group of the pure braids on the monodromy data (Figure 11). This action is computed in [13], to which we refer. For a counter-clockwise loop around 0, we have to transform (x0 , x1 , x∞ ) by the action of the braid β12 , where β1 : (x0 , x1 , x∞ ) → (−x0 , x∞ − x0 x1 , x1 ), β12 : (x0 , x1 , x∞ ) → (x0 , x1 + x0 x∞ − x1 x02 , x∞ − x0 x1 ).


331

Figure 10.

Figure 11.

The analytic continuation of the branch y(x; x0 , x1 , x∞ ) is the new branch y(x; x0 , x1 + x0 x∞ − x1 x02 , x∞ − x0 x1 ). For a counter-clockwise loop around 1, we need the braid β22 , given by β2 : (x0 , x1 , x∞ ) → (x∞ , −x1 , x0 − x1 x∞ ), β22 : (x0 , x1 , x∞ ) → (x0 − x1 x∞ , x1 , x∞ + x0 x1 − x∞ x12 ). The analytic continuation of y(x; x0 , x1 , x∞ ) is the new branch y(x; x0 −x1 x∞ , x1 , x∞ + x0 x1 − x∞ x12 ). A generic loop P1 \{0, 1, ∞} is represented by a braid β, which is a product of factors β1 and β2 . The braid β acts on (x0 , x1 , x∞ ) and gives a new triple β β β β β β (x0 , x1 , x∞ ) and a new branch y(x; x0 , x1 , x∞ ). On the other hand, y(x; x0 , x1 , x∞ ) is the branch of a transcendent which has analytic continuation on the universal covering of P1 \{0, 1, ∞}. We still denote this transcendent by y(x; x0 , x1 , x∞ ), where x is now regarded as a point in the universal covering. A loop transforms x to a new point x in the covering. The transcendent at x is y(x ; x0 , x1 , x∞ ). Let β be the corresponding braid. We have β ) = y(x ; x0 , x1 , x∞ ). y(x; x0 , x1 , x∞ β

β

(44)

332

DAVIDE GUZZETTI

Let σ , a be associated to (x0 , x1 , x∞ ) according to Theorem 2. Let x ∈ D(σ ). β β β At x we have y(x; x0 , x1 , x∞ ) = y(x; σ, a). Let σ β , a β = a(σ β ; x0 , x1 , x∞ ) be β β β associated to (x0 , x1 , x∞ ). If D(σ ) ∩ D(σ β ) is not empty and x also belongs to β β β / D(σ β ), it belongs to one D(σ β ), then y(x; x0 , x1 , x∞ ) = y(x; σ β , a β ) at x. If x ∈ β β β and only one of the domains D(±σ β + 2n) and y(x; x0 , x1 , x∞ ) = y(x; ±σ β + β β β 2n, a˜ β ) at x, where a˜ β = a(±σ β + 2n; x0 , x1 , x∞ ). We note, however, that if β σ = 1, it may happen that x lies in the strip between B(σ β ) and B(2 − σ β ), where there may be poles (see the beginning of Section 5). In this case, we are not able to describe the analytic continuation (actually, the new branch may have a pole in x). But in this case, we can slightly change arg x in such a way that x falls in a domain D(±σ β + 2n). As an example, let us start at x ∈ D(σ ); we perform the loop γ1 around 1 and 2 go back to x. If x also belongs to D(σ β2 ), the transformation is γ1 : y(x; σ, a) → 2 2 2 2 / D(σ β2 ) but x belong to one of the D(±σ β2 + 2n) we have y(x; σ β2 , a β2 ). If x ∈ 2 2 y(x; σ, a) → y(x; ±σ β2 + 2n, a˜ β2 ). Again, let us start at x ∈ D(σ ); we perform the loop γ0 around 0 and we go back to x. The transformation of (σ, a) according to the braid β1 is (σ β1 , a β1 ) = (σ, ae−2πiσ ) 2

2

(45)

as it follows from the fact that x0 is not affected by β12 , then σ does not change, and β2

β2

β2

from the explicit computation of a(σ, x0 1 , x1 1 , x∞1 ) through Theorem 2 (we will do it at the end of Section 9). Therefore, the effect of γ0 is γ0 : y(x; σ, a) → y(x; σ β1 , a β1 ) = y(x; σ, ae−2πiσ ). 2

2

Since we are considering a loop around 0, it makes sense to consider is as a loop in C\{0} ∩ {|x| < }. The loop is x → x = e2πi x. Suppose that also x ∈ D(σ ). Then we can represent the analytic continuation on the universal covering as y(x; σ, a) → y(x ; σ, a). On the other hand, according to (44), we must have 2 2 y(x ; σ, a) = y(x; σ β1 , a β1 ). This is immediately verified because y(x ; σ, a) = a[x ]1−σ 1 + O |x |δ = ae−2πiσ x 1−σ (1 + O(|x|δ )) ≡ y(x; σ, ae−2πiσ ). Thus Theorem 1 is in accordance with the analytic continuation obtained by the action of the braid group. 7. Singular Points x = 1, x = ∞ (Connection Problem) In this section we restore the notation σ (0) and a (0) to denote the parameters of Theorem 1 near the critical point x = 0. We describe now the analogs of Theorem 1 near x = 1 and x = ∞. The three critical points 0, 1, ∞ are equivalent thanks to the symmetries discussed in [30] and [13].

333


Figure 12.

(a) Let x=

1 y(x) := y(t). ˆ t

1 , t

(46)

ˆ is a solution of PVIµ Then y(x) is a solution of PVIµ (variable x) if and only if y(t) (variable t). The singularities 0 and ∞ are exchanged. Theorem 1 holds for y(t) ˆ at t = 0 with some parameters σ , a that we call now σ (∞) , a (∞) . Then, we go back to y(x) and find a transcendent y(x; σ (∞) , a (∞) ) with the behavior 1 (∞) (∞) (∞) σ (∞) 1+O , x→∞ (47) y(x; σ , a ) = a x |x|δ in D(M; σ (∞) ; θ1 , θ2 , σ˜ ) s.t. |x| > M, := x ∈ C\{∞} e−θ1 σ

(∞)

|x|−σ˜ |x −σ

(∞)

| e−θ2 σ

(∞)

, 0 < σ˜ < 1 ,

(48)

where M > 0 is sufficiently large and 0 < δ < 1 is small (Figure 12). (b) Let x = 1 − t,

y(x) = 1 − y(t). ˆ

(49)

ˆ satisfies PVIµ . Theorem 1 holds for y(t) ˆ at y(x) satisfies PVIµ if and only if y(t) t = 0 with some parameters σ , a that we now call σ (1) and a (1) . Going back to y(x), we obtain a transcendent y(x; σ (1) , a (1) ) such that (1)

y(x, σ (1) , a (1) ) = 1 − a (1) (1 − x)1−σ (1 + O(|1 − x|δ )),

x→1

(50)

in D(; σ (1); θ1 , θ2 , σ˜ ) s.t. |1 − x| < , := x ∈ C\{1}

(1) e−θ1 σ |1 − x|σ˜ |(1 − x)σ | e−θ2 σ , 0 < σ˜ < 1 .

(51)

334

DAVIDE GUZZETTI

Consider a branch y(x; x0 , x1 , x∞ ). The symmetries in (a) and (b) affect the monodromy data, according to the following formulae proved in [13]: 1 y(t; ˆ x∞ , −x1 , x0 − x1 x∞ ), t y(x; x0 , x1 , x∞ ) = 1 − y(t; ˆ x1 , x0 , x0 x1 − x∞ ), y(x; x0 , x1 , x∞ ) =

1 , t x = 1 − t. x=

(52) (53)

We are ready to solve the connection problem for the transcendents of Theorem 1, so extending the result of [13]. We recall that we always assume that 0 σ (i) 1, i = 0, 1, ∞; otherwise we write ±σ (i) + 2n, n ∈ Z. We consider a transcendent y(x; σ (0) , a (0) ). We choose a point x ∈ D(σ (0)). At x there exists a unique branch y(x; x0 , x1 , x∞ ) whose analytic continuation in D(σ (0)) is precisely y(x; σ (0) , a(σ (0) )), where the triple of monodromy data (x0 , x1 , x∞ ) corresponds to σ (0), a (0) according to Theorem 2. If we increase the absolute value of the point and we keep arg x constant, we obtain a new point X = |X| exp{i arg x}, where |X| is large. The branch y(x; x0 , x1 , x∞ ) is also defined in X, because we have not changed arg x. According to (52), we compute σ (∞) , a (∞) from the data (x∞ , −x1 , x0 − x1 x∞ ) by the formulae of Theorem 2. Therefore, if X ∈ D(M; σ (∞) ), the analytic continuation of y(x; x0 , x1 , x∞ ) = y(x; σ (0) , a (0) ) at X is y(X; σ (∞) , a (∞) ). We observe that if 0 σ (∞) < 1, it is always possible to choose X ∈ D(M; σ (∞) ), provided that |X| is large enough. But for σ (∞) = 1 we have a restriction on the argument of the points of D(M; σ (∞) ) given by a set B(σ (∞) ) analogous to (31). This implies that X may not be chosen in D(M; σ (∞) ) for any value of |X|. In this case, we can choose X in one of the domains D(M; σ (∞) ), D(M; −σ (∞) ), D(M; 2 − σ (∞) ), D(M; σ (∞) − 2). See Figure 13. This is almost

Figure 13.

335


always possible, except for the case when arg x lies in the strip between B(σ (∞) ) and B(2−σ (∞)), where there may be movable poles (see the discussion about these strips at the beginning of Section 5). We recall that a (∞) depends on (x∞ , −x1 , x0 − x1 x∞ ) but it is also affected by the choice of ±σ (∞) + 2n. Thus, we write below a (∞) (±σ (∞) + 2n). We conclude that the analytic continuation of y(x; x0 , x1 , x∞ ) = y(x; σ (0) , a (0) ) at X is either y(X; σ (∞) , a (∞) (σ (∞) )), or y(X; −σ (∞) , a (∞) (−σ (∞) )), or y(X; 2 − σ (∞) , a (∞) (2 − σ (∞) )), or y(X; σ (∞) − 2, a (∞) (σ (∞) − 2)), provided that X is not in the strip where there may be poles. If X falls in the strip, this is not actually a limitation, because we can slightly change arg x in such a way that x is still in D(σ (0)) and X falls into D(M; σ (∞) ) ∪ D(M; −σ (∞) ) ∪ D(M; 2 − σ (∞) ) ∪ D(M; σ (∞) − 2). In the same way we treat the connection problem between x = 0 and x = 1. We repeat the same argument taking (53) into account. We remark again that for σ (1) = 1 it is necessary to consider the union of D(σ (1)), D(−σ (1)), D(2 − σ (1)), D(σ (1) − 2) to include all possible values of arg(1 − x). 8. Proof of Theorem 1 We recall that PVIµ is equivalent to the Schlesinger equations for the 2×2 matrices A0 (x), Ax (x), A1 (x) of (32): [Ax , A0 ] dA1 [A1 , Ax ] dA0 = , = , dx x dx 1−x [Ax , A0 ] [A1 , Ax ] dAx = + . dx x 1−x

(54)

We look for solutions satisfying A0 (x) + Ax (x) + A1 (x) =

−µ 0 0 µ

:= −A∞ ,

µ ∈ C, 2µ ∈ / Z,

tr(Ai ) = det(Ai ) = 0. Now let A(z, x) :=

Ax A1 A0 + + . z z−x z−1

We have explained that y(x) is a solution of PVIµ if and only if A(y(x), x)12 = 0. The system (54) is a particular case of the system 2

dAµ = [Aµ , Bν ]fµν (x), dx ν=1

n

n2 n1 n2

1 dBν =− [Bν , Bν ] + [Bν , Aµ ]gµν (x) + [Bν , Bν ]hνν (x), dx x µ=1 ν =1

ν =1

(55)

336

DAVIDE GUZZETTI

where the functions fµν , gµν , hµν are meromorphic with poles at x = 1, ∞ and ν Bν + µ Aµ = −A∞ (here the subscript µ is a label, not the eigenvalue of A∞ !). System (54) is obtained for hµν = 0, fµν = gµν = bν /(aµ − xbν ), n2 = 2, a1 = b2 = 1, b1 = 0 n1 = 1, and B1 = A0 ,

B2 = Ax ,

A1 = A1 .

We prove the analogous result of [33], p. 262, in the domain D(; σ ; θ1 , θ2 , σ˜ ) for σ ∈ / (−∞, 0) ∪ [1, +∞): LEMMA 1. Consider matrices Bν0 (ν = 1, . . . , n2 ), A0µ (µ = 1, . . . , n1 ) and A, to be independent of x and such that

Bν0 + A0µ = −A∞ , ν

µ

Bν0

= A,

eigenvalues(A) =

ν

σ σ ,− , 2 2

σ ∈ / (−∞, 0) ∪ [1, +∞).

Suppose that fµν , gµν , hµν are holomorphic if |x| < , for some small < 1. For any 0 < σ˜ < 1 and θ1 , θ2 real, there exists a sufficiently small 0 < < such that the system (55) has holomorphic solutions Aµ (x), Bν (x) in D(; σ ; θ1 , θ2 , σ˜ ) satisfying #Aµ (x) − A0µ # C|x|1−σ1 ,

#x −A Bν (x)x A − Bν0 # C|x|1−σ1 .

Here C is a positive constant and σ˜ < σ1 < 1. Important remark. There is no need to assume here that 2µ ∈ / Z. The theorem holds true for any value of µ. If in the system (55) the functions fµν , gµν , hµν are chosen in such a way as to yield Schlesinger equations for the Fuchsian system / Z is still not necessary, provided that the matrix of PVIµ , the assumption 2µ ∈ R in (22) is considered as a monodromy datum independent of the deformation parameter x. Proof. Let A(x) and B(x) be 2 × 2 matrices holomorphic on D(; σ ) (we omit θ1 , θ2 , σ˜ ) and such that #A(x)# C1 ,

#B(x)# C2

on D(; σ ).

Let f (x) be a holomorphic function for |x| < . Let σ2 be a real number such that σ˜ < σ2 < 1. Then, there exists a sufficiently small < such that for

337


x ∈ D(; σ ), we have ±A ±A x B(x)x ∓A C2 |x|−σ2 , x A(x)x ∓A C1 |x|−σ2 , −A A −A A 1−σ2 x dsA(s)s B(s)s f (s)x , C1 C2 |x| L(x) −A A −A A 1−σ2 x ds s B(s)s A(s)f (s)x , C1 C2 |x| L(x)

where L(x) is a path in D(; σ ) joining 0 to x. To prove the estimates, we observe that σ

σ

1

1

θ1

σ˜

#x A # = #x diag( 2 ,− 2 ) # = max{|x σ | 2 , |x σ |− 2 } e 2 σ |x|− 2 ,

in D(; σ ).

Note here the importance of the bound |x σ | e−θ2 σ in the definition of D(; σ ): it 1 determines the above estimates of #x A # because it ensures that |x −σ | 2 is dominant. If this were not true, the lemma would fail and Theorem 1 could not be proved. Now we estimate #x A A(x)x −A # #x A ##A(x)##x −A # eθ1 σ C1 |x|−σ˜ = eθ1 σ |x|σ2 −σ˜ C1 |x|−σ2 . Thus, if is small enough (we require σ2 −σ˜ e−θ1 σ ) we obtain #x A A(x)x −A # C1 |x|−σ2 . We turn to the integrals. We choose a real number σ ∗ such that 0 σ ∗ σ˜ and we choose a path L(x) from 0 to x, represented in Figure 14. For σ = 0, L(x) is

Figure 14. Path of integration.

338

DAVIDE GUZZETTI

given by arg(s) = a log |s| + b, b = arg x −

a=

σ − σ ∗ log |x|, σ

σ − σ ∗ , σ |s| |x|.

For σ = 0, we choose L(x) with σ ∗ = σ and arg(s) = arg(x). Note that on the ∗ ∗ L(x) we have |s σ | = |x σ |(|s|σ /|x|σ ). Then we compute −A A −A A x dsA(s)s B(s)s f (s)x L(x)

A −A s s dsx A(s)x B(s) f (s) x x L(x) ∗ |s|−σ |ds| −σ ∗ . eθ1 σ |x|−σ˜ C1 C2 max |f (x)| |x| |x|}. Of course, 1 for x → 0 and Y (z, x) → Yˆ (z). U (z, x) = I + O z But now observe that 1 1 −1 I +O z−A∞ zR , z → ∞. Yˆ (z) = U (z, x) Y (z, x) = I + O z z Then Yˆ (z) ≡ YˆN (z). Finally, for z → 1, ˆ 1 (I + O(z − 1))(z − 1)J Cˆ 1 Y (z, x) = U (x, z)YˆN (z) = U (x, z)G = G1 (x)(I + O(z − 1))(z − 1)J Cˆ 1 . This implies C1 ≡ Cˆ 1 and then M1 = Cˆ 1−1 e2πiJ Cˆ 1 .

(76)

Here we have chosen a monodromy representation for (69) by fixing a base-point and a basis in the fundamental group of P1 as in Figure 15. M0 , M1 , Mx , M∞ are the monodromy matrices for the solution (72) corresponding to the loops γi , i = 0, x, 1, ∞. M∞ M1 Mx M0 = I . The result (76) may also be proved simply observing that M1 becomes Mˆ 1 as x → 0 in D(σ ) because the system (70) is obtained from (69) when z = x and z = 0 merge and the singular point z = 1 does not move. x may converge to 0 along spiral paths (Figure 15). We recall that the braid βi,i+1 changes the monodromy matrices of dY /dz = ni=1 Ai (u)/(z − ui )Y −1 , Mk → Mk for any k = i, i + 1 according to Mi → Mi+1 , Mi+1 → Mi+1 Mi Mi+1 (see [13]). Therefore, if arg(x) increases of 2π as x → 0 in (69), we have M0 → Mx ,

Mx → Mx M0 Mx−1 ,

M1 → M1 .

If follows that M1 does not change and then M1 ≡ Mˆ 1 = Cˆ 1−1 e2πiJ Cˆ 1 ,

(77)

where Mˆ 1 is the monodromy matrix of (73) for the loop γˆ1 in the basis of Figure 15. Now we turn to Y˜ (z). Let Y˜ (z, x) := x −A Y (xz, x), and by definition Y˜ (z, x) → ˜ Y (z) as x → 0. In this case, −A x (A0 + Ax )x A − A x −A A1 x A ˜ dY˜ (z, x) = + Y (z, x) := F˜ (z, x)Y˜ (z, x). dx x x − 1z Proceeding by successive approximations as above, we get Y˜ (z, x) = V (z, x)Y˜ (z), ∞

dx1 . . . dxn F˜ (z, x1 ) . . . F˜ (z, xn ) → I, V (z, x) = I + n=1

L(x)

L(xn−1 )

for x → 0 uniformly in x ∈ D(σ ) and in z in every compact subset of {z | |z| < 1/|x|}.

350

DAVIDE GUZZETTI

Figure 15. (1): Branch cuts and loops for the Fuchsian system associated PVIµ . (2) Branch cuts and loops when x → 0. (3) Branch cuts and loops for the research system before and x → 0.

Let’s investigate the behavior of Y˜ (z) as z → ∞ and compare it to the behavior of YÑ (z). First we note that x −A YˆN (xz) = x −A (I + O(xz))(xz)A Cˆ 0 → zA Cˆ 0

for x → 0.

Then

−1 = x −A U (xz, x)x A → Y˜ (z)Cˆ 0−1 z−A . x −A Y (xz, x) x −A YˆN (xz)

On the other hand, from the properties of U (z, x) we know that x −A U (xz, x)x A is holomorphic in every compact subset of {z | |z| > 1} and x −A U (xz, x)x A = I + O( 1z ) as z → ∞. Thus U˜ (z) := limx→0 x −A U (xz, x)x A exists uniformly in every compact subset of {z | |z| > 1} and U˜ (z) = I + O(1/z), z → ∞. Then Y˜ (z) = U˜ (z)zA Cˆ 0 ≡ YÑ (z)Cˆ 0 ,

351


as we wanted to prove. Finally, the above result implies z z ˆ A , x YÑ C0 Y (z, x) = x V x x  ˜ 0 (I + O(z/x))x −J zJ C˜ 0 Cˆ 0 x A V xz , x G     = G0 (x)(I + O(z))zJ C˜ 0 Cˆ 0 , z → 0, = z  ˜ 1 O − 1 z − 1 J C˜ 1 Cˆ 0 x A V xz , x G   x x  = Gx (x)(I + O(z − x))(z − x)J C˜ 1 Cˆ 0 ,

z → x.

Let M˜ 0 , M˜ 1 denote the monodromy matrices of YÑ (z) in the basis of Figure 13, then M0 = Cˆ 0−1 C˜ 0−1 e2πiJ C˜ 0 Cˆ 0 = Cˆ 0−1 M˜ 0 Cˆ 0 , Mx = Cˆ 0−1 C˜ 1−1 e2πiJ C˜ 1 Cˆ 0 = Cˆ 0−1 M˜ 1 Cˆ 0 .

(78) (79)

The same result may be obtained observing that from −A x A0 x A x −A Ax x A x −A A1 x A −A d(x −A Y (xz, x)) x Y (xz, x) (80) = + + dz z z−1 z − x1 we obtain the system (71) as z = 1/x and z = ∞ merge (Figure 15). The singularities z = 0, z = 1, z = 1/x of (80) correspond to z = 0, z = x, z = 1 of (69). The poles z = 0 and z = 1 of (80) do not move as x → 0 and 1/x converges to ∞, in general along spirals. At any turn of the spiral the system (80) has new monodromy matrices according to the action of the braid group M1 → M∞ ,

−1 M∞ → M∞ M1 M∞ ,

but M0 → M0 , Mx → Mx . Hence, the limit Y˜ (z) still has monodromy M0 and Mx at z = 0, x. Since Y˜ = YÑ Cˆ 0 we conclude that M0 and Mx are (78) and (79). In order to find the parameterization y(x; σ, a), in terms of (x0 , x1 , x∞ ), we have to compute the monodromy matrices M0 , M1 , M∞ in terms of σ and a and then take the traces of their products. In order to do this, we use the formulae (77), (78), (79). In fact, the matrices M˜ i (i = 0, 1) and Mˆ 1 can be computed explicitly because a 2 × 2 Fuchsian system with three singular points can be reduced to the hyper-geometric equation, whose monodromy is completely known. Before going on with the proof, we recall that in the proof of Theorem 1 we defined a = −(1/4s) (or a = s for σ = 0). LEMMA 3. The Gauss hyper-geometric equation dy d2 y − α0 β0 y = 0 + [γ0 − z(α0 + β0 + 1)] 2 dz dz is equivalent to the system 1 1 dJ 0 0 0 1 = + J, dz z −α0 β0 −γ0 z − 1 0 γ0 − α0 − β0 z(1 − z)

(81)

(82)

352

DAVIDE GUZZETTI

where

J=

y (z − 1) dy dz

.

LEMMA 4. Let B0 and B1 be matrices of eigenvalues 0, 1−γ , and 0, γ −α−β−1, respectively, such that B0 + B1 = diag(−α, −β), Then

B0 =

α(1+β−γ ) α−β β(β+1−γ ) 1 α−β r

α = β.

α(γ −α−1) r α−β β(γ −α−1) α−β

α(γ −α−1)

,

B1 =

α−β

−(B0 )12

−(B0 )21

β(β+1−γ ) α−β

for any r = 0. We leave the proof as an exercise. The following lemma connects Lemmas 3 and 4: LEMMA 5. The system (82) with α0 = α,

β0 = β + 1,

γ0 = γ ,

α = β

is gauge-equivalent to the system B0 B1 dX = + X, dz z z−1

(83)

where B0 , B1 are given in Lemma 4. This means that there exists a matrix 1 0 G(z) := (α−β)z+β+1−γ α−β 1 z α(1+α−γ (1+α−γ )r )r such that X(z) = G(z)J(z). It follows that (83) and the corresponding hypergeometric equation (81) have the same Fuchsian singularities 0, 1, ∞ and the same monodromy group. Proof. By direct computation. ✷ Note that the form of G(z) ensures that if y1 , y2 are independent solutions of the hyper-geometric equation, then a fundamental matrix of (83) may be chosen to be y1 (z) y2 (z) . X(z) = ∗ ∗ We also observe that if we re-define r1 := r

α(γ − α − 1) , α−β

353


the matrices G(z), B0 , B1 are not singular except for α = β. Actually, we have α(β+1−γ ) r1 α−β B0 = αβ(β+1−γ )(γ −1−α) 1 β(γ −α−1) , (α−β)2

r1

α(γ −α−1) B1 =

−r1

α−β

−(B0 )21

G(z) =

α−β

β(β+1−γ ) α−β

1

α((α−β)z+β+1−γ ) 1 β−α r1

, 0 − rz1

.

The form of B0 , B1 of Lemma 4 will correspond to the matrices define in Lemma 2 in general, while the form of B0 , B1 above will correspond to (64) and (66) of Lemma 2 (with r1 → r). For this reason, we must disregard the matrices (65), (67) when we prove Theorem 1. Now we compute the monodromy matrices for the systems (70), (71) by reduction to an hyper-geometric equation. We first study the case σ ∈ / Z. Let us start σ with (70). With the gauge Y (1)(z) := z− 2 Yˆ (z), we transform (70) in 0 A − σ2 I (1) A1 dY (1) = + Y . (84) dz z−1 z We identify the matrices B0 , B1 with A − σ2 I and A01 , with eigenvalues 0, −σ and 0, 0, respectively. Moreover σ σ σ 0 . A1 + A − I = diag −µ − , µ − 2 2 2 Thus σ σ , β = −µ + , 2 2 γ = σ + 1; α − β = 2µ = 0 by hypothesis.

α =µ+

The parameters of the correspondent hyper-geometric equation are α0 = µ +

σ , 2

β0 = 1 − µ +

σ , 2

γ0 = σ + 1.

From them we deduce the nature of two linearly independent solutions at z = 0. / Z (σ ∈ / Z) the solutions are expressed in terms of hyper-geometric Since γ0 ∈ functions. On the other hand, the effective parameters at z = 1 and z = ∞ are, respectively: σ , 2 σ β1 := β0 = 1 − µ + , 2 γ1 := α0 + β0 − γ0 + 1 = 1,

α1 := α0 = µ +

σ , 2 σ β∞ := α0 − γ0 + 1 = µ − , 2 γ∞ := α0 − β0 + 1 = 2µ. α∞ := α0 = µ +

354

DAVIDE GUZZETTI

Since γ1 = 1, at least one solution has a logarithmic singularity at z = 1. Also note that γ∞ = 2µ, therefore logarithmic singularities appear at z = ∞ if 2µ ∈ Z\{0}. For the derivations which follows, we use the notations of the fundamental paper by Norlund [29]. To derive the connection formulae we use the paper of Norlund when logarithms are involved. Otherwise, in the generic case, any textbook of special functions (like [25]) may be used. / Z. This means σ = ±2µ + 2m, m ∈ Z. First case: α0 , β0 ∈ We can choose the following independent solutions of the hyper-geometric equation: At z = 0, y1(0) (z) = F (α0 , β0 , γ0 ; z),

(85)

y2(0) (z) = z1−γ0 F (α0 − γ0 + 1, β0 − γ0 + 1, 2 − γ0 ; z), where F (α, β, γ ; z) is the well-known hyper-geometric function (see [29]). At z = 1, y1(1) (z) = F (α1 , β1 , γ1 ; 1 − z),

y2(1) (z) = g(α1 , β1 , γ1 ; 1 − z).

Here g(α, β, γ ; z) is a logarithmic solution introduced in [29], and γ ≡ γ1 = 1. At z = ∞, we consider first the case 2µ ∈ / Z, while the resonant case will be considered later. Two independent solutions are 1 (∞) −α0 , y1 = z F α∞ , β∞ , γ∞ ; z 1 (∞) −β0 . y2 = z F β0 , β0 − γ0 + 1, β0 − α0 + 1; z Then, from the connection formulas between F (. . . ; z) and g(. . . ; z) of [25] and [29] we derive [y1(∞) , y2(∞) ] = [y1(0) , y2(0) ]C0∞ , 0 −β0 )(1−γ0) e−iπα0 (1+α (1−β0)(1+α0 −γ0 ) C0∞ = 0 −β0 )(γ0 −1) eiπ(γ0 −α0 −1) (1+α (α0 )(γ0 −β0 )

0 −α0 )(1−γ0 ) e−iπβ0 (1+β (1−α0 )(1+β0−γ0 ) 0 −α0 )(γ0 −1) eiπ(γ0 −β0 −1) (1+β (β0 )(γ0 −α0 )

[y1(0) , y2(0) ] = [y1(1) , y2(1) ]C01 , (2−γ0 ) π sin(π(α0 +β0 )) 0 − sin(πα 0 ) sin(πβ0 ) (1−α0 )(1−β0 ) . C01 = (2−γ0 ) 0) − (γ0−α(γ − (1−α0 )(1−β0) 0 )(γ0−β0 ) We observe that 1 F σ σ (1) Y (z) = I + + O 2 zdiag(−µ− 2 ,µ− 2 ) , z → ∞ z z diag(0,−σ ) ˆ −1 ˆ ˆ 0 (I + O(z))z G0 C0 , z → 0 = G ˆ 1 (I + O(z − 1))(z − 1)J Cˆ 1 , = G

z → 1,

,

355


ˆ −1 ˆ ˆ 0 ≡ T of Lemma 2; namely G where G 0 AG0 = diag(σ/2, −σ/2). By direct substitution in the differential equation, we compute the coefficient F 0 (A01 )12 (A1 )11 1−2µ σ 2 − (2µ)2 1 −r 0 . = , where A F =− 1 1 −1 (A01 )21 8µ 0 r (A1 )22 1+2µ Thus, from the asymptotic behavior of the hyper-geometric function (F (α, β, γ ; 1z ) ∼ 1, z → ∞) we derive (∞) σ 2 −(2µ)1 (∞) y1 (z) r 8µ(1−2µ) y2 (1) . Y (z) = ∗ ∗ From

Y

(1)

(z) ∼

we derive Y

(1)

(z) =

1 ∗

z−σ ∗

ˆ −1 ˆ G 0 C0 ,

y1(0) (z) y2(0) (z) ∗ ∗

z → 0,

(86)

ˆ ˆ −1 G 0 C0 .

Finally, observe that u ωu + vr ˆ G1 = u v r for arbitrary u, v ∈ C,

u = 0,

and

ω :=

σ 2 − (2µ)2 . 8µ

We recall that y2(1) = g(α1 , β1 , 1; 1 − z) ∼ ψ(α1 ) + ψ(β1 ) − 2ψ(1) − iπ + log(z − 1), |arg(1 − z)| < π, as z → 1. We can choose u = 1 and a suitable v, in such a way that the asymptotic behavior of Y (1) for z → 1 is precisely realized by (1) y1 (z) y2(1) (z) ˆ (1) C1 . Y (z) = ∗ ∗ Therefore we conclude that the connection matrices are: σ 2 −(2µ)2 (C0∞)12 (C0∞ )11 r 8µ(1−2µ) ˆ0 , Cˆ 0 = G σ 2 −(2µ)2 (C0∞ )21 r 8µ(1−2µ) (C0∞)22 σ 2 −(2µ)2 (C0∞)12 (C0∞ )11 r 8µ(1−2µ) −1 ˆ 0 Cˆ 0 ) = C01 . Cˆ 1 = C01 (G σ 2 −(2µ)2 (C0∞ )21 r 8µ(1−2µ) (C0∞)22

356

DAVIDE GUZZETTI

It’s now time to consider the resonant case 2µ ∈ Z\{0}. The behavior of Y (1) at z = ∞ is 1 σ σ F Y (1)(z) = I + + O 2 zdiag(−µ− 2 ,µ− 2 ) zR , z z 3 5 1 0 R12 , for µ = , 1, , 2, , . . . , R= 0 0 2 2 2 3 5 1 0 0 , for µ = − , −1, − , −2, − , . . . R= R21 0 2 2 2 and the entry R12 is determined by the entries of A01 . For example, if µ = 12 , we can compute R12 = (A01 )12 = −r

σ2 − 1 4

(and F12 arbitrary); if µ = − 12 we have R21 = (A01 )21 = −

1 σ2 − 1 r 4

(and F21 arbitrary); if µ = 1 we have R12 = −r

σ 2 (σ 2 − 4) . 32

Since σ ∈ / Z, R = 0. This is true for any 2µ ∈ Z\{0}. Note that the R computed here coincides (by isomonodromicity) to the R of the system (69). Therefore, there is a logarithmic solution at ∞. Only C0∞ and thus Cˆ 0 and ˆ C1 change with respect to the nonresonant case. We will see in a while that such matrices disappear in the computation of tr(Mi Mj ), i, j = 0, 1, x. Therefore, it is not necessary to know them explicitly, the only important matrix to know being C01 , which is not affected by resonance of µ. This is the reason why the formulae of Theorem 2 hold true also in the resonant case. Second case: α0 , β0 ∈ Z, namely σ = ±2µ + 2m, m ∈ Z. The formulae are almost identical to the first case, but C01 changes. To see this, we need to distinguish four cases. (i) σ = 2µ + 2m, m = −1, −2, −3, . . .. We choose y2(1) (z) = g0 (α1 , β1 , γ1 ; 1 − z). Here g0 (z) is another logarithmic solution of [29]. Thus (−m)(−2µ−m+1) 0 (−2µ−2m) . C01 = (1−2µ−2m) 0 − (1−m−2µ)(−m) As usual, the matrix is computed from the connection formulas between the hypergeometric functions and g0 that the reader can find in [29].

357


(ii) σ = 2µ + 2m, m = 0, 1, 2, . . .. We choose y1(2) = g(α1 , β1 , γ1 ; 1 − z). Thus (m+1)(2µ+m) 0 (2µ+2m) . C01 = (2µ+2m+1) − (2µ+m)(m+1) 0 (iii) σ = −2µ + 2m, m = 0, −1, −2, . . .. We choose y2(1) (z) = g0 (α1 , β1 , γ1 ; 1 − z). Thus (1−m)(2µ−m) 0 (2µ−2m) C01 = . (1+2µ−2m) 0 − (2µ−m)(1−m) (iv) σ = −2µ + 2m, m = 1, 2, 3, . . .. We choose y2(1) (z) = g(α1 , β1 , γ1 ; 1 − z). Thus (m)(m+1−2µ) 0 (2m−2µ) . C01 = (2m+1−2µ) − (m+1−2µ)(m) 0 Note that this time r 0 1−2µ F = 0 0 in the case σ = ±2µ (i.e. m = 0) because A01 has a special form in this case. Then in Cˆ 0 the elements σ 2 − (2µ)2 (C0∞ )12 , 8µ(1 − 2µ)

σ 2 − (2µ)2 (C0∞)22 8µ(1 − 2µ)

must be substituted, for m = 0, with 1 (C0∞)12 , 1 − 2µ

1 (C0∞ )22 . 1 − 2µ

We turn to the system (71). Let Y˜ be the fundamental matrix (74). With the ˆ −1 ˜ ˆ gauge Y (2)(z) := G 0 (YN (z)G0 ) we have ˜ B0 B˜ 1 dY (2) = + Y (2), dz z z−1 σ σ σ s − σ4 s −1 0 ˆ −1 0 ˆ 4 4 4 ˆ ˆ ˜ ˜ , B1 = G Ax G0 = σ . B0 = G A0 G0 = σ − σ4 − σ4 − 4s 4s This time the effective parameters at z = 0, 1, ∞ are σ σ σ α1 = − , α∞ = − , α0 = − , 2 2 2 σ σ σ β1 = + 1, β∞ = , β0 = + 1, 2 2 2 γ1 = 1, γ∞ = σ. γ0 = 1,

358

DAVIDE GUZZETTI

If follows that both at z = 0 and z = 1 there are logarithmic solutions. We skip the derivation of the connection formulae, which is done as in the previous cases, with some more technical complications. Before giving the results, we observe that 1 σ σ (2) zdiag( 2 ,− 2 ) , z → ∞ Y (z) = I + O z −1 ˜ 0 (1 + O(z))zJ C0 , z → 0 ˆ0 G = G J ˜ ˆ −1 = G 0 G1 (1 + O(z − 1))(z − 1) C1 ,

ˆ 0 , i = 0, 1. Then where Ci := C˜ i G 1 2π i −1 ˆ −1 ˆ 0 (C0 ) C0 G M˜ 0 = G 0 , 0 1

M˜ 1 =

z → 1,

ˆ 0 (C1 )−1 G

1 0

2π i 1

ˆ −1 C1 G 0 .

So, we need to compute Ci , i = 0, 1. The result is )12 (C0∞)11 σ σ+1 4s (C0∞ C0 , , C1 = C01 C0 = (C0∞ )21 σ σ+1 4s (C0∞ )22 where

= C01

(β0 −α0 ) eiπα0 0 (β0 )(1−α0 ) = (α0 −β0 ) (1−α )(β 0 0 ) iπβ0 eiπβ0 − (β0−α e (α0 )(1−β0) 0 +1) π 0 − sin(πα0 ) . −iπα0 0) − sin(πα −e π

)−1 (C0∞

,

The case σ ∈ Z interests us only if σ = 0, otherwise σ ∈ / C\{(−∞, 0) ∪ [1, +∞)}. We observe that the system (70) is precisely the system for Y (2)(z) with the substitution σ → −2µ. In the formulae for xi2 , i = 0, 1, ∞ we only need C01 , with α0 = µ. which is obtained from C01 ˆ −1 ˜ ˆ As for the system (71), the gauge Y (2) = G 0 Y G0 yields 0 s 0 1−s , B˜ 1 = . B˜ 0 = 0 0 0 0 ˆ 0 is the matrix such that Here G 0 1 −1 ˆ ˆ . G0 AG0 = 0 0 The behavior of Y (2) (z) is now 1 (2) zJ , Y (z) = I + O z

z→∞

˜˜ −1 (1 + O(z))zJ C , z → 0 = G 0 0 ˜ ˜ (1 + O(z − 1))(z − 1)J C , = G 1

1

z → 1.

359


˜˜ is the matrix that puts B˜ in Jordan form, for i = 0, 1. Y (2) can be computed Here G i i explicitly: 1 s log(z) + (1 − s) log(z − 1) (2) Y (z) = . 0 1 ˜˜ = diag(1, 1/s), then C = 1 0. In the same way we find C = If we choose G 0 0 1 0 s 1 0 . 0 1−s To prove Theorem 2, it is now enough to compute −1 2πiJ ) e C01 ), 2 − x02 = tr(M0 Mx ) ≡ tr(e2πiJ (C01 −1 2πiJ e C01 ), 2 − x12 = tr(Mx M1 ) ≡ tr((C1 )−1 e2πiJ C1 C01 −1 2πiJ 2 = tr(M0 M1 ) ≡ tr((C0 )−1 e2πiJ C0 C01 e C01 ). 2 − x∞

Note the remarkable simplifications obtained from the cyclic property of the trace ˆ 0 disappear). The fact that Cˆ 0 and Cˆ 1 disappear implies (for example, Cˆ 0 , Cˆ 1 and G that the formulae of Theorem 2 are derived for any µ = 0, including the resonant cases. Thus, the connection formulae in the resonant case 2µ ∈ Z\{0} are the same of the nonresonant case. The final result of the computation of the traces is: (I) Generic case: 2(1 − cos(π σ )) = x02 , 1 1 2 + F (σ, µ)s + = x12 , f (σ, µ) F (σ, µ)s 1 1 −iπσ 2 2 − F (σ, µ)e = x∞ s− , f (σ, µ) F (σ, µ)e−iπσ s

(87)

where f (σ, µ) =

2 cos2 ( π2 σ ) 4 − x02 ≡ 2 , 2 −x x x cos(π σ ) − cos(2π µ) x1 + x∞ 0 1 ∞

F (σ, µ) = f (σ, µ)

)4 16σ ( σ +1 2 . (1 − µ + σ2 )2 (µ + σ2 )2

(II) σ ∈ 2Z, x0 = 0. 2(1 − cos(π σ )) = 0,

4 sin2 (π µ)(1 − s) = x12 ,

2 4 sin2 (π µ)s = x∞ .

2 = −x12 exp(±2π iµ). Four cases (III) x02 = 4 sin2 (π µ). Then (33) implies x∞ which yield the values of σ not included in (I) and (II) must be considered:

360

DAVIDE GUZZETTI 2 (III)1 x∞ = −x12 e−2πiµ

σ = 2µ + 2m, m = 0, 1, 2, . . . , (m + 1)2 (2µ + m)2 2 x1 . s = 162µ+2m (µ + m + 12 )4 2 = −x12 e2πiµ (III)2 x∞

σ = 2µ + 2m, m = −1, −2, −3, . . . , 2µ+2m 4 π4 s = 16 µ + m + 12 × 4 cos (π µ) × (−2µ − m + 1)2 (−m)2 x12

−1

.

2 (III)3 x∞ = −x12 e2πiµ

σ = −2µ + 2m, m = 1, 2, 3, . . . , (m − 2µ + 1)2 (m)2 x12 . s = 16−2µ+2m (−µ + m + 12 )4 2 = −x12 e−2πiµ (III)4 x∞

σ = −2µ + 2m, m = 0, −1, −2, −3, . . . , −2µ+2m π4 1 4 16 − µ + m + × s = 2 cos4 (π µ) −1 × (2µ − m)2 (1 − m)2 x12 . We recall that a in y(x; σ, a) is a = −(1/4s) in general, and a = s for σ = 0. To compute σ and s in the generic case (I) for a given triple (x0 , x1 , x∞ ), we solve the system (87). It has two unknowns and three equations and we need to prove that it is compatible. Actually, the first equation 2(1− cos(π σ )) = x02 always has solutions. Let us choose a solution σ0 (±σ0 + 2n, ∀n ∈ Z are also solutions). Substitute it in the last two equations. We need to verify that they are compatible. Instead of s and 1/s, write X and Y . We have the linear system in two variable X, Y 1 F (σ0 ) X f (σ0 )x12 − 2 F (σ0 ) = . 2 2 − f (σ0 )x∞ Y F (σ0 )e−iπσ0 F (σ1 0 ) e−iπσ0 The system has a unique solution if and only if 1 F (σ0 ) F (σ0 ) = 0. 2i sin(π σ0 ) = det F (σ0 )e−iπσ0 F (σ1 0 ) e−iπσ0 / Z. The condition is not restrictive, because for σ even we This happens for σ0 ∈ turn to the case (II), and σ odd is not in C\[(−∞, 0) ∪ [1, +∞)]. The solution is


361

then 2 −iπσ0 e ) 2(1 + e−iπσ0 ) − f (σ0 )(x12 + x∞ , −2πiσ 0 − 1) F (σ0 )(e 2 f (σ0 )e−iπσ0 (e−iπσ0 x12 + x∞ ) − 2e−iπσ0 (1 + e−iπσ0 ) . Y = F (σ0 ) e−2πiσ0 − 1 Compatibility of the system means that XY ≡ 1. This is verified by direct computation. It follows from this construction that for any σ solution of the first equation of (87), there always exists a unique s which solves the last two equations. To complete the proof of Theorem 2 (points (i), (ii), (iii)), we just have to compute the square roots of the xi2 (i = 0, 1, ∞) in such a way that (33) is satisfied. For example, the square root of (I) satisfying (33) is π σ , x0 = 2 sin 2 ! 1 1 , F (σ, µ)s + √ x1 = √ f (σ, µ) F (σ, µ)s ! i 1 −i π2σ F (σ, µ)se −√ , x∞ = √ πσ f (σ, µ) F (σ, µ)se−i 2

X =

which yields (i), with F (σ, µ) = f (σ, µ)(2G(σ, µ))2 . We remark that in case (II) only σ = 0 is in C\{(−∞, 0) ∪ [1, +∞)}. If µ integer in (II), the formulae give (x0 , x1 , x∞ ) = (0, 0, 0). The triple is not admissible, and direct computation gives R = 0 for the system (84). This is the case of commuting monodromy matrices with a 1-parameter family of rational solutions of PVIµ . The last remark concerns the choice of (64), (66) instead of (65), (67). The reason is that at z = 0 the system (84) has solution corresponding to (85). This is true for any σ = 0 in C\{(−∞, 0) ∪ [1, +∞)}, also for σ → ±2µ. Its behavior ˆ 0 = T of (64), (66) but not of (65), (67). is (86), which is obtainable from the G See also the comment following Lemma 5. Remark. In the proof of Theorem 2 we take the limits of the system and of the rescaled system for x → 0 in D(σ ). At x we assign the monodromy M0 , M1 , Mx characterized by (x0 , x1 , x∞ ) and then we take the limit proving the theorem. If we start from another point x ∈ D(σ ) we have to choose the same monodromy M0 , M1 , Mx , because what we are doing is the limit for x → 0 in D(σ ) of the matrix coefficient A(z, x; x0 , x1 , x∞ ) of the system (69) considered as a function defined on the universal covering of C0 ∩ {|x| < }. Proof of Remark 2 of Section 4. We prove that a(σ ) = 1/16a(−σ ), namely 1 1 a=− . s(σ ) = s(−σ ) 4s

362

DAVIDE GUZZETTI

Given monodromy data (x0 , x1 , x∞ ) the parameter s corresponding to σ is uniquely determined by 1 1 2 + F (σ )s + = x12 , f (σ ) F (σ )s 1 1 −iπσ 2 2 − F (σ )e = x∞ s− . f (σ ) F (σ )e−iπσ s We observe that f (σ ) = f (−σ ) and that the properties of the Gamma function (1 − z)(z) =

π , sin(π z)

(z + 1) = z(z)

imply F (−σ ) = 1/F (σ ). Then the value of s corresponding to −σ is (uniquely) determined by s F (σ ) 1 2+ + = x12 , f (σ ) F (σ ) s s F (σ )e−iπσ 1 2 2− = x∞ − . f (σ ) F (σ )e−iπσ s We conclude that s(−σ ) = −(1/s(σ )).

✷

Proof of formula (45). We are ready to prove formula (45), namely β12 : (σ, a) → (σ, ae−2πiσ ). For σ = 0, we have x0 = 0 and β12 : (0, x1 , x∞ ) → (0, x1 , x∞ ). Thus, a=

2 2 x∞ x∞ → ≡ a. 2 2 x12 + x∞ x12 + x∞

For σ = ±2µ + 2m, we consider the example σ = 2µ + 2m, m = 0, 1, 2, . . .. 2 H (σ )e2πiµ, where The other cases are analogous. We have s = x12 H (σ ) = −x∞ the function H (σ ) is explicitly given in Theorem 2(III). Then 2 H (σ )e2πiµ → −x12 H (σ )e2πiµ = −se2πiµ . β1 : s = −x∞

Then β12 : s → se4πiµ ⇒ a → ae−4πiµ ≡ ae−2πiσ . For the generic case (I) (σ ∈ / Z, σ = ±2µ + 2m) recall that 1 = x12 f (σ ) − 2, F (σ )s 1 2 = 2 − x∞ f (σ ) F (σ )e−iπσ s + −iπσ F (σ )e s

F (σ )s +


363

has a unique solution s. Also observe that β1 : x∞ → x1 . Then the transformed parameter β1 : s → s β1 satisfies the equation 1 F (σ )e−iπσ s β1 = 2 − x12 f (σ ) 1 . ≡ − F (σ )s + F (σ )s

F (σ )e−iπσ s β1 +

Thus s β1 = −eiπσ s. This implies β12 : s → se2πiσ ⇒ a → ae−2πiσ .

✷

We finally prove the proposition stated at the end of Section 4. Proof. Observe that both y(x) and y(x; σ, a) have the same asymptotic behavior for x → 0 in D(σ ). Let A0 (x), A1 (x), Ax (x) be the matrices constructed from y(x) and A∗0 (x), A∗1 (x), A∗x (x) constructed from y(x; σ, a) by means of formulae (21). It follows that Ai (x) and A∗i (x), i = 0, 1, x, have the same asymptotic behavior as x → 0. This is the behavior of Lemma 1 of Section 8 (adapted to our case). From the proof of Theorem 2, it follows that A0 (x), A1 (x), Ax (x) and A∗0 (x), A∗1 (x), A∗x (x) produce the same triple (x0 , x1 , x∞ ). The solution of the Riemann–Hilbert problem for such a triple is unique, up to conjugation of the Fuchsian systems. / Z, the conjugation Therefore, Ai (x) and A∗i (x), i = 0, 1, x are conjugated. If 2µ ∈ is diagonal. If 2µ ∈ Z and R = 0, then Ai (x) = A∗i (x). Putting [A(z; x)]12 = 0 and [A∗ (z; x)]12 = 0, we conclude that y(x) ≡ y(x; σ, a).

10. Proof of Theorem 3 The elliptic representation was derived by Fuchs in [14]. In the case of PVIµ , the representation is discussed at the beginning of Subsection 5.1. Here we study the solutions of (34). We let x → 0. If τ > 0 and

u

(88)

4ω < τ, 1 we expand the elliptic function in Fourier series (39). The first condition τ > 0 is always satisfied for x → 0 because 4 1 ln |x| + ln 2 + O(x), x → 0. π π Therefore, in the following we assume that |x| < < 1 for a sufficiently small . We look for a solution u(x) of (34) of the form τ (x) = −

u(x) = 2ν1 ω1 (x) + 2ν2 ω2 (x) + 2v(x),

364

DAVIDE GUZZETTI

where v(x) is a (small) perturbation to be determined from (34). We observe that u(x) ν1 ν2 v(x) = + τ (x) + 4ω1 (x) 2 2 2ω1 (x) i i F1 (x) v(x) ν1 ν2 + − ln x − + . = 2 2 π π F (x) 2ω1 (x) Note that for x → 0, F1 (x)/F (x) = −4 ln 2 + g(x), where g(x) = O(x) is a convergent Taylor series starting with x. Thus, condition (88) becomes (2 + ν2 ) ln |x| − C(x, ν1 , ν2 ) − 8 ln 2 < ν2 arg(x) < ( ν2 − 2) ln |x| − C(x, ν1 , ν2 ) + 8 ln 2,

(89)

where C(x, ν1 , ν2 ) = [

πv + 4 ln 2 ν2 + π ν1 + O(x)]. ω1

We expand the derivative of ℘ appearing in (34) u ∂ ℘ ; ω1 , ω2 ∂u 2 πu 3 ∞ nπ u n2 e2πinτ π 3 cos 4ω1 π πu sin − = ω1 n=1 1 − e2πinτ 2ω1 2ω1 sin3 4ω 1 3 ∞ 2 2πinτ ne 1 π in π u −in π u e 2ω1 − e 2ω1 + = 2πinτ 2i ω1 n=1 1 − e i πu −i π u π 3 e 4ω1 + e 4ω1 . + 4i i πu −i π u 3 2ω1 e 4ω1 − e 4ω1 πu −i 4ω

Now we come to a crucial step in the construction: we collect e term, which becomes 4πi u 2πi u π 3 e 4ω1 + e 4ω1 4i 3 . 2πi u 2ω1 e 4ω1 − 1 2πi

u

1

in the last

The denominator does not vanish if |e 4ω1 | < 1. From now on, this condition is added to (88) and reduces the domain (89). The expansion of ∂/∂u℘ becomes u ∂ ℘ ; ω1 , ω2 ∂u 2 ∞ iπn[−ν1 +(2−ν2 )τ − ωv ] 1 1 π 3 n2 e 2iπn[ν1 +ν2 τ + ωv ] 1 − 1 + e = 2πinτ 2i ω1 n=1 1−e 2πi[ν1 +ν2 τ + ωv ] πi[ν1 +ν2 τ + ωv ] 1 + e 1 π 3e . + 4i 3 πi[ν1 +ν2 τ + v ] ω1 2ω1 −1 e

365


We observe that xC xC eiπCτ = C eCg(x) = C (1 + O(x)), x → 0, for any C ∈ C. 16 16 Hence, u e−iπν1 2−ν2 −iπ ωv eiπν1 ν2 iπ ωv ∂ 1, ℘ ; ω1 , ω2 = F x, 2−ν x e x e 1 , ∂u 2 16 2 16ν2 where 3 ∞ π n2 en(2−ν2 )g(x) 1 n 2nν g(x) 2n F (x, y, z) = 1 2n y e 2 z − 1 + 2i ω1 (x) n=1 1 − eg(x) x 2n 16 3 2ν2 g(x) 2 e z + eν2 g(x)z π . + 4i 2ω1 (x) (eν2 g(x)z − 1)3 The series converges for |x| < and for |y| < 1, |yz| < 1; this is precisely (88). However, we require that the last term is holomorphic, so we have to further impose |eν2 g(x)z| < 1. On the resulting domain |x| < , |y| < 1, |eν2 g(x)z| < 1, F (x, y, z) is holomorphic and satisfies F (0, 0, 0) = 0. The condition

−iπν

1

e v

ν2 g(x) 2−ν2 −iπ ω1

|y| < 1, |e z| < 1 is 2−ν x e

< 1, 16 2

ν g(x) eiπν1 ν iπ v

e 2 x 2 e ω1

< 1,

16ν2 namely ν2 ln |x| − C(x, ν1 , ν2 ) < ν2 arg(x) < ( ν2 − 2) ln |x| − C(x, ν1 , ν2 ) + 8 ln 2, which is more restrictive than (89). but

−iπν 1

e v

2−ν2 −iπ ω1

e

< 1,

162−ν2 x

(90)

For ν2 = 0, any value of arg(x) is allowed,

ν g(x) eiπν1 ν iπ v

2 ω1

e 2 x e

0 such that |L(x, y, z)| c(|x| + |y| + |z|), |J(x, y, z, v, w)| c(|x| + |y| + |z|), |J(x, y, z, v2 , w2 ) − J(x, y, z, v1 , w1 )| c(|x| + |y| + |z|)(|v2 − v1 | + |w2 − w1 |),

(93) (94) (95)

for |x|, |y|, |z|, |v|, |w| < . Proof. Let’s prove (94). J(x, y, z, v, w) 1 d J(λx, λy, λz, v, w) dλ × = 0 dλ 1 1 ∂J ∂J (λx, λy, λz, v, w) dλ + y (λx, λy, λz, v, w) dλ + ×x 0 ∂x 0 ∂y 1 ∂J (λx, λy, λz, v, w) dλ. +z 0 ∂z

370

DAVIDE GUZZETTI

Moreover, for δ small, ∂J (λx, λy, λz, v, w) = ∂x

|ζ −λx|=δ

J(ζ, λy, λz, v, w) dζ , (ζ − λx)2 2π i

which implies that ∂J/∂x is holomorphic and bounded when its arguments are less than . The same holds true for ∂J/∂y and ∂J/∂z. This proves (94), c being a constant which bounds |∂J/∂x|, |∂J/∂y| |∂J/∂z|. The inequality (93) is proved in the same way. We turn to (95). First we prove that for |x|, |y|, |z|, |v1 |, |w1 |, |v2 |, |w2 | < there exist two holomorphic and bounded functions ψ1 (x, y, z, v1 , w1 , v2 , w2 ), ψ2 (x, y, z, v1 , w1 , v2 , w2 ) such that J(x, y, z, v2 , w2 ) − J(x, y, z, v1 , w1 ) = (v2 − v1 )ψ1 (x, y, z, v1 , w1 , v2 , w2 ) + + (w1 − w2 )ψ2 (x, y, z, v1 , w1 , v2 , w2 ).

(96)

In order to prove this, we write J(x, y, z, v2 , w2 ) − J(x, y, z, v1 , w1 ) 1 d = J(x, y, z, λv2 + (1 − λ)v1 , λw2 + (1 − λ)w1 ) dλ dλ 0 1 ∂J (x, y, z, λv2 + (1 − λ)v1 , λw2 + (1 − λ)w1 ) dλ + = (v2 − v1 ) 0 ∂v 1 ∂J (x, y, z, λv2 + (1 − λ)v1 , λw2 + (1 − λ)w1 ) dλ + (w2 − w1 ) 0 ∂w =: (v2 − v1 )ψ1 (x, y, z, v1 , w1 , v2 , w2 ) + + (w2 − w1 )ψ2 (x, y, z, v1 , w1 , v2 , w2 ). Moreover, for small δ, ∂J (x, y, z, v, w) = ∂v

|ζ −v|=δ

J(x, y, z, ζ, w) dz , (ζ − v)2 2π i

which implies that ψ1 is holomorphic and bounded for its arguments less than . We also obtain ∂J/∂v(0, 0, 0, v, w) = 0, then ψ1 (0, 0, 0, v1 , w1 , v2 , w2 ) = 0. The proof for ψ2 is analogous. We use (96) to complete the proof of (95). Actually, we observe that 1 d ψi (λx, λy, λz, v1 , w1 , v2 , w2 ) dλ ψi (x, y, z, v1 , w1 , v2 , w2 ) = 0 dλ 1 1 1 ∂ψi ∂ψi ∂ψi dλ + y dλ + z dλ = x ∂z 0 ∂x 0 ∂y 0 and we conclude as in the proof of (94).

✷


371

We solve the system of integral equations by successive approximations. We can choose any path L(x) such that 0 < ν ∗ < 2. Here we choose ν ∗ = 1. For convenience, we put eiπν1 e−iπν1 , B := . 162−ν2 16ν2 Therefore, for any n 1 the successive approximations are A :=

v0 = w0 = 0, 1 wn (x) = L(s, As 2−ν2 , Bs ν2 ) + L(x) t

+ J(s, As 2−ν2 , Bs ν2 , vn−1 (s), wn−1 (s)) ds, 1 wn (s) ds. vn (x) = L(x) s

(97) (98)

SUB-LEMMA 4. There exists a sufficiently small < such that for any n 0 the functions vn (x) and wn (x) are holomorphic in the domain ˜ 0 such that |x| < , |Ax 2−ν2 | < , |Bx ν2 | < . D( ; ν1 , ν2 ) := x ∈ C They are also correctly bounded, namely |vn (x)| < , |wn (x)| < for any n. They satisfy n (2c)n P (ν2 )2n |x| + |Ax 2−ν2 | + |Bx ν2 | , (99) n! n (2c)n P (ν2 )2n |x| + |Ax 2−ν2 | + |Bx ν2 | , (100) |wn − wn−1 | n! where P (ν2 ) := P (ν2 , ν ∗ = 1) and c is the constant appearing in Sub-Lemma 3. Moreover xdvn /dx = wn . Proof. We proceed by induction. 1 1 2−ν2 ν2 L(s, As w1 (s) ds. , Bs ) ds, v1 = w1 = s L(x) L(x) s |vn − vn−1 |

It follows from Sub-Lemma 2 and (93) that w1 (x) is holomorphic for |x|, |Ax 2−ν2 |, |Bx ν2 | < . From (92) and (93), we have 1 |L(s, As 2−ν2 , Bs ν2 )| |ds| |w1 (x)| |s| cP (ν2 )(|x| + |Ax 2−ν2 | + |Bx ν2 |) 3cP (ν2 ) < on D( ; ν1 , ν2 ), provided that is small enough. By Sub-Lemma 2, also v1 (x) is holomorphic for |x|, |Ax 2−ν2 |, |Bx ν2 | < and x(dv1 /dx) = w1 . By (92) we also have |v1 (x)| cP (ν2 )2 (|x| + |Ax 2−ν2 | + |Bx ν2 |) 3cP (ν2 )2 <

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DAVIDE GUZZETTI

on D( ; ν1 , ν2 ). Note that P (ν2 ) 1, so (100) (99) are true for n = 1. Now we suppose that the statement of the sub-lemma is true for n and we prove it for n + 1. Consider |wn+1 (x) − wn (x)|

1 J(s, As 2−ν2 , Bs ν2 , vn , wn ) − =

L(x) s

2−ν2 ν2 , Bs , vn−1 , wn−1 ) ds

. − J(s, As By (95) the above is 1 c (|s| + |As 2−ν2 | + |Bs ν2 |)(|vn − vn−1 | + |wn − wn−1 |)|ds|. |s| L(x) By induction this is (2c)n P (ν2 )2n 2c n!

L(x)

1 (|s| + |As 2−ν2 | + |Bs ν2 |)n+1 |ds| |s|

(2c)n P (ν2 )2n P (ν2 ) (|x| + |Ax 2−ν2 | + |Bx ν2 |)n+1 2c n! n+1 (2c)n+1 P (ν2 )2(n+1) (|x| + |Ax 2−ν2 | + |Bx ν2 |)n+1 . (n + 1)! This proves (100). Now we estimate |wn+1 (s) − wn (s)| |ds| |vn+1 (x) − vn (x)|

L(x) n+1

(2c)

P (ν2 )2n+1 (n + 1)!

n+1

L(x)

1 (|s| + |As 2−ν2 | + |Bs ν2 |)n+1 |ds| |s|

2(n+1)

(2c) P (ν2 ) (|x| + |Ax 2−ν2 | + |Bx ν2 |)n+1 (n + 1)(n + 1)! (2c)n+1 P (ν2 )2(n+1) (|x| + |Ax 2−ν2 | + |Bx ν2 |)n+1 . (n + 1)!

This proves (99). From Sub-Lemma 2, we also conclude that wn and vn are holon = wn . Finally, we see that morphic in D( , ν1 , ν2 ) and x dv dx |vn (x)|

n

|vk (x) − vk−1 (x)|

k=1

exp{2cP 2 (ν2 )(|x| + |Ax 2−ν2 | + |Bx ν2 |)} − 1 exp{6cP 2 (ν2 ) } − 1

373


and the same for |wn (x)|. Therefore, if is small enough we have |vn (x)| < , |wn (x)| < on D( , ν1 , ν2 ). ✷ Let us define w(x) := lim wn (x)

v(x) := lim vn (x), n→∞

n→∞

if they exist. We can also rewrite v(x) = lim vn (x) = n→∞

∞

(vn (x) − vn−1 (x)).

n=1

We see that the series converges uniformly in D( , ν1 , ν2 ) because

∞

(vn (x) − vn−1 (x))

n=1

∞

(2c)n P (ν2 )2n n=1

n!

(|x| + |Ax 2−ν2 | + |Bx ν2 |)n

= exp{2cP 2 (ν2 )(|x| + |Ax 2−ν2 | + |Bx ν2 |)} − 1. The same holds for wn (x). Therefore, v(x) and w(x) define holomorphic functions in D( , ν1 , ν2 ). From Sub-Lemma 4, we also have x(dv(x)/dx) = w(x) in D( , ν1 , ν2 ). We show that v(x), w(x) solve the initial integral equations. The left-hand side of (97) converges to w(x) for n → ∞. Let us prove that the right-hand side also converges to 1 L(s, As 2−ν2 , Bs ν2 ) + J(s, As 2−ν2 , Bs ν2 , v(s), w(s)) ds. L(x) s We have to evaluate the following difference:

1

J(s, As 2−ν2 , Bs ν2 , v(s), w(s)) ds −

L(x) s

1 2−ν2 ν2 J(s, As , Bs , vn (s), wn (s)) ds

. − L(x) s From (95), the above is 1 (|s| + |As 2−ν2 | + |Bs ν2 |)(|v − vn | + |w − wn |)|ds|. c |s| L(x)

(101)

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DAVIDE GUZZETTI

Now we observe that |v(x) − vn (x)|

∞

|vk − vk−1 |

k=n+1

=

∞

(2c)k P (ν2 )2k (|x| + |Ax 2−ν2 | + |Bx ν2 |)k k! k=n+1

(|x| + |Ax

2−ν2

| + |Bx |) ν2

n+1

∞

(2c)k+n+1 P (ν2 )2(k+n+1) k=0

(k + n + 1)!

×

× (|x| + |Ax 2−ν2 | + |Bx ν2 |)k . The series converges. Its sum is less than some constant S(ν2 ) independent of n. We obtain |v(x) − vn (x)| S(ν2 )(|x| + |Ax 2−ν2 | + |Bx ν2 |)n+1 . The same holds for |w − wn |. Thus, (101) is 1 2cS(ν2 ) (|s| + |As 2−ν2 | + |Bs ν2 |)n+2 |ds| |s| L(x) 2cS(ν2 )P (ν2 ) (|x| + |Ax 2−ν2 | + |Bx ν2 |)n+2 . n+2 Namely,

1

J(s, As 2−ν2 , Bs ν2 , v(s), w(s)) ds −

L(x) s

2cS(ν2 )P (ν2 ) n+2 1 2−ν2 ν2 J(s, As (3 ) . , Bs , vn (s), wn (s)) ds

− n+2 L(x) s In a similar way, the right-hand side of (98) is

S(ν2 )P (ν2 ) n+1

1

(3 ) .

s (w(s) − wn (s)) ds n+1 Therefore, the right-hand sides of (97) and (98) converge on the domain D(r, ν1 , ν2 ) for r < min{ , 1/3}. We finally observe that |v(x)| and |w(x)| are bounded on D(r). For example, |v(x)| (|x| + |Ax

2−ν2

| + |Bx |) ν2

∞

(2c)k+1 P (ν2 )2(k+1)

k=0 ν2 k

× (|x| + |Ax 2−ν2 | + |Bx |) =: M(ν2 )(|x| + |Ax 2−ν2 | + |Bx ν2 |),

(k + 1)!

×

375


where the sum of the series is less than a constant M(ν2 ). We have proved Lemma 6. ✷ We note that the proof of Lemma 6 only makes use of the properties of L and J, regardless of how these functions have been constructed. The structure of the integral equations implies that v(x) is bounded (namely |v(x)| = O(r)). Now, we come back to our case, where L and J have been constructed from the Fourier expansion of elliptic functions. We need to check if (90) and D(r, ν1 , ν2 ) have nonempty intersection. This is true, indeed D(r) is contained in (90), because in (90) the term (π v/2ω1 ) is O(r), while in D(r, ν1 , ν2 ) the term ln r appear, and r is small. To conclude the proof of Theorem 3, we have to work out the explicit series (13). In order to do this, we observe that w1 and v1 are series of the type

cpqr (ν2 )x p (Ax 2−ν2 )q (Bx ν2 )r , (102) p,q,r0

where cpqr (ν2 ) is rational in ν2 . This follows from w1 (x) = L(x) L(s, As 2−ν2 , Bs ν2 ) ds and from the fact that L(x, Ax 2−ν2 , Bx ν2 ) itself is a series (102) by construction, with coefficients cpqr (ν2 ) which are rational functions of ν2 . The same holds true for J. We conclude that wn (x) and vn (x) have the form (102) for any n. This implies that the limit v(x) is also a series of type (102). We can reorder such a series to obtain (13). Consider the term cpqr (ν2 )x p (Ax 2−ν2 )q (Bx ν2 )r , and recall that by definition B = 1/162 A. We absorb 16−2r into cpqr (ν2 ) and we study the factor Aq−r x p+(2−ν2 )q+ν2 r = Aq−r x p+2q+(r−q)ν2 . We have three cases: (1) r = q, then we have x p+2q =: x n ,

n = p + 2q.

(2) r > q, then we have 1 ν2 r−q 1 ν2 m x x =: x n , x p+2q A A

n = p + 2q,

(3) r < q, then we have q−r m q−r p+2r 2−ν2 n 2−ν2 =: x Ax , Ax A x

m = q − r.

n = p + 2r,

m = q − r.

This brings a series of the type (102) to the form (13). The proof of Theorem 3 is complete. A system of integral equations similar to the one we considered here was first studied by Shimomura in [37] and [19].

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DAVIDE GUZZETTI

Acknowledgements I am grateful to B. Dubrovin for many discussions and advice. I would like to thank A. Bolibruch, A. Its, M. Jimbo, M. Mazzocco, and S. Shimomura for fruitful discussions. The author is supported by a fellowship of the Japan Society for the Promotion of Science (JSPS).

References 1. 2. 3.

4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Anosov, D. V. and Bolibruch, A. A.: The Riemann–Hilbert Problem, Publ. Steklov Institute of Mathematics, 1994. Balser, W., Jurkat, W. B. and Lutz, D. A.: Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations, J. Math. Anal. Appl. 71 (1979), 48–94. Balser, W., Jurkat, W. B. and Lutz, D. A.: On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities, SIAM J. Math. Anal. 12 (1981), 691–721. Birman, J. S.: Braids, Links, and Mapping Class Groups, Ann. of Math. Stud. 82, Princeton Univ. Press, 1975. Bolibruch, A. A.: On movable singular points of Schlesinger equation of isomonodromic deformation, Preprint, 1995; On isomonodromic confluences of Fuchsian singularities, Proc. Steklov Inst. Math. 221 (1998), 117–132; On Fuchsian systems with given asymptotics and monodromy, Proc. Steklov Inst. Math. 224 (1999), 98–106. Dijkgraaf, R., Verlinde, E. and Verlinde, H.: Topological strings in d < 1, Nuclear Phys. B 352 (1991), 59–86. Dubrovin, B.: Integrable systems in topological field theory, Nuclear Phys. B 379 (1992), 627– 689. Dubrovin, B.: Geometry and Itegrability of topological-antitopological fusion, Comm. Math. Phys. 152 (1993), 539–564. Dubrovin, B.: Geometry of 2D topological field theories, In: Lecture Notes in Math. 1620, Springer, New York, 1996, pp. 120–348. Dubrovin, B.: Painlevé trascendents in two-dimensional topological field theory, In: R. Conte (ed.), The Painlevé Property, One Century Later, Springer, New York, 1999. Dubrovin, B.: Geometry and analytic theory of Frobenius manifolds, math.AG/9807034, 1998. Dubrovin, B.: Differential geometry on the space of orbits of a Coxeter group, math.AG/9807034, 1998. Dubrovin, B. and Mazzocco, M.: Monodromy of certain Painlevé-VI trascendents and reflection groups, Invent. Math. 141 (2000), 55–147. Fuchs, R.: Uber lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singularen Stellen, Math. Annal. 63 (1907), 301–321. Gambier, B.: Sur des équations differentielles du second ordre et du premier degré dont l’intégrale est à points critiques fixes, Acta Math. 33 (1910), 1–55. Guzzetti, D.: Stokes matrices and monodromy for the quantum cohomology of projective spaces, Comm. Math. Phys. 207 (1999), 341–383. Guzzetti, D.: Inverse problem and monodromy data for three-dimensional Frobenius manifolds, J. Math. Phys. Anal. Geom. 4 (2001), 245–291. Its, A. R. and Novokshenov, V. Y.: The Isomonodromic Deformation Method in the Theory of Painlevé Equations, Lecture Notes in Math. 1191, Springer, New York, 1986. Iwasaki, K., Kimura, H., Shimomura, S. and Yoshida, M.: From Gauss to Painlevé, Aspects Math. 16, Vieweg, Braunschweig, 1991.


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379

Spectrum Localization of Infinite Matrices M.I. GIL’ Department of Mathematics, Ben Gurion University of the Negev, PO Box 653, Beer-Sheva 84105, Israel. e-mail: [email protected] (Received: 4 September 2000; in final form: 1 June 2001) Abstract. The paper deals with linear operators in a separable Hilbert space represented by infinite matrices with compact off diagonal parts. Bounds for the spectrum are established. In particular, new estimates for the spectral radius are proposed. These results are new even in the finite-dimensional case. Also applications to integral, differential and integro-differential operators are discussed. Mathematics Subject Classifications (2000): 47A10, 47A55, 15A9, 15A18. Key words: finite and infinite matrices, spectrum localization, integral, differential and integrodifferential operators.

1. Introduction and Preliminaries A lot of papers and books are devoted to the spectrum of compact operators, mainly relating to the distributions of the eigenvalues, cf. [6, 10, 11] and references therein. However, in many applications, for example, in stability theory and numerical analysis, bounds for eigenvalues are very important. But the bounds are investigated considerably less than the distributions. Let H be a separable complex Hilbert space with a scalar product (·, ·), the norm · and the unit operator I . Let {ek }∞ k=1 be an orthogonal normed basis in H . Everywhere below A is a bounded linear operator in H represented by a matrix with the entries aj k = (Aek , ej )

(j, k = 1, 2, . . .).

(1.1)

In the sequel V , W and D denote the upper triangular, lower triangular, and diagonal parts of A, respectively: (V ek , ej ) = 0 for all j > k; (V ek , ej ) = aj k for j < k, (W ek , ej ) = 0 for all j < k; (W ek , ej ) = aj k for j > k, (Dek , ej ) = 0 for j = k (j, k = 1, 2, . . .). (Dek , ek ) = akk ,

(1.2)

So A = D + V + W . Throughout this paper it is assumed that V and W are compact operators. This research was supported by the Israel Ministry of Science and Technology.

(1.3)

380

M.I. GIL’

Numerous integral operators can be represented by matrix A under condition (1.3). In the present paper we derive bounds for the spectrum of A. The paper is organized as follows. In Section 2 we prove the main result of the paper – Theorem 2.1 on the spectrum of matrix A. Section 3 is devoted to finite matrices. In particular, in the case of matrices which are ‘near’ to triangular, we improve the Cassini Ovals Theorem and, the Frobenius estimate for the spectral radius, cf. [9]. Section 4 deals with infinite matrices having Hilbert–Schmidt off-diagonal components. Besides, we supplement the well-known estimates for the spectral radius [7]. In Section 5 we consider matrices whose off-diagonals are Neumann–Schatten operators. Section 6 is devoted to the nonlinear spectrum. Localization of spectra of unbounded operators is examined in Section 7. Illustrative examples are collected in Section 8. Here we consider an integral operator, a nonselfadjoint differential operator and an integro-differential one. Besides, our results, supplement the well-known ones, cf. [1, 2, 8, 11] and references therein. LEMMA 1.1. Under conditions (1.3), operators V and W are quasinilpotent. That is, lim n W n = 0. lim n V n = 0, n→∞

n→∞

Proof. For any natural m introduce the orthogonal projectors Pm and Qm by Pm h =

m

(h, ek )ek

(h ∈ H ) and

Qm = I − Pm .

k=1

Simple calculation shows that Pk V Pk = V Pk , Qk W Qk = W Qk and (Pk+1 − Pk )V (Pk+1 − Pk ) = (Qk+1 − Qk )W (Qk+1 − Qk ) = 0 for all natural k. Now the required result is due to Lemma 3.2.2 ([3]).

✷

2. The Main Result In the sequel, σ (B) denotes the spectrum of a linear operator B, rs (B) is its spectral radius, and ρ(λ, D) =

inf |λ − ajj |

j =1,2,...

(λ ∈ C).

Clearly, σ (D) is the closure of the set {akk }∞ k=1 . THEOREM 2.1. Let the conditions [(D − λI )−1 V ]k ck ρ −k (λ, D)

381

SPECTRUM LOCALIZATION OF INFINITE MATRICES

and [(D − λI )−1 W ]k dk ρ −k (λ, D)

(λ ∈ / σ (D))

(2.1)

hold, where ck , dk (k = 1, 2, . . .) are nonnegative numbers with the properties √ k k ck → 0, dk → 0 (k → ∞). (2.2) Then for any µ ∈ σ (A), we have either µ ∈ σ (D), or ∞ j,k=1

ck dj k+j ρ (µ, D)

1.

(2.3)

To prove this theorem we need the following lemma: LEMMA 2.2. Let A˜ be a bounded linear operator in H of the form A˜ = I + V˜ + W˜ ,

(2.4)

where operators V˜ and W˜ are quasinilpotent. If, in addition, the condition ∞ k+j ˜ k ˜ j (−1) V W < 1 θA ≡

(2.5)

j,k=1

is fulfilled, then operator A˜ is boundedly invertible. Proof. We have A˜ = I + V˜ + W˜ = (I + V˜ )(I + W˜ ) − V˜ W˜ . Since W˜ and V˜ are quasinilpotent, the operators I + V˜ and I + W˜ are invertible: (I + V˜ )−1 =

∞

(−1)k V˜ k ,

(I + W˜ )−1 =

k=0

∞ (−1)k W˜ k . k=0

Thus, I + V˜ + W˜ = (I + V˜ )[I − (I + V˜ )−1 V˜ W˜ (I + W˜ )−1 ](I + W˜ ) = (I + V˜ )(I − BA˜ )(I + W˜ ), where BA˜ = (I + V˜ )−1 V˜ W˜ (I + W˜ )−1 . But V˜ (I + V˜ )−1 =

∞

(−1)k−1 V˜ k ,

k=1

So BA˜ =

∞ j,k=1

(−1)k+j V˜ k W˜ j .

(I + W˜ )−1 =

∞ k=1

(−1)k−1 W˜ k .

382

M.I. GIL’

If (2.5) holds, then BA˜ < 1. So A˜ −1 = (I + W˜ )−1 (I − BA˜ )−1 (I + V˜ )−1 is bounded. ✷ Proof of Theorem 2.1. We have A − λI = D + W + V − λ = (D − λ)(I + (D − λ)−1 W + (D − λ)−1 V ). Let µ = amm for all natural m. Due to (2.1) and (2.2), operators (D − µ)−1 V and (D − µ)−1 W are quasinilpotent and ∞ ∞ ck dj k+j −1 k −1 j . (−1) ((D − µ) V ) ((D − µ) W ) k+j ρ (µ, D) j,k=1 j,k=1 Assume that ∞ j,k=1

ck dj k+j ρ (µ, D)

< 1.

Then due to the previous lemma, A − µI is invertible. This contradiction proves the required result. ✷ Note that in the case of a triangular matrix Theorem 2.1 yields the exact result: σ (A) is the closure of the set {akk , k = 1, 2, . . .}. COROLLARY 2.3. Let V = 0 and W = 0. Then under conditions (2.1), (2.2), for any µ ∈ σ (A), ρ(µ, D) z(A) where z(A) is the unique positive root of the equation ∞

ck dj z−j −k = 1.

(2.6)

j,k=1

Indeed, the required result is due to comparison of (2.3) with (2.6). To estimate z(A), consider the equation ∞

bk z−k = 1,

(2.7)

k=1

where the coefficients bk are nonnegative and have the property θ0 ≡ 2 sup j bj < ∞. j

LEMMA 2.4. The unique positive root z0 of Equation (2.7) satisfies the estimate z0 θ0 .

383


Proof. Set in (2.7) z = x −1 θ0 . We have 1=

∞

bk θ0−k x k .

(2.8)

k=1

But ∞

bk θ0−k

k=1

∞

2−k = 1

k=1

and therefore the unique positive root x0 of (2.8) satisfies the inequality x0 1. ✷ Hence, z0 = θ0 x0−1 θ0 . As claimed. Note that the latter lemma generalized the well-known result for algebraic equations. Rewrite Equation (2.6) as ∞

b˜k zk = 1 with b˜j =

k=2

j −1

cj −m dm (j 2).

(2.9)

m=1

Now the previous lemma gives j z(A) ψ(A) ≡ 2 sup b˜j .

(2.10)

j 2

3. Finite Matrices In this section A = (aj k ) is a complex (n × n)-matrix (2 n < ∞) and · is the Euclidean norm in a Euclidean space Cn . Introduce the numbers p Cn−1 (p = 1, . . . , n − 1) and γn,0 = 1. γn,p = (n − 1)p Here p

Cn−1 =

(n − 1)! (n − p − 1)!p!

are the binomial coefficients. Evidently, for n > 2 2 = γn,p

1 (n − 2)(n − 3) . . . (n − p) p−1 (n − 1) p! p!

(p = 1, 2, . . . , n − 1).

For an (n × n)-matrix B, denote jn (B) =

n−1

γn,k N k (B),

k=0

where N(B) is the Frobenius (Hilbert–Schmidt) norm of B: N 2 (B) = Trace B ∗ B.

384

M.I. GIL’

THEOREM 3.1. Let A be an (n × n)-matrix. Then for any µ ∈ σ (A) there is an integer m n, such that either µ = amm , or n−1 γn,j γn,k N k (V )N j (W ) 1. |µ − amm |k+j j,k=1

(3.1)

Proof. Due to Lemma 17.1.2 [4], for any nilpotent matrix V˜ V˜ j γn,j N j (V˜ ).

(3.2)

Clearly, (D − λI )−1 V is the nilpotent operator. Due to (3.2) ((D − λI )−1 V )j γn,j N j ((D − λI )−1 V ) γn,j (D − λI )−1 j N j (V ) γn,j N j (V˜ )ρ −j (λ, D). Similarly, ((D − λI )−1 W )j γn,j N j (W )ρ −j (λ, D). Now the required result follows from Theorem 2.1.

✷

COROLLARY 3.2. Let zn (A) be the unique nonnegative root of the algebraic equation n−1

γn,k γn,j z2(n−1)−j −k N j (V )N k (W ) = z2(n−1) .

(3.4)

j,k=1

Then for any eigenvalue µ of A there is an integer m n, such that |µ − amm | zn (A) ψn (A),

(3.5)

where

j −1 j N j −k (V )N k (W )γn,k γn,j −k . ψn (A) = 2 max j n

k=1

In particular, the spectral radius of A subordinates the inequalities rs (A) max |akk | + zn (A) max |akk | + ψn (A). k=1,...,n

k=1,...,n

Indeed, Equation (3.4) is equivalent to n−1 γn,j γn,k N k (V )N j (W ) = 1. zj +k j,k=1

(3.6)

385


The required result is now due to Corollary 2.3 and inequality (2.10). Note that inequalities (3.6) improve the Frobenius estimate rs (A) max

j =1,...,n

n

|aj k |,

k=1

cf. [9], Section 2.2.1, if n

zn (A) < max

j =1,...,n

|aj k |

or

ψn (A) < sup

n

j =1,...,n k=1, k=j

k=1, k=j

|aj k |.

Put n

Pj =

|aj k |

(j = 1, . . . , n).

k=1, k=j

As is well known, the spectrum of A lies in the union of the Cassini ovals {µ ∈ C : |(aii − µ)(ajj − µ)| Pj Pi }

(i = j ; i, j = 1, . . . , n)

([9], Ch. 3, Section 2.4.2). Let A be upper-triangular: aj k = 0 (j > k). Then Theorem 3.1 gives the exact result σ (A) = nj=1 ajj . At the same time, if n > 2, the Cassini ovals give us the greater set. So Theorem 3.1, improves the mentioned result for matrices, which are ‘near’ to the triangular ones. 4. Matrices with Hilbert–Schmidt Off-Diagonals Again, let N(·) be the Hilbert–Schmidt norm. Assume that N 2 (V ) =

∞ ∞

|aj k |2 < ∞,

j =−∞ k=j +1

N 2 (W ) =

j −1 ∞

|aj k |2 < ∞. (4.1)

j =−∞ k=−∞

That is, V and W are Hilbert–Schmidt operators (HSO). LEMMA 4.1. Under condition (4.1), let µ ∈ σ (A). Then either µ ∈ σ (D) or ∞

N k (V )N j (W ) √ 1. ρ j +k (µ, D) j !k! j,k=1

(4.2)

Proof. Due to Lemma 2.2.1 ([3]), j

V0

N j (V0 ) √ j!

(4.3)

386

M.I. GIL’

for any quasinilpotent HSO V0 . But under (4.1), the operator (D − λI )−1 V is a quasinilpotent HSO for any regular point λ of D. Thus N j ((D − λI )−1 V ) √ j! N j (V ) (D − λI )−1 j N j (V ) √ j √ . j! ρ (λ, D) j !

((D − λI )−1 V )j

Similarly, ((D − λI )−1 W )j

N j (W ) √ . ρ j (λ, D) j !

Now the required result follows from Theorem 2.1.

(4.4) ✷

Furthermore, according to relations (4.3), (4.4), Corollary 2.3 and inequality (2.10) yield: THEOREM 4.2. Under condition (4.1), let V = 0, W = 0. Then any µ ∈ σ (A) satisfies the inequality ρ(µ, D) zH (A), where zH (A) is the unique positive root of the equation ∞ N k (V )N j (W ) = 1, √ zk+j k!j ! j,k=1

(4.5)

i.e., σ (A) lies in the closure of the union of the discs {z ∈ C : |z − akk | zH (A)}

(k = 1, 2, . . .).

Besides, zH (A) ψH (A), where j −1 N j −k (V )N k (W ) j . ψH (A) = 2 sup √ k!(j − k)! j =1,2,... k=1 Under (4.1), Theorem 4.1 gives rs (A) sup |akk | + zH (A) sup |akk | + ψH (A). k=1,2,...

(4.6)

k=1,2,...

Let sup

∞

j =1,2,... k=1

|aj k | < ∞.

(4.7)

Then the following well-known estimate is valid: rs (A) sup

∞

j =1,2,... k=1

|aj k |

(4.8)

387


([7], Theorem 13.2). So under (4.7), inequalities (4.6) improve (4.8), provided that zH (A) < sup

∞

j =1,2,... k=1, k=j

|aj k |

or

ψH (A) sup

∞

j =1,2,... k=1, k=j

|aj k |.

5. Matrices with Neumann–Schatten Off-Diagonals In this section it is assumed that V and W belong to the Neumann–Schatten ideal C2r for some integer r > 1, i.e., Nr (V ) ≡ Trace[(V ∗ )r V r ]1/2r < ∞ and Nr (W ) ≡ Trace[(W ∗ )r W r ]1/2r < ∞.

(5.1)

So N1 (.) = N(.) is the Hilbert–Schmidt norm. LEMMA 5.1. Under condition (5.1), let µ ∈ σ (A). Then either µ ∈ σ (D), or ∞ r−1

i+rj

(W )Nrm+rk (V ) √ 1. m+i+r(j +k) (µ, D) j !k! ρ i,m=0 j,k=1 Nr

(5.2)

/ σ (D) Proof. Due to Corollary 2.3.3 of [3], the operator (D − λI )−1 V with λ ∈ is quasinilpotent and, clearly, it is in C2r . Hence, it follows that ((D − λI )−1 V )r is a HSO. So according to (4.3) ((D − λI )−1 V )rk

N rk ((D − λI )−1 V ) N1k ([(D − λI )−1 V ]r ) r . √ √ k! k!

Consequently, ((D − λI )−1 V )m+rk Nrm+rk ((D − λI )−1 V ) √ k! N m+rk (V ) (D − λI )−1 m+rk N m+rk (V ) √ √ k! ρ m+rk (λ, D) k!

(5.3)

for m < r, k = 1, 2, . . . . Similarly, ((D − λI )−1 W )m+rk

N m+rk (W ) √ . ρ m+rk (λ, D) k!

Now the required result is due to Theorem 2.1. Furthermore, according to relations (5.3), (5.4) Corollary 2.3 we get

(5.4) ✷

388

M.I. GIL’

THEOREM 5.2. Under condition (5.1), let V = 0, W = 0. Then any µ ∈ σ (A) satisfies the inequality, ρ(µ, D) zr (A), where zr (A) is the unique positive root of the equation r−1 ∞ N rk+i (V )N rj +m (W ) = 1. √ zi+m+r(k+j ) k!j ! i,m=0 j,k=1

That is, σ (A) lies in the closure of the union of the discs {z ∈ C : |z − akk | zr (A)} (k = 1, 2, . . .). Clearly, Lemma 2.4 gives us an estimate for zr (A) and the latter theorem gives the estimate for the spectral radius, which is similar to (4.6). 6. The Nonlinear Spectrum To apply our results to unbounded operators, in this section we are going consider the nonlinear spectrum. Let now the diagonal operator depend on a complex parameter λ: D = D(λ) and ajj = aj (λ) where aj (λ) (j = 1, 2, . . .) are entire functions. Put A(λ) = D(λ) + W + V , where W and V are constant upper and lower quasinilpotent matrices defined as in (1.2), again. So in a given orthogonal normed basis {ek } (A(λ)ej , ej ) = aj (λ),

(A(λ)ej , ek ) = aj k = const

(j = k).

We will say that λ is a regular value of A(·) if A(λ) has the inverse bounded operator. The complement of the set of all regular values to the closed complex plane is the spectrum of A(·) and is denoted by σ˜ (A(·)). It is simple to see that the spectrum σ˜ (D(·)) of the diagonal matrix D(·) coincides with the closure of the set of all the roots of aj (λ), j = 1, 2, . . . . For a regular λ of D(·), D −1 (λ) = 1/ρ0 (λ), where ρ0 (λ) ≡

inf |ak (λ)|.

k=1,2,...

Assume that for a regular λ of D(·) and any natural k, [D −1 (λ)V ]k ck ρ0−k (λ),

[D −1 (λ)W ]k dk ρ0−k (λ).

(6.1)

LEMMA 6.1. Under conditions (6.1) and (2.2), let µ ∈ σ˜ (A(·)). Then, either µ ∈ σ˜ (D(·)), or ∞

ck dj k+j j,k=1 ρ0 (µ)

1.

The proof of this lemma is exactly the same as the proof of Theorem 2.1.

(6.2)

389


THEOREM 6.2. Under conditions (6.1), let V = 0, W = 0 and z(A) be the unique positive root of Equation (2.6). Then for any µ ∈ σ˜ (A(·)), the inequality ρ0 (µ) z(A) is true. Proof. The required result is due to comparison of (6.2) with (2.6). ✷ Furthermore, Lemma 6.1 and inequality (3.2) yield the following corollary: COROLLARY 6.3. Let W and V be an HSO. Then for any µ ∈ σ˜ (A(·)), either there is an integer m, such that am (µ) = 0, or ∞ N k (V )N j (W ) 1. √ j +k ρ (µ) j !k! j,k=1 0

Moreover, Theorem 6.2 implies COROLLARY 6.4. Let W = 0 and V = 0 be an HSO. Then ρ0 (µ) zH (A) ψH (A) for any µ ∈ σ˜ (A(·)). We remind that zH (A) and ψH (A) are defined in Section 4. In the case (5.1) we also can easily derive results similar to Corollaries 6.3 and 6.4. 7. Spectrum Localization of Unbounded Operators Again, let {ek }∞ k=1 be an orthogonal normed basis in H . Introduce a normal unbounded operator S by Sh =

∞

λk (S)(h, ek )ek

(h ∈ Dom(S))

(7.1)

k=1

with the eigenvalues λk (S) and a domain Dom(S), assuming that S has the compact inverse operator S −1 =

∞

λ−1 k (S)(·, ek )ek .

k=1

Let Q be a linear generally unbounded operator in H , represented by a matrix (qj k )∞ j,k=1 ;

qj k = (Qej , ek )

with the zero diagonal (Qej , ej ) = qjj = 0 (j = 1, 2, . . .). It is assumed that QS −1 is a compact operator.

(7.2)

390

M.I. GIL’

In the sequel we investigate the operator T ≡ S + Q. Define operators V and W by (V ek , ej ) = qj k λ−1 k (S) if j < k,

and

(V ek , ej ) = 0 if j k;

(W ek , ej ) = qj k λ−1 k (S)

and

(W ek , ej ) = 0 if j k.

if j > k

(7.3)

Due to (7.2), operators V and W defined by (7.3) are compact. Denote D(λ) = diag[a1 (λ), a2 (λ), . . .]

with aj (λ) = 1 − λλ−1 j (S).

(7.4)

Obviously, T − λI = S + Q − λI = (I + QS −1 − S −1 λ)S = [D(λ) + W + V ]S.

(7.5)

Put A(λ) = I + QS −1 − S −1 λ = D(λ) + W + V . So under the consideration ρ0 (λ) = ρS (λ) where ρS (λ) ≡

inf |1 − λλ−1 k (S)|.

k=1,2,...

Now Lemma 6.1 implies: LEMMA 7.1. With notations (7.3) and (7.4), let conditions (6.1) and (2.2) be fulfilled. Then, for any µ ∈ σ (T ), either µ ∈ σ (S), or ∞

ck dj k+j j,k=1 ρS (µ)

1.

This result is due to Lemma 6.1 and relation (7.5). THEOREM 7.2. With notations (7.3) and (7.4), let conditions (6.1) and (2.2) be fulfilled. Then for any µ ∈ σ (T ), the inequalities ρS (µ) z(A) ψ(A) are true, where z(A) is the unique positive root of (2.6) and ψ(A) is defined by (2.10). This result is due to Theorem 6.2. The latter theorem and relations (4.3), (4.4) yield: COROLLARY 7.3. With notations (7.3), let W = 0, V = 0 be HSO. Then ρS (µ) zH (A) for any µ ∈ σ (T ), where zH (A) is the unique positive root of Equation (4.6). That is, σ (T ) lies in closure of the the union of the sets {µ ∈ C : |1 − µλ−1 k (S)| zH (A)}

(k = 1, 2, . . .).

Besides, according to (2.10), zH (A) ψH (A), where ψH (A) is defined by (4.7). In the case (5.1) we also can easily derive results similar to Corollary 7.3.

391


8. Examples 8.1. AN INTEGRAL OPERATOR In space H = L2 [0, 1], let us consider an operator A defined by

1

(Au)(x) = u(x) +

(0 x 1),

K(x, s)u(s) ds

(8.1)

0

where K is a complex valued Hilbert–Schmidt kernel. Take the orthonormal basis ek (x) = e2πikx

(0 x 1, k = 0, ±1, ±2, . . .).

(8.2)

Let K(x, s) =

∞

(8.3)

bj k ek (x)ej (s)

j,k=−∞

be the Fourier expansion of K with the Fourier coefficients bj k . Obviously, aj k ≡ (Aek , ej ) = bj k

(j = k),

and

ajj ≡ (Aej , ej ) = 1 + bjj .

Assume that infj |1 + bjj | > 0. According to (1.2) under consideration we have ∞ ∞

N 2 (V ) =

|bj k |2 < ∞,

N 2 (W ) =

j =−∞ k=j +1

j −1 ∞

|bj k |2 < ∞.

j =−∞ k=−∞

Now we can apply Theorem 4.2. 8.2. A NONSELFADJOINT DIFFERENTIAL OPERATOR In space H = L2 [0, 1], let us consider an operator T defined by du(x) d2 u(x) + l(x)u(x) + w(x) 2 dx dx (0 < x < 1, u ∈ Dom(T ))

(T u)(x) = −

(8.4)

with the domain Dom(T ) = {h ∈ L2 [0, 1] : h ∈ L2 [0, 1], h(0) = h(1), h (0) = h (1)}.(8.5) Here w(·), l(·) ∈ L2 [0, 1] are bounded scalar-valued functions. So the periodic boundary conditions u(0) = u(1),

u (0) = u (1)

(8.6)

392

M.I. GIL’

are imposed. With orthonormal basis (8.2), let l =

∞

l˜k ek

w=

and

k=−∞

∞

w˜ k ek

k=−∞

(w˜ k = (w, ek ), l˜k = (l, ek ))

(8.7)

be the Fourier expansions of l and w, respectively. Omitting simple calculations, we have (T ek , ej ) = iπ k w˜ j −k + l˜j −k

(k = j )

and (T ek , ek ) = π 2 k 2 + iπ k w˜ 0 + l˜0 . Take Dom(S) = Dom(T ) and define operator S by (7.1) with eigenfunctions (8.2) and λk (S) = π 2 k 2 + iπ k w˜ 0 + l˜0 assuming that infk |λk (S)| > 0. Take a matrix Q = (qj k )∞ j,k=−∞ with qj k = iπ k w˜ j −k + l˜j −k

(j = k)

and

qjj = 0.

According to (7.4), under consideration, we have N 2 (V ) =

∞ k−1

|(π 2 k 2 + iπ k w˜ 0 + l˜0 )−1 (iπ k w˜ j −k + l˜j −k )|2

k=−∞ j =−∞

=

∞ −1

|(π 2 k 2 + iπ k w˜ 0 + l˜0 )−1 (iπ k w˜ m + l˜m )|2

k=−∞ m=−∞

2

∞

|π 2 k 2 + iπ k w˜ 0 + l˜0 |−2 π |k|2

∞

|w˜ m |2 +

m=−∞

k=−∞

+2

−1

|(π 2 k 2 + iπ k w˜ 0 + l˜0 )−1 |2

k=−∞

−1

|l˜m |2 < ∞

m=−∞

since w, l ∈ L2 . Similarly, N (W ) = 2

∞ ∞

|(π 2 k 2 + iπ k w˜ 0 + l˜0 )−1 (iπ k w˜ j −k + l˜j −k )|2 < ∞.

k=−∞ j =k+1

So V and W are HSO. Now we can apply Corollary 7.3.

393


8.3. AN INTEGRO - DIFFERENTIAL OPERATOR In space H = L2 [0, 1] consider the operator d2 u(x) + w(x)u(x) + dx 2

1 + K(x, s)u(s) ds (u ∈ Dom(T ), 0 < x < 1)

(T u)(x) = −

(8.8)

0

with the domain defined by (8.5). So the periodic boundary conditions (8.6) hold. Again, K is a Hilbert–Schmidt kernel and w is a bounded scalar-valued function. Take the orthonormal basis (8.2). Let (8.3) and (8.7) be the Fourier expansions of K and of w, respectively. Obviously, (T ek , ej ) = w˜ j −k + bj k

(j = k)

and

(T ek , ek ) = π 2 k 2 + w˜ 0 + bkk .

Define S by (7.1) with eigenfunctions (8.2) and λk (S) = π 2 k 2 + w˜ 0 + bkk assuming that λk (S) = 0 for any integer k. Take a matrix Q = (qj k )∞ j,k=−∞ with qj k = w˜ j −k + bj k

(j = k)

and

qjj = 0.

According to (7.4) N (V ) = 2

k−1 ∞

|(π 2 k 2 + w˜ 0 + bkk )−1 (w˜ j −k + bj k )|2 < ∞.

k=−∞ j =−∞

Similarly, N (W ) = 2

∞ ∞

|(π 2 k 2 + w˜ 0 + bkk )−1 (w˜ j −k + bj k )|2 < ∞.

k=−∞ j =k+1

Thus, V and W are Hilbert–Schmidt operators. Now we can apply Corollary 7.3. References 1. 2. 3. 4. 5.

Edmunds, D. E. and Evans, W. D.: Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1990. Egorov, Y. and Kondratiev, V.: Spectral Theory of Elliptic Operators, Birkhäuser-Verlag, Basel, 1996. Gil’, M. I.: Norm Estimations for Operator-Valued Functions and Applications, Marcel Dekker, New York, 1995. Gil’, M. I.: Stability of Finite and Infinite Dimensional Systems, Kluwer Acad. Publ., Dordrecht, 1998. Horn, R. A. and Johnson, Ch. R.: Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991.

394 6. 7. 8. 9. 10. 11.

M.I. GIL’

König, H.: Eigenvalue Distribution of Compact Operators, Birkhäuser-Verlag, Basel, 1986. Krasnosel’skii, M. A., Lifshits, J. and Sobolev, A.: Positive Linear Systems. The Method of Positive Operators, Heldermann-Verlag, Berlin, 1989. Locker, J.: Spectral Theory of Nonselfadjoint Two Point Differential Operators, Math. Surveys Monogr. 73, Amer. Math. Soc., Providence, 1999. Marcus, M. and Minc, H.: A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, 1964. Pietsch, A.: Eigenvalues and s-Numbers, Cambridge Univ. Press, Cambridge, 1987. Prössdorf, S.: Linear Integral Equations, Itogi Nauki i Tekhniki 27, VINITI, Moscow, 1998, (Russian).

Mathematical Physics, Analysis and Geometry 4: 395–396, 2001.

395

Contents of Volume 4 Volume 4

No. 1

2001

VLADIMIR VASILCHUK / On the Law of Multiplication of Random Matrices

1–36

A. BEN AMOR and PH. BLANCHARD / Smoothing Properties of the Heat Semigroups Associated to Hamiltonians Describing Point Interactions in One and Two Dimensions

37–49

RANIS N. IBRAGIMOV / On the Tidal Motion Around the Earth Complicated by the Circular Geometry of the Ocean’s Shape Without Coriolis Forces

51–63

TAMARA GRAVA / From the Solution of the Tsarev System to the Solution of the Whitham Equations

65–96

Volume 4

No. 2

2001

ABEL KLEIN and ANDREW KOINES / A General Framework for Localization of Classical Waves: I. Inhomogeneous Media and Defect Eigenmodes

97–130

ATTILIO MEUCCI / Toda Equations, bi-Hamiltonian Systems, and Compatible Lie Algebroids 131–146 SERGEI KUKSIN and ARMEN SHIRIKYAN / Ergodicity for the Randomly Forced 2D Navier–Stokes Equations 147–195 Volume 4

No. 3

2001

NAKAO HAYASHI and PAVEL NAUMKIN / On the Modified Korteweg– De Vries Equation 197–227 R. DEL RIO and B. GRÉBERT / Inverse Spectral Results for AKNS Systems with Partial Information on the Potentials 229–244 DAVIDE GUZZETTI / Inverse Problem and Monodromy Data for ThreeDimensional Frobenius Manifolds 245–291

396 Volume 4

CONTENTS OF VOLUME 4

No. 4

2001

DAVIDE GUZZETTI / On the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painlevé VI Equation 293–377 M.I. GIL’ / Spectrum Localization of Infinite Matrices

379–394

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Mathematical Physics, Analysis and Geometry - Volume 2

Analysis and mathematical physics

Analysis and Mathematical Physics

Differential geometry, analysis, and physics

Reviews in Mathematical Physics - Volume 4

Stochastic analysis and mathematical physics

Trends in differential geometry, complex analysis and mathematical physics

Global analysis in mathematical physics

Operator Theory, Analysis and Mathematical Physics

Operator Theory, Analysis and Mathematical Physics

Analysis and Mathematical Physics (Trends in Mathematics)

P-adic analysis and mathematical physics

Operator Theory, Analysis and Mathematical Physics

Quantum Physics (Berkeley Physics Course, Volume 4)

Algebraic Geometry and Analysis Geometry

Global analysis: Differential forms in analysis, geometry, and physics

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Geometry, Topology and Physics

Geometry and quantum physics

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Mathematical Physics, Analysis and Geometry - Volume 4

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Mathematical Physics, Analysis and Geometry - Volume 3

Mathematical Physics, Analysis and Geometry - Volume 12

Mathematical Physics, Analysis and Geometry - Volume 13

Mathematical Physics, Analysis and Geometry - Volume 1

Mathematical Physics, Analysis and Geometry - Volume 5

Mathematical Physics, Analysis and Geometry - Volume 9

Mathematical Physics, Analysis and Geometry - Volume 11

Mathematical Physics, Analysis and Geometry - Volume 10

Mathematical Physics, Analysis and Geometry - Volume 2

Analysis and mathematical physics

Analysis and Mathematical Physics

Differential geometry, analysis, and physics

Reviews in Mathematical Physics - Volume 4

Stochastic analysis and mathematical physics

Trends in differential geometry, complex analysis and mathematical physics

Global analysis in mathematical physics

Operator Theory, Analysis and Mathematical Physics

Operator Theory, Analysis and Mathematical Physics

Analysis and Mathematical Physics (Trends in Mathematics)

P-adic analysis and mathematical physics

Operator Theory, Analysis and Mathematical Physics

Quantum Physics (Berkeley Physics Course, Volume 4)

Algebraic Geometry and Analysis Geometry

Global analysis: Differential forms in analysis, geometry, and physics

Geometry and Physics

Geometry, Topology and Physics

Geometry and quantum physics

Differential geometry and physics

Geometry and Physics

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