This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
, we obtain
Vf E .9'(X).
Now 9'(X) is dense in .9''(X). Therefore S = c e C. This proves the lemma.
Some other subgroups of G(l) will be needed in proving that the map G(l) - Sp(l) is an epimorphism. They are
i. the finite group generated by the Fourier transforms in one of the coordinates (some base of the vector space X having been fixed); ii. the group consisting of automorphisms of .9''(X) of the form
f -. e°Qf, where Q is a real quadratic form mapping X -. R; iii. the group consisting of automorphisms of .9''(X) of the form
f' -* f, where f (x) =
det T f'(Tx), T an automorphism of X.
Each of these groups has a restriction to .9'(X) that gives a group of automorphisms of .9'(X) and a restriction to .*'(X) that gives a group of unitary (that is, isometric and invertible) transformations of .i*'(X). The following definition uses these properties. Definition 1.2.
Let A be the collection of elements A each consisting of
1°) a quadratic form X Q+ X
R, whose value at (x, x') e X Q+ X is
A (x, x') = Z - f(x)dIx.
The equivalence is deduced from the Fourier inversion formula; the lemma follows.
To compute compositions of the SA, we will find an explicit expression for SA(e°`° ), where (p' is a second-degree polynomial. This is made possible by the following definition.
Choose linear coordinates in X such that d'x = dx' A ... A dx' and choose the dual coordinates in X*. The following Definition 2.3.
notions are independent of this choice. Let cp be a real function, twice differentiable: cp : X -+ R.
1,§1,2
11
Hess,,((p) denotes the hessian of cp, the determinant of its second derivatives.
Alternatively this is the determinant of the quadratic form X-3
Inertz((p) denotes the index of inertia of this form. It is defined") when 0. Clearly
Hess((p)
Inert(-q') = I - Inert(q), arg Hess((p) = n Inert(g) mod 2n. This formula makes possible the definition arg Hess((p) = n Inert(q).
(2.1)
Thus, for example, [Hess (9)]112 =
(2.2)
IHess((p)I112itnert(N).
If op is a real quadratic form,
rp: X a x i-- I, where R = 1R: X - X*, then Hess((p) and Inert((p) will be denoted Hess(R) and Inert(R). Hess(R) is
the determinant of the symmetric matrix R. Inert(R) is the number of negative eigenvalues of R. Clearly
Inert(R) = Inert(R-1),
[Hess(R)]1J2[Hess(-R-1)]112 = it.
(2.3)
Let 9' be a real second-degree polynomial. Let A e A be such that Hessx.(cp'(x') + A(x, x')) A 0. Denote by 9(x) the critical value of the polynomial LEMMA 2.2.
Xax't--* A (x,x') + V (x'); rp is a second-degree polynomial. We have SA(e'11)
= A(A)[Hessx.(cp' +
Remark 2.1.
A)]-1l2e"°.
(2.4)
This lemma assumes v/i > 0. Up to this point, it was
sufficient to assume v/i real and nonzero.
Proof. We know that 'It is the number of negative eigenvalues of the linear symmetric operator dx F--. dcp_
12
I,§1,2
exp [ - 2] dx =
2n.
J
L
Therefore if c e C and I arg p I< n1/2, x JexP[-1(X
Iarg f
a-
1a I\ (v'
x) ev
vex '
1a1
P + vaxJ aQ (v, x)
- [eXp
\
v
\Ox, ap )] a
since, by Taylor's formula, for every polynomial P: function f : X
P (p
- C,
+ v ax)f (x)
Y P.
P
P(P) (vex) f (x)
X/
(v, p, x),
C and every
I,§1,3
22
_
[exp!(. ap)] [P(P)f (x)]
The bijection a- r-+ a+ can then be defined by the relation a+
_ exp-1
(v, x, p)
[
v
a
a\
ax' ap
)]a-(v, p, x)
This is what the lemma asserts.
Let a be a differential operator that can be expressed as in (3.1) and (3.2). It is defined by the polynomial a° in (1/v, x, p) that satisfies (3.4). We say that a is the differential operator associated to the Definition 3.1.
polynomial a°.
Theorem 3.1 will describe the transform SaS-1 of a by Sc Sp2(1); Lemma 1.1 has already dealt with the case in which a° is linear in (x, p). The proof of this theorem will use the following properties.
If a and b are the operators associated to the polynomials a° and b°, then the operator
LEMMA 3.2.
c=ab is associated to the polynomial co, where
c°(v, x, p) _ }[exp 2v \ay' ap) [a°(v, x, P)b°(v, y, q)]
Proof.
2v
Cax' aqM (3.5)
Y='-
e-n If b°(v, x, p) only depends on p, then the polynomial co associated
to c = ab is
)][a4 (v, x, P)b°(P)] c°(v, x, P) _ [exp - 2v ax' ap {[exp _
{[exp
2v 1
(ax a
p + aq)]
[a+(v, x,
P)b°(q)]} 9-P
a
2v ax ap)]
[a°(v, x,
P)b°(q)}q=r
Similarly, if h°(v, x, p) only depends on x, then the polynomial associated
to c = ab is
23
I,§1,3
c°(v, x, P) =
{[ex-(- , a aY
P
)] [a°(v, x, P)b°(Y)]Lx
.
Thus if b+(x,p) = b'(x)b"(p), then the polynomial associated to c = ab is c°(v, x, P) = [exp - 2v ax
aq)] {Lexp 2v (P' a-)]
[a°(v, x, p)b'(y)] 1)y=x
-
{[exp
2v
ax' aq)
[a°(v, x,
2v
(ay aq)
+ 2v
(ay' P)]
}y_}4=p
This is (3.5) since, by (3.4),
[exp
- 2v (ax' 4)] [b
b°(Y, q)
This implies lemma 3.2, which has the following obvious consequence: LEMMA 3.3.
The operator
c = 2(ab + ba)
is associated to the polynomial
x, p) _ cosh
1
o
a
[2V(8Y'2
1
2v
o
a
ax
aq)]
[a°(v, x, P)b°(v, y, q)]lq=p y=z
If b is linear in (y, q), then
(,)]2[ao(v,x,p)bo(v,y,q)] Oy, ap -
= 0,
from which follows
cosh[
] a°b° = a°b°
therefore we have the following lemma. LEMMA 3.4.
(ab + ba).
If b is linear in (x, p), then the operator associated to a° b° is
I,§1,3
24
This lemma enables us to prove the following theorem. THEOREM 3.1.
The transform SaS-1 of a by S is the differential operator
associated to the polynomial a° o s -I [S E Sp2(l); s is the image of S in Sp(l)].
Proof. Let b be a differential operator associated to a polynomial b° that is linear or affine in (x, p); lemma 1.1 shows that theorem 3.1 holds
for b. To prove the theorem by induction on the degree of a° in (x, p), it suffices to prove that, if the theorem holds for a°, then it holds for a°b°. Since the theorem holds for a° and b°, the operators associated to the polynomials
a°b° and (a°b°) o s-1 = (a° o s-1)(b° o s-1) are, respectively, by lemma 3.4,
4(ab + ba); 2(SaS-1SbS-1 + SbS-1SaS-1) = 4S(ab + ba)S-1. The theorem thus holds for a°b°, which completes the proof.
We supplement this by a theorem about adjoint operators. Definition 3.2.
(f 19) =
Recall that.,Y(X) has a scalar product:
jf(x)d1x
bf, 9 E Af (X),
x
where g(x) is the complex conjugate of g(x). Two differential operators a and b are said to be adjoint if (af I 9) = (f I b9)
df, 9 E -*'(X).
(3.6)
Two differential operators a and b associated to two polynomials a° and b° are adjoint if and only if THEOREM 3.2.
b°(v,x, p) = a°(v,x, p) Proof. b
VvCiR, xcX, pEX*.
It is clear that (3.6) is equivalent to
(v, p, x) = a+(v, x, p),
that is to say, since v is pure imaginary, to
(3.7)
25
I,§1,3-1,§2,0
\l -2v (ax,
P/
)Jb°(v, x, P)
r
1
exp -2v (-a
)]QVx, p),
a ax, a
P
and hence to (3.7).
Theorems 3.1 and 3.2 obviously have the following corollary. COROLLARY 3.1.
If a* is the adjoint of a, then VS E Sp2(l), Sa*S-1 is the
adjoint of SaS -1. Theorem 3.2 clearly has the following corollary, which will be important later. COROLLARY 3.2.
The operator a associated to a polynomial a° is self-
adjoint if and only if the polynomial a° is real valued Vv e iR, x e X, p e X*.
§2. Maslov Indices; Indices of Inertia; Lagrangian Manifolds and Their Orientations 0. Introduction
Historical account. Following V. C. Buslaev [3], [11], §1 has defined a Maslov index mod4 on Sp2(l) by (2.15) and has connected it by (2.20) to an index of inertia that is a function of a pair of elements of Sp(l). On the other hand, V. 1. Arnold [1], [11] defined another Maslov index
on the covering space of the lagrangian grassmannian A(l) of Z(l); this index is connected to the preceding one and to a second index of inertia that is a function of a triple of points of A(l). J. M. Souriau [16] has given a variant of the definition of the Maslov index that is considered in this section.
Chapter I, §3. and chapter II use these two Maslov indices and a third index of inertia, which is a function of an element of Sp(l) and a Summary.
point of A(l). We review and modify the various definitions of these indices (Arnold's,
section 5; Maslov's, section 6; Buslaev's, section 7) so as to clarify their
properties (sections 4-8). In §3 those properties that will be used in chapter II are set forth. First of all we must recall and supplement the topological properties of Sp(l) and A(l) (theorem 3). To study these properties we follow Arnold in employing a hermitian structure on Z(l) (sections 1 and 2).
I,§2,1
26
1. Choice of Hermitian Structures on Z(1)
Let (. I ) be the scalar product defining a hermitian structure on Z(1); clearly
Im(zIz')= -Im(z'Iz) is a symplectic structure on Z(1). Now in §1,1, a symplectic structure [ , - ] was defined on Z(1). Restriction to X defines a homeomorphism between the set of hermitian structures ( I -) on Z(1) such that LEMMA I.I.
ix = X*,
Im(z I z') = [z, z'],
(1.1)
and the set of euclidean structures on X. Proof. (i) The restriction to X of a hermitian structure on Z(1) satisfying (1.1) is euclidean since
[x, X] = 0
Vx, X' E X.
Observe that, by (1.1),
(z I z') = [iz, z] + i[z, z'], and in particular
(XIX')= [ix, X]
Vx, X, C- X.
Hence
ix =
lax z E X.
2
Ox a
(ii) A given (- I ) on X defines
by (1.4), the restriction of i to X,
i,: X -> X*; the restriction of i to X i2: X* -+ X,
because i2 = -il' since i2
= -1;
hence the automorphism i of Z(1),
27
1,§2,1-1,§2,2
(1.5)
i(x, P) = (i2P, ilx); finally, by (1.2), the hermitian structure on Z(1).
The restriction to X of hermitian structures on Z(1) satisfying (1.1) is thus an injective mapping of the set of such structures into the set of euclidean structures on X. (iii) It is bijective. Indeed, the automorphism i of Z(1) defined by the given (
I
i2 = -1,
) on X, that is, by (1.5), satisfies
[iz, z'] = [iz', z],
since `i1 = i1, `i2 = i2; the function ( is clearly linear in z e C', and
I
), which (1.2) defines on Z(1),
satisfies (z', z) = (z', z), hence is sesquilinear, satisfies Ix + iyI2 = IxI2 + IyJ2, and indeed defines a hermitian structure.
Lemma 1.1 has as the following corollary.
The set of hermitian structures on Z(1) satisfying (1.1) is an open convex cone. It is therefore connected. LEMMA 1.2.
Remark 1. We choose arbitrarily one of these hermitian structures on Z(1), which we shall use to define topological notions (the Maslov indices). By the preceding lemma, these notions will not depend on this choice.
2. The Lagrangian Grassmannian A(1) of Z(1) Definition 2.1 A subspace of Z(1) is called isotropic when the restriction of [ , ] to this subspace is identically zero, that is, by (1.1), when the res-
triction of the hermitian structure on Z(1) is a euclidean structure on this subspace. Every orthonormal frame of an isotropic subspace of dimension k is thus composed of vectors orthogonal in Z(1); hence k < 1.
Definition 2.2. The isotropic subspaces of maximal dimension I are called lagrangian subspaces; the collection of lagrangian subspaces A(1) is called the lagrangian grassmannian:
XandX*EA(1).
28
I,§2,2
Let A E A(l) and let r be an orthonormal frame of A. It is a frame of Z(1): the elements of Z(1) (respectively A) are linear combinations with complex (respectively real) coefficients of the vectors that make up r. Let U(1) denote the group of unitary automorphisms u of Z(l) (that is, uu* = e, where u* = 'u, and e is the identity). By (1.1), U(1) C Sp(1).
Further let A' E A(l) and let r' be an orthonormal frame of A'. There is a unique element u in U(l) such that
r = ur' from which follows A = ua.'.
The group U(1) thus acts transitively on A(l). The same holds a fortiori for Sp(l); whence 1°) of the lemma below, where St(l) [respectively, 0(l)] denotes the stabilizer of X* in Sp(l) [respectively, U(1)], that is, the subgroup of s such that sX * = X *. Now 0(l) is clearly the orthogonal group. Lemma 2.3 characterizes St(l); part 2 shows why the stabilizer of X* interests us more than that of X. LEMMA 2.1.
1 ° We have
A(l) = Sp(l)/St(l) = U(1)/0(1).
(2.1)
2°) Let W(l) be the set of symmetric elements w in U(1), that is, the set of elements w such that 'w = w; thus w c- W(1) means w = 'w = w-'. The diagram U(l) a
u-
U'u = w E W(l)
U(1)/0(1) = A(l) a2 = UX* defines a natural homeomorphism
Then z c- a. is equivalent to z + w(A)z = 0.
I,§2,2
29
Let
z = x + iy, where x and y c X. Assume 10 sp(w(A),
where sp(w) is the spectrum of w, a 0-chain of the unit circle S'. Then
z e 2 is equivalent toy = i
e + w(A)x,
e - w(it)
(2.5)
where i(e + w(A)]/[e - w(.)] is a real symmetric matrix (that is, equal to its transpose). 3°) dim(.1 n A') is the multiplicity of 1 in sp(w(,)w-1(.1')). Remark 2.1. Part 3 is preparation for the topological definition of the Maslov index (section 5).
The proof of lemma 2.1 is based on the following lemma. Writing u E U(l) in terms of its eigenvectors and eigenvalues, the proof of lemma 2.2 is clear.
1°) Let u c- U(l). A necessary and sufficient condition for u e W(l) is that all of its eigenvectors can be chosen to be real. 2°) Every surjective mapping
LEMMA 2.2.
F : S1 - S' (S' is the unit circle in C) defines a surjective mapping W(l) a w i-+ F(w) E W(l).
The diagram (2.2) defines a mapping (2.3) since, if u and uv c- U(l) have the same image in A(l) = U(l)/O(l), then v c- 0(l), and so Proof of lemma 2.1,2°).
uv `(uv) = uv `v `u = U 'U.
By lemma 2.2,2°), given w c- W(l), there exists some u c- W(l) such that w = u2. Then w = u 'u, and so the map (2.3) is surjective. Since a. = uX *, where u c- U(l), the condition z c- ) means u-'z c- X*, or
Re(u-lz) = 0, or u-1z + u-12 = 0, or z + w2 = 0. The map (2.3) is therefore injective.
I,§2,2
30
Proof of lemma 2.1,3°).
Let w = w(,1), w' = w(A'). Then A n A' is given
by the equations
z+w2=0,
z+w'2=0;
that is, r) A':
w-'z =w'-'z,
z=
WE.
Let T be the analytic subspace of Z(l) given by the equation
T: w-'z = w'-'z. Then dim, T = k, where k is the multiplicity of 1 in sp(ww'-'). The equation
of An A' in Tjs
z+wz=0. By lemma 2.2,2°), there exists a u c- W(l) such that
-w=U2 =u-'u. Thus the equation of A n A' in T may be written
uz=iii. The isomorphism T=)
therefore maps A n A' onto the real part Rk of Ck, and so
dim, A n A' = k. The stabilizer St(l) of X * in Sp(l) has the following properties: 1°) The elements s of St(l) are characterized as follows:
LEMMA 2.3.
s(x', P) = (x, P) is equivalent to
x = slx',
P = 'Si'(P' + s2x'),
(2.6)
where sl is an arbitrary automorphism of X and s2 = 's2 is an arbitrary symmetric morphism X -+ X*. 2°) An element s of St(l) is the projection of two elements S of Sp2(l) defined by
31
I,§2,2-I,§2,3
(Sf)(W ) _ ,/d et s1 1
[e("12)Cx',s=x'>f(x')]z
(2.7)
=sI,X
Remark 2.2. We denote by St2(1) the subgroup of Sp2(1) whose projection onto Sp(1) is St(1). By Remark 2.5 in §1, St2(1) is the set of S E Sp2(1) that
act pointwise on .°(X): the value of Sf at a point x of X depends only on the behavior of fat a point x' of X (in fact on the value of f at x'). The elements of the stabilizer of X* in the group of autornorphisms of the vector space Z(1) are the mappings (x', p') i-- (x, p) Proof of 1 °).
defined by
x = six',
p = s*(p + s2x'),
where sl and s* are automorphisms of X and X* and s2 is a morphism X -a X*. These elements belong to Sp(1) when t -1 s = S1
t S2 = S2.
Proof of 2°). Formula (2.7) defines an automorphism S of Y'(X) that belongs to Sp2(1) by Remark 2.4 in §1. Clearly
x-(Sf) = S[f -s1x'],
I
ax(Sf) =
S[tsl1 (-'V ax'
+ S2x'J1 f].
Hence, for any a in .sd (§1,1),
la
a°(X,- )(Sf) vax
= Sao(s1x', ts11
la v8x
+ S2x'
f,
that is, by (2.6),
S -1 aS is associated to a° o s,
and so s is the natural image in Sp(1) of ± S E Sp2(1). 3. The Covering Groups of Sp(1) and the Covering Spaces of A(1)
The properties of these covering groups and spaces (2) follow from properties of n1 [Sp(1)] and n1 [A(1)], which are obtained by studying n1 [U(1)]. Here ik denotes the kth homotopy group, (see Steenrod [17]; we note that N. Steenrod uses the expression symplectic groups in a different sense than we do.) 2See Steenrod [17], 1.6, 14.1.
I,§2,3
32
1°) The inclusion O(l) c St(l) induces an isomorphism
LEMMA 3.1.
rzk[O(1)]
' nk[St(l)]
t1k E N.
2°) The inclusion U(l) c Sp(l) induces an isomorphism lrk[U(1)]
Vk E N.
it,k[Sp(l)]
3°) The morphism nl [U(1)]
3Y
1 --r
27ri
d(det u) EZ det u
(3.1)
is a natural isomorphism: Z.
ir,[U(1)]
Proof of 1°). The elements s of St(l) are characterized by (2.6); those for which s2 = 0 form a subgroup GL(l) of St(l). The inclusions
O(l) c GL(l) c St(l) induce natural morphisms ltk[O(1)]
itk[GL(l)]
7rk[St(l)].
The second morphism is an isomorphism, since
St(l) = GL(l) x R", where n = 1(1 + 1)/2. It has to be shown that i is an isomorphism. Now GL(l) acts transitively on the set Q+ of positive definite quadratic forms on X, and O(l) is the stabilizer of one of them. Hence
GL(l)/O(l) = Q+, where Q+ is convex.
The exactness of the homotopy sequence of this fibration (see Steenrod [17], 17.3, 17.4) proves that i is indeed an isomorphism. Proof of 2°).
U(l) c Sp(l),
The inclusions
St(l) n U(l) = 0(1) c St(l)
define a mapping (see Steenrod [17], 17.5) of the fibration
U(l)/O(l) = A(l) into Sp(l)/St(l) = A(1); its restriction to A(l) is the identity. This mapping induces a morphism of
33
I,§2,3
the homotopy sequences of these two fibrations (see Steenrod [17], 17.3, 17.11, 17.5):
nk+I[A(l)] -4 nk[O(l)] 4 nk[U(l)] p' nk[A(l)] ... no[o(l)] l i°
1 io
Ii,
i i°
i i°
7rk+i[A(l)] 4 n,[St(l)] - nk[Sp(l)] P rzk[A(l)] ... no[St(l)] This diagram, in which the lines are exact, is thus commutative. Since the mappings io are isomorphisms, it follows that the mappings it are necessarily isomorphisms. (Steenrod [17], 25.2, proves part of this by other means.) We denote by det (that is, determinant) the epimorphism Proof of 3°).
U(l) 3 u E- det u c S'
(3.2)
and by SU(l) its kernel. Since u c SU(l) when det u = 1, we have
U(l)/SU(l) = S'; (3.2) is the natural projection of U(l) onto S'. The homotopy sequence of this fibration contains the following, which is thus exact:
it1[SU(l)] -4 ni[U(l)] -°' rrl[S1] * no[SU(l)];
(3.3)
here p is induced by the morphism (3.2). Since SU(l) is connected, iro[SU(l)] is trivial. Let us compute n1[SU(l)]. SU(l) acts transitively on the sphere 521-1:Izl = 1.
The stabilizer of the vector (1, 0, ... , 0) in C' is SU(l - 1); thus SU(l)/SU(l - 1) = S21-1
The homotopy sequence of this fibration contains the following, which is thus exact: it2 [S21-1]
nl [SU(l - 1)] '' 7r1 [SU(l)]
_v
nl [Su i]
where 7x1[521-'] and n2[S21-'] are trivial for l 3 2 (see Steenrod [17], 21.2). Thus i' is an isomorphism. Now 7r, [SU(1)] is trivial, since SU(1) is trivial, and so
it1[SU(l)] is trivial.
(3.4)
I,§2,3
34
Since 1r1 [S U(1)] and no [S U(1)] are trivial in the exact sequence (3.3), p is
an isomorphism. Now 9G1[S1]3FH-
I frd- EZ
is an isomorphism. The composition of p, which is induced by (3.2), with this isomorphism is an isomorphism n1 [U(1)] -+ Z, which clearly is defined by (3.1). 1 °) The composition of the natural isomorphism n [A(1)] n1 [W(1)] [cf. (2.3)] and the morphism LEMMA 3.2.
1
nl[j'1'(l)]-3
7ri
f
ddetw)EZ
(3.5)
is a natural isomorphism (Arnold [1]): [A(1)] ^' Z. 2°) The fibration Sp(l)/St(l) = A(1) defines a monomorphism
p : Z '=' it1 [Sp(1)] - nl (A(1)] = Z,
(3.6)
which is multiplication by 2 on Z. Proof of 1 °).
The homeomorphism (2.3) allows us to define
det ). = det w e S 1;
(3.7)
the mapping
A(l)a2"detA c- S'
(3.8)
is clearly an epimorphism. By (2.2) we have
det A = det z u, if i = uX *. Hence for all u e U(1) det(uA) = det 2 u det A.
'(3.9)
The mapping (3.8) thus defines a fibration on which U(1) permutes the fibers. The fibration is A(1)/SA(l) = S 1,
I§23
35
where SA(1) denotes the variety in A(l) defined by the equation SA(l) : det w = 1.
The exactness of the homotopy sequence of this fibration proves the exactness of the sequence 71,
[SA(l)] -4 n, [A(l)] A n, [S1].
(3.10)
Since
p: n, [A(l)] - n, [S1] is induced by
det:W(l) - Si, p is an epimorphism. Let us compute n, [SA(l)]. SU(1) acts transitively on SA(1) and X * E SA(l). The stabilizer of X * in SU(1) is SO(1), the connected component of the identity element of O(l), and so we have the fibration SU(1)/SO(l) = SA(l).
The exactness of its homotopy sequence implies the exactness of
n, [SU(l)] 4 n, [SA(1)] 4 no[SO(l)], where n, [SU(1)] and no [SO(1)] are trivial, by (3.4) and the fact that SO(1) is connected. Thus n, [SA(1)] is trivial, and so pin (3.10) is an isomorphism Now p is induced by the mapping (3.8). Taking its composition with the isomorphism
7t, [S'] 3Fr-,
1
2ni
dzEZ z
we obtain an isomorphism 7r, [A(1)] - Z, defined by (3.5). In (3.6) the isomorphism Z ^- n, [Sp(l)] is the composition of the isomorphisms defined by parts 2 and 3 of lemma 3.1. The isomorphism n, [A(l)] Z is the composition of the isomorphism (3.5) and the one that induces the homeomorphism of A(l) and W(1). Proving part 2 of lemma 3.2 is thus the same as proving the following: Proof of 2°).
The fibration U(1)/0(1) = W(l) induces a morphism
1,§2,3
36
P: Z
i i[U(1)] - n [W(l)] "' Z
(3.11)
that is multiplication by 2 on Z. This fibration is defined by the mapping U(1) 9 u "_w = u `u, which satisfies det w = (det u)2. Since the isomorphisms entering into (3.11) are defined by (3.1) and (3.5), we have p:
Z3
1
Yd(det u)1--.
2ni
det u
1
27ci
fd(detu)'cZ. (det u)
This morphism p: Z - Z is evidently multiplication by 2. The two preceding lemmas have the following theorem as an immediate consequence. It is clearly independent of the hermitian structure on Z(1) used above.
a and Q are the generators of n,[Sp(l)] and n, [A(l)] whose natural images in Z are 1.
Definition 3.
1°) Sp(l) has a unique covering group, Spq(l), of order q (q = 1, 2, ... , .x) [namely: the number of points having the same projection onto Sp(l) is q]; a acts on Spq(I); a' does not act as the identity on Spq(1) unless r = 0 mod q. 2°) A(l) has a unique covering space, Aq(l), of order q; fi acts on A,(1); (lr l' does not act as the identity on Aq(I) unless r = 0 mod q. THEOREM 3.
3°) Spq(l) acts transitively on A2,(1): (aSq)) .2q = Sq(Q2A2q) = f32(SgA2q), where A2q E A2q(I),
Sq E Spq(1).
(3.12)
Example 3.1. A2(0 is the set of oriented (in the euclidean sense) lagrangian subspaces of Z(1); Sp(l) acts on A2(1).
Sp2(1) acts on A4(l). This result is essential for the theory of asymptotic expansions (chapter II). Example 3.2.
Notation 3. Denote by s the projection of Sc Spq(l) onto Sp(l); by A the projection of Aq c- NO onto A(l); by A2 the projection of A2q E A2,(1) onto A2(1); by e the identity element of Sp(l); and by E the identity element of Spq(1).
I,§2,3-I,§2,4
37
Let us choose an element X of A,,(1) projecting onto X * in A(l); XQ denotes its projection onto A9(1). 4. Indices of Inertia
Definition of the index of inertia Inert(,, A', ).") of a triple (A, A', A") of elements of A(1) pairwise transverse. We have
A0+A'=A'
A"=A" $+A=Z(1).
The conditions Z' E A',
Z E A,
Z" E
Z + Z' + Z" = 0
therefore define three isomorphisms
zE
1% A" 3 Z" a--I Z' E A'
whose product is the identity. By (4.1)
[z, z'] = [z', z"] = [z", Z];
(4.3)
this number is the value of a quadratic form at z e A (at z' E A' or at z" e A'). The isomorphisms (4.2) transform each one of these forms into the others. Thus they all have the same index of inertia, which will be denoted Inert(A, A',1").
LEMMA. The quadratic form
A3z -+ [z, z'] ER is nondegenerate (that is, has no zero eigenvalues). Proof. C E A,
Since
Take a second triple C' E A',
C" e A" such that C + C' + C" = 0.
A', and A" are lagrangian, the bilinear form
(Z, r) i- [Z, 4'] = [yr', Z"] = [Z", 4] = [C, Z'] = [Z ,
y r"]
y
= [S", Z]
is symmetric and, hence, is the polar form of the quadratic form Z H [Z, Z'].
1,§2,4
38
If this were degenerate, then there would exist z -A 0 such that
zeA,
[z, C] = 0 VC'eA,
that is,
zeA n A'. contrary to hypothesis.
Thus Inert(-, , ) has the following properties:
Inert(.,
A") = Inert(A', A", 1) = l - Inert(2,).", A').
(4.4)
Inert( , , ), which is defined when its arguments are pairwise transverse, is locally constant on its domain of definition. Inert(sA, sA', s2") = Inert(A, 2', A")
Vs E Sp(l).
(4.5)
Formulas (2.11) and (2.14) of 1,§1 defined Inert(s, s', s") for
s s' s" = e.
s, s', s" E Sp(l)\Esp,
(4.6)
The following relation exists between these two indices of inertia: THEOREM 4.
Under assumption (4.6),
Inert(s, s', s") = Inert(sX *, X *, s"-1 X *).
(4.7)
The condition s 0 Esp is equivalent to the following: sX* and X* are transverse. Remark 4.1.
Proof. We return to the notation of I,§1,2 (specifically that of lemma 2.3), setting
s = SA,
s = SA ,
Sri = SA,. ;
sA : (x', p') h-. (x, p) means p = Px - `Lx',
p' = Lx - Qx';
sA., : (x, p) i - (x", p") means p" = P"x" - `L"x,
The equations of the subspaces
A=sX*,
J' = X*,
s"-1X*
are then 1: p = Px,
2':x, = 0,
p
_
_Qx
p = L"x" - Q"x.
39
I,§2,4
Condition (4.1), Z" _ (x", p") E A",
Z = (x, p) E A,
Z + Z" E A',
may be written
z" = (-X, Q"x),
z = (x, Px),
from which follows
[z", z] = 0.
(4.10) (4.11)
Then
Inert(;., X, X*) = KI[a,(-1)] - KI[a*, (1)]. Remark 4.3.
Proof.
Lemma 5.1 will use this decomposition of Inert.
Let
Z = x + iy e 2,
Z' E X,
Z" E X*
be such that
z+z'+z"=0, where x, y e X. Clearly
(4.12)
I,§2,4
41
i = - x,
z" = - iy,
[z,, z"] = Im(z' I z") = - (x
v)
Let w be the natural image of A under (2.3). By (2.5)
y= i e+ wx e-w
(e is the identity).
Since w is unitary and symmetric, i(e + w)/(e - w) is real and symmetric. Hence Inert(1, X, X*) is the index of inertia of the quadratic form
ie + w
X=3xi-' -I
e-w
x ER,
that is, the number of positive eigenvalues of the real symmetric matrix i(e + w)/(e - w). Now the positive real axis is the image of the half-circle in S', where Im z < 0, under the homographic transformation S 3 (exp i0)
i--* i
1 + exp iO - E R.
(4.13)
1 - exp i0
We orient it positively: it forms a chain x in S'. Since (4.13) transforms the eigenvalues of w into those of i(e + w)/(e - w),
Inert(1, X, X*) = KI[y, sp(w)]. Let r be a chain in S' with boundary
ar = sp(w) - sp(ie). Then
Inert(;., X, X*) = KI[x, ar] _ -KI[r, OZ]
= KI[r, (-1)] - KI[r, (1)]. Let a and a* be 1-chains in S' such that a - r and a* - r are defined as follows:
a(a - r) = sp(ie) - sp(e), a - r belongs to the arc 0 E [0, n/2] in S'
a(Q* - r) = sp(ie) - sp(-e), a* - r belongs to the arc 0 E [n/2, n] in S'.
(exp i0) t;
I,§2,4-I,§2,5
42
Since
KI[ r - z, (- 1)] = KI[Q* - T, (1)] = 0, the preceding expression for Inert(A, X, X*) is equivalent to (4.12). Now or and rr* satisfy conditions (4.10) and (4.11) and are determined by them up to the addition of 1-cycles y and y* in S' such that y - y* belongs to the half-circle in S', where Im z > 0. Since y and y* are then homologous,
KI[y, (-1)] = KI[;,*, (01 Thus (4.12) holds for any pair (a, a*) satisfying (4.10) and (4.11).
5. The Maslov Index m on A(1) We are going to supplement Arnold's definition [1], but first we shall use the following preliminary definition.
Definition 5.1 of the index M on A4(1). Let (v,, v') be an element of A,,(l) x A,,(1) = A',(1), that is, a pair of elements of A.,(1). Let v and w(v) be the natural images of v., in A(1) and W(l) [cf. (2.3)], and let v' and w' = w(v') be the images of v.. By lemma 2.1, 3°), the condition that v and v' be transverse can be expressed as (1) 0 sp(ww'-');
sp(ww'-1), which we denote sp(v, v'), is a 0-chain in S' (see Lefschetz [9]). The mapping F n (vim, vim)
- sp(v,
V)
maps the arc r onto a 1-chain in S1 denoted sp(r). If
ar = 0, tip) - (µ., Nx), then
asp(r) = sp(, ti') - sp(µ, µ'). Suppose
,Z and 2' are transverse, u and µ' are transverse; then by lemma 2.1, 3°), (1) Ojasp(r)I.
43
1,§2,5
Thus KI[sp(r), (1)] is defined. It is an integer depending only on the homotopy class of r, that is, on OF. We denote it
p ., p.] = KI[sp(r), (1)] e Z.
M[A.
(5.2)
By Remark 1, M is independent of the choice of the hermitian structure on Z(l) used in its definition. Remark.
Properties of M. M is defined under hypothesis (5.1). M is locally constant on its domain of definition. M has the obvious additive property
+ M[ux,
v
,
ti's] =
vti> v ], (5.3)
which implies
Ar] = 0. M has the invariance property
M[S%c,SA':'; i
,p' ] =
(5.5)
dS E
If /3 is the generator of n, [A (1)] (cf. definition 3), then
fl`Ate; µx, µx] = M[A,;x;µ.,µ'] + r - r'
dr,r'EZ. (5.6)
Proof of (5.5).
Assuming hypothesis (5.1), s1. and sip' are transverse, and
the mapping
S r-4 M[S)., S;",; per,, ux] e Z is defined and locally constant. Therefore it is constant since Sp,c (1) is connected. Proof of (5.6).
Choose the arc r to be differentiable and such that
Clearly one can define 1 continuous functions
O:F=(v,x,vj- U;(vz,,v.)ER,
j = 1,
1,
such that sp(v, v')
(expi9,); O (/1'A, /x'A ) = O
A) mod 27r.
I,§2,5
44
Definition (5.2) of M gives
AlA.,2,]
2nfd6'
2n
jrd(0j)
ddet(ww'-1)
1
2ni J r det(ww - 1)
f d(det w)
1
2iri
r
det w
1r
d(det w')
2ni J r -d-et w'
_
-r
r
by (3.5) and definition 3. Hence (5.6) follows by the additive property (5.3) of M. In order to recover assumption (4.11) of lemma 4.2, we need the following definitions.
Definition 5.2
X., is the point of A,,,(1) that projects onto X in A(1) and
that may be joined to X* by an arc y of Ax (l) whose spectrum sp(y) belongs to the half-circle in S', where lm(z) > 0. Definition 5.3.
The Maslov index is the function m given by
m(;,, 4) = M(2,, 4; X *, X.). This allows us to formulate lemma 4.2 as follows. LEMMA 5.1.
Inert(2,
2',
For any triple 2")
= m(A.,
,)
,
A of elements of A,,,(I),
m(2., 1 ) + m(2,, 2 ,,
).
(5.7)
Proof. By (5.6), the right-hand side of (5.7) depends only on (2, A', 2"). By Remark 4.2 and the invariance property (5.5) it suffices to establish the following special case of (5.7):
Inert(2, X, X*) = m(;.,, X.) - m(1.,, X*) + m(X,,, X*).
(5.8)
Definition 5.3 of m and the additive property (5.3) of M give
X* , X.]
m(1.,,, X.) = m(2,
,
X*) - m(X., X*) = M(1.z, X*;
X*].
Definition 5.1 of these two values of M makes use of two arcs r and r* of A'(1) such that
iar = (2,w, Xx) - (X*, XJ,
OF* = (A,, X*) - (X', X*).
45
1,§2,5
We choose these arcs as cartesian products,
r*=y* x x*,,,
f=y x x,,,,,
where y and y* are then arcs of AW(1) such that
X,*
ny=gym
oy* = L -X".
We have
a(y-y*)=XX Definition 5.2 of X,,allows us to choose y and y* such that sp(y - y*) belongs to the half-circle in S', where Im(z) ? 0. Denote a = sp(y),
a* = sP(y*)
Since the homeomorphism
w(X) _ -e,
defined by (2.3) has the values
w(X*) = e,
definition 5.1 of M gives M[A
,
X*, Xj = KI[a, (-1)],
M[A, X**; X., X*] = KI[a*, (1)] The formula (5.8) to be proved is thus identical to formula (4.12), which
holds because or - a* = sp(y - y*) satisfies condition (4.11). LEMMA 5.2. We have
m(L, X.) + m(7.'', n.") = 1. Proof.
Substitute the expression (5.7) for Inert into (4.4).
The following lemma supplements lemma 5.1. LEMMA 5.3. Every function n:Oq, A) i-+ n(2q, Aq)
and A' transverse, with values in an abelian group, locally constant on its domain of definition, such that, for pairwise transverse ., A', A", defined for Aq, Aq e Aq(1),
(5.9)
1,§2,5
46
).q) + n()
,
)) = 0,
(5.10)
is identically zero. Proof. By (5.10), n is constant in a neighborhood of any pair (:tq, 2q), transverse or not. Therefore n is constant on Aq (l), which is connected. By (5.10), its value is 0.
We have proved the following theorem. THEOREM 5.
1°) The Maslov index (definitions 5.1-5.3) is the only function
M ( ; ., m:(A ),2x)F-'m().
)EZ
defined on pairs of transverse elements of A. (1) that is locally constant on its
domain and makes the following decomposition of the index of inertia possible:
Inert(A, )',.2.") = m(A,,, 2x) - m(2,°, Ate) + m(A', A
(5.7)
).
2°) Taking into account definition 5.2, this function has the following properties:
m(da A') + m(2', A,°) = 1;
(5.9)
m(s), SA') = m(Am, Ax)
VS E Sp. (1);
m(l'2,,, /I''A') = m(J.., %;°) + r - r'
m(X*,, Xj = 0,
Vr, r' E Z;
Xm) = 1.
(5.11) (5.12) (5.13)
Proof of 1 °). Lemmas 5.1 and 5.3.
Proof of (5.9).
Lemma 5.2.
Proof of (5.11) and (5.12). Proof of (5.13).
Definition 5.3 of m, (5.5), and (5.6).
Definition 5.3 of m and (5.4); (5.9).
Formula (5.7) clearly implies the following: If A, A', are four pairwise transverse elements of A(l), then Remark 5.1.
2"'
Inert(A, 2', 2") - Inert(., A', 2) + Inert(., %", ti") - Inert(,', A", A"') = 0. Remark 5.2.
M[2 x.,
A
We call any transverse pair (p., µ,) E AI(1) such that
; p, , µz] = m(1.
,
A')
VA, A' E Ax(1)
47
I,§2,5-1,§2,6
a basis. For example, (X *, X_) is a basis. By the additive property (5.3) of M, the condition that (pm, i') be a basis can be expressed as
m(pm, t4) = 0. 6. The Jump of the Maslov Index m(),,,, , d' ) at a Point (2, A'), Where
dim 2n2'= 1 Maslov [11] defined his index, up to an additive constant, by the expression of its jump across the hypersurface YA2 in A'(1), which is the set of pairs (L , ).) of nontransverse elements of A.(1). Theorem 6 will make the expression for this jump explicit.
First of all let us establish some properties of U(1). By y we denote a differentiable arc of U(1): 7:1- 1, 119 0 F- U (O) E U (I),
ue = du 00.
Let exp(i/(0)) be a simple eigenvalue of u(O) and let z(O) be a corresponding eigenvector. Then LEMMA 6.1.
IIZ(e)I
% = (u_1(o)uZ(o)Iz(o)).
Recall that if U is a Lie group with generic element u, infinitesimal transformations Xk, and Maurer-Cartan forms CO, then Remark 6.1.
u-1 du (hence, in particular, u-1ue) may be written
u-1 du = Iwk(u, du)Xk; k
see E. Cartan [4]. Remark 6.2.
Since U(1) is the unitary group, (1/i)u-1(O)u9 is an arbitrary
I x l self-adjoint matrix characterizing the vector ug tangent to U(1) at u(0).
Proof. Since exp(i>!i(O)) is a simple eigenvalue, k(0) is a differentiable function of 0, and the vector z(0), which is defined up to a scalar factor, may be chosen to be a differentiable function of 6. Then differentiating the relation
u(0)z(0) = z(0)exp(ii(0))
I,§2,6
48
and multiplying on the left by (1/i)u-'(0), we obtain
lu
luez
+ l zo = -u- ' zo exp(io) + z(6)Oe t
i
Taking the scalar product with z eliminates ze and gives (6.1) because (u-ize exp(ii) I z) = (Z e I exp(- i/i) uz) _ (z e l z) since u is unitary.
Denote by Eu the set of u e U(1) such that (1) e sp(u); recall that (1) denotes the point of S' c C with coordinate 1. The following lemma has the sole purpose of interpreting the assumption of lemma 6.3 geometrically.
If u(o) E Eu, (1) is a simple value of sp u(o), and z(o) is a corresponding eigenvector, then the condition that uo define a direction tangent to Eu at u(o) takes the form LEMMA 6.2.
(u-'ouozozo)) 1
= 0.
(6.2)
Proof. u(o) is a regular point of E,,,. By lemma 6.1, every direction tangent to Eu at u(o) satisfies (6.2). The hyperplane of directions satisfying (6.2) thus contains the tangent plane to Eu at u(o); but Eu is a hypersurface in U(l).
If the arc y in U(1) intersects E. only at the point u(o), if the eigenvalue I of u(o) is simple, and if the corresponding eigenvector z(o) LEMMA 6.3.
satisfies
(U'O)UZOZO))
0
(i.e., if y is not tangent to Eu), then
KI[sp y, (1)] = sign( U-'(o)uoz(o) I I z(o) Proof.
.
Let exp(iir(O)) be the eigenvalue of u(O) near 1. Evidently
KI[spy,(1)] = sign/e(o) if Le(o) j4 0, so (6.3) follows from (6.1).
(6.3)
49
I,§2,6
Notation. EA3
EA2 denotes the set of nontransverse (1t, X):
AZ(1).
F denotes a differentiable arc in A2(1): [-1, 1[3 0 i-- (,(6), A'(B)) e A2(1).
w(9) and w'(B) denote the natural images of ,(O) and )'(O) E A(1) in W(1): see (2.3). LEMMA 6.4.
If the arc F in A2(1) intersects EA2 only at the point (L(o),
A'(o)) and if
dim ),(o) n ).'(o) = 1,
z(o) E ).(o) n ti (o),
(w'-'(o)z(o)lz(o)) 9 I
nw''-' (o)z(o) z(o)
then
KI[sp T', (1)] = sign
WOW-1(o)z(o) I l
z(o))
f
_ (ww'-'o)zo)Izo))}. 1
(6.4)
Proof. The image of the arc r E A2(1) in U(1) is the arc y : [-1, 11 -3 B
u(O) = w(O)w' 1(B).
By definition,
KI[spF, (1)] = KI[spy, (1)]. By lemma 2.1,3°), intersects Ev only at the point A = o, and 1 is a simple value of sp u(o). By lemma 2.1,2°), z(o) + w(o)z(o) = o,
z(o) + w'(o)z(o) = o,
so z(o) - u(o)z(o).
Now u-1ue
= u-1wew-lu - wew'-1;
I,§2,6
50
thus, because u is unitary, (u-'(o)uez(o)I z(o)) = (wew-'(o)z(o)j z(o)) - (wBw'-'(o)z(o)Iz(o))
Hence, by lemma 6.3,
KI[spy,(1)] = sign{(WW
'(o)z(o) I z(o))
- (ww'1(o)z(o) Iz(o)l,
and (6.4) follows. LEMMA 6.5.
Let
0 Hz(0) be differentiable mappings of [-1, 11 into A(l) and Z(l) such that Z(O) E 2(B).
Then Iwew-1zIz)
( Proof.
= [ze,z(0)]
By lemma 2.1,2°),
z+wa=o, so
zo + wee + wee = o. Hence (ze I z) -
(wow-1 z I z)
+ (wee I z) = 0,
where (wze I z) = (Ze I
w-1z)
_ -(291 Z) _ -(Ze II Z).
From (1.1), (6.5) follows.
The left-hand side of (6.4) can be expressed in terms of the Maslov index by (5.2) and definition 5.3; the right-hand side of (6.4) can be expressed in terms of the symplectic form [ , ] by lemma 6.5. Hence the following theorem, which will allow us to establish theorem 3.2 of §3.
51
I,§2,6-I,§2,7
THEOREM 6.
Let
[-1, 1]30(Z,z')eZ2(l) be two differentiable mappings such that
zEA,
z'E)
for each 0;
z = z' : 0 for 0 = 0;
:i and 2' are transverse for 0 : 0;
the function
[-1, 1] -3 0H[z,z']ER vanishes only at 0 = 0 and vanishes only to first order. Then there exists a constant c e Z such that C
for [z, z'] < 0
1 +cfor[z,z']>0. 7. The Maslov Index on Spm(1); the Mixed Inertia Let
S, S', S" E Sp.(l)\Esp" be such that
SS'S" = E (the identity element); let s, s', s" be their projections onto Sp(l). Formula (4.7) of theorem 4.1, which relates the definitions of the inertia on A(l) and Sp(l), and formula (5.7) of theorem 5, which relates the inertia on A(l) to the Maslov index, yield, by the invariance property of the Maslov index (5.11),
Inert(s, s', s") = m(SX*, X*) - m(S'-1X*, X*) + m(S"X*, X*). We rewrite this formula as (7.3), by virtue of the following definition:
Definition 7.1 of the Maslov index on Sp,(l). If S e SpJ)\Esp. we define
m(S) = m(SX*, X*).
(7.2)
This definition makes sense by Remark 4.1: S Esp. is equivalent to the condition that SX* and X* be transverse. Let us supplement this formula with a lemma analogous to lemma 5.3:
1,§2,7
52
LEMMA 7.
Every function
n:Si-+n(S) defined for S e Spq(l)\Espg,
with values ip an abelian group, locally constant on its domain of definition,
such that
n(S) - n(S'-1) + n(S") = 0 when SS'S" = E, is identically zero.
This lemma, combined with theorem 5 and (3.12) gives the following theorem. THEOREM 7.1.
1°) The Maslov index defined by (7.2) is the only function
Z
m:
that is locally constant on its domain and that makes the following decomposition of the index of inertia possible: under hypothesis (7.1),
Inert(s, s', s") = m(S) - m(S'-1) + m(S").
(7.3)
2°) This function has the following properties:
M(S) + m(S- `) = 1,
(7.4)
m(a'S) = m(S) + 2r,
(7.5)
where a is the generator of 7r, [Sp(l)] (definition 3). Definition 7.2 of the mixed inertia. Let
s e Sp(l)\Esp; A, 2' a A(l), transverse to X* and such that 2 = s2'.
(7.6)
The mixed inertia is defined by the formula
Inert(s, 1, A') = Inert(sX*, X*, A) = Inert(X*, s-1X*, 1'),
(7.7)
the last two terms being equal because of the invariance (4.5) of the inertia. The following theorem is evident. THEOREM 7.2.
Under assumption (7.6),
I,§2,7-I,§2,8
53
Inert(s-', A', A) = I - Inert(s, A, i.'), Inert(s, A, A') = m(S) - m(A.z X*00 ) + m(A', X*o)
(7.8)
(7.9)
if A,, = SA',, and if s is the projection of S E Sp,,(1).
Now we express the mixed inertia in terms of the inertia of a quadratic form, as we did for Inert(s, s', s") (§l,definition 2.4) and Inert(., A', A") (§2,4). This result will be used in section 10. Under assumption (7.6), we have s = sA(§1,1) and, by (2.5), the equation of A' is p' = cp;,,(x'), where cp' is a quadratic form on X. Then
THEOREM 7.3.
Inert(s, A, A') = Inert(A(o, ) + (p'( )). Proof.
(7.10)
By (7.7) and the definition of Inert(., A', A") (section 4),
Inert(s, A, A') = Inert(X*, s-'X*, 1.')
is the inertia of the quadratic form x' i-- [z, z']
for z = (x, p), i = (x', p'),
(X + X', p + p')
S(X', P) E X*,
(X, P) E X*,
Relations (7.11) may be rewritten
x=o,
p' =-A(o,x'),
p+p'=cp.,,(x+x'),
whence
x = o,
p = Ax.(o, x') +
This implies
[z, z'] = = cp'(x') - 1 < p', x'> + const.
(9.4)
If s = sA 0 Esp(,), then p = Ax, p' _ -Ax, by §1,(1.11), and so
tp(x) = cp'(x') + A(x, x') + const, (9.5)
p = (px = Ax, LEMMA 9.
p' = (px, = -Ax..
Let SA e Sp(l)\Esp; let z' _ (x', p') e V'\E,,, be such that
sAZ' = z = (x, p) e V\E,,, where V = sAV'. Evidently these relations define a diffeomorphism x' -+ x of the local coordinates of V and V'. Its jacobian takes the form d'x _ Hessx.[A(x, x') + cp'(x')]
d'x'
A2(A)
(9.6)
(In the calculation of Hess,,, x and x' are viewed as independent.) Proof.
By (9.5), the diffeomorphism x' f-- x satisfies
cp', (x') + As (x, x') = 0. Then (9.6) follows, because, from §1,1,
A2(A) = det(-Ax:x,,).
10. q-Orientation (q = 1 , 2, 3, ... , co) The notions defined in this section enable us to supply a complement to formula (9.6). This is theorem 10, which will be crucial in the sequel. By a q-orientation of A E A(l) we mean a choice of a it2q E A2q(l) with natural projection A. A q-oriented lagrangian manifold, denoted Vq, is a lagrangian manifold V together with a continuous mapping V -3 Z H A2q(Z) E A2q(1)
57
I,§2,10
that, when composed with the natural mapping A2q(1) - A(1),
gives the mapping V -3 z f--' A(z) e A(1), where ).(z) is the tangent 1-plane to V at z.
Each element of Spq(1) maps a q-oriented lagrangian manifold Vq into another. A q-orientation of V is characterized by the values taken by the locally constant function V\Ey
zf
m(Xzq, 2q) E Z2,-
By (5.12), a change in this q-orientation is equivalent to the addition of a constant E Z2q to the function m.
The statement of theorem 10 will be simplified by the following definition. Definition 10. The argument of d'x at a point of Vq, where the tangent 1-plane is A2q e A2q(l), is defined by the formula arg d`x = rcm(XZq, ).Zq) mod 2q7[.
(10.1)
For example, by (5.13),
argd'x = 0 on X2y.
(10.2)
Let SA E Spq(1)\Espq
(definition 8),
and let Vq be a q-oriented lagrangian manifold in Z(1); denote Vq = SA Vq .
Let ) 2q and 2q be the tangent planes to V9 and Vq at z' and z = sAZ'. In
formula (8.5), taking S = SA and s = SA, we have, by (8.6) and §1, definition 1.2,
m(S) = m(A) =
2
n
arg(t (A)).
By theorem 7.3 and equation (9.3) for ).'we obtain that Inert(s, inertia of the symmetric matrix
is the
I,§2,10-I,§3,1
58
z
x,; x [A(x, x') + T'(x')] By §1,definition 2.3 of arg Hess, this is
1 arg Hessx, [A(-x, x') + p'(x')]. n
Thus, by (8.4), formula (8.5) may be written m(Xzq, 2q) - m(XZq, A'2q) _
argHessx.[A + lp]
- 1 arg A2 (A) mod 2q, whence, by definition 10, we have the following theorem. THEOREM 10.
If Vq and Vy are two q-oriented lagrangian manifolds such
that Vq = SAV4, where SA e Spq(l)\Espq,
then not only are the two terms of (9.6) equal, but so are their arguments mod 2gir.
It is the special case q = 2 of this theorem that will be used (see §3, corollary 3).
§3. Symplectic Spaces 0. Introduction
Symplectic geometry makes it possible to state the preceding results in the following form, which will be used in chapter II. Z(l) has been provided with a symplectic structure, and moreover with a particular frame consisting of a pair (X, X*) of transverse lagrangian I-planes. It is important to state conclusions that are independent of this choice of a particular frame. 1. Symplectic Space Z A symplectic space Z consists of R21 together with a symplectic form, that is, a form
59
I,§3,1
x Rz1a(z,z')"[z,z']ER that is bilinear, alternating, real valued, and nondegenerate. We then have
[z, z'] = - [z', z]; [z, z'] = 0 for a given z and all z' R21 only if z = 0. Z(l) is provided with the symplectic structure defined in §1,1; the isomorphism of the symplectic structures of Z and Z(l) is obvious and has the following consequences.
The subspaces of Z on which [ , ] vanishes identically are called isotropic ; their dimension is 0,
c
c a constant c- Z. Considering (3.14), (3.10) follows.
4. q-Symplectic Geometries
If R is a q-frame, clearly the group
R-' Spq(l)R = Spq(Z) acts on Z and its lagrangian manifolds while preserving their q-orientations; this group is independent of R by (3.1). F. Klein defined a geometry by specifying a manifold and a Lie group acting on that manifold. Each of the groups Spq(Z), where q c (1, 2, ... , cc }, thus defines a geometry on Z which it is convenient to call the q-symplectic geometry.
Conclusion
This chapter in §1 defined a unitary representation of the group Sp2(!) in the Hilbert space K(X), where dim X = 1. As a result of §2, in §3 the study of this group was subsituted by that of the isomorphic group Sp2(Z) that defines the 2-symplectic geometry of Z, where dim Z = 21. Earlier §2
66
I,Conclusion
and §3 detailed the properties of the index of inertia and the Maslov index, already introduced in §1. The interest of these various properties is to make possible the definition and study of a new structure, lagrangian analysis, which is the subject of the next chapter.
I I Lagrangian Functions; Lagrangian Differential Operators Introduction Summary.
In Chapter I we studied only differential operators with
polynomial coefficients and functions defined on all of R' = X. The aim of chapter II is to extend those results. In §2, we consider a symplectic space Z of dimension 21 provided with a 2-symplectic geometry. We define
lagrangian functions and lagrangian distributions, on which Sp2(Z) acts locally;
lagrangian operators, more general than differential operators, which are transformed by Sp2(Z) like differential operators with polynomial coefficients (see I,§l,theorem 3.1).
Each lagrangian function U is defined on a lagrangian manifold V in Z; in each 2-frame R, U has an expression UR, a function with formal values defined on V\ER. In each frame R, a lagrangian operator has an expression aR
=
+ V, X,1Ia-lJ1
aR
V ax
that is a formal differential operator of order < oc, acting locally on the UR.
A change of frame
S = RR"' e Sp2(l) (see 1,§3,3)
is a local operator that transforms UR, into UR =
aR' into aR = SQR'S-1,
and hence transforms aR, UR' into aR UR = S(aR. UR.).
It follows that lagrangian operators act on lagrangian functions and on lagrangian distributions.
All the expressions aR of a single lagrangian operator a are readily obtained from a single formal function a° defined on Z. In geometry, a change of frame leaves invariant (or changes linearly)
the value of a scalar (or vector) function at a point; but an essential
68
II,Introduction-11,§1, 1
characteristic of lagrangian analysis is the following: at each point z of V, the group Sp2(l) of changes of frame acts on the germs of expressions UR of a lagrangian function U and not on the values at z of these expressions. U. = SUR, may have singularities on ER, but has none on ER.\ER, n ER, whereas the singularities of UR, are on ER.. Thus the singularities of the expressions UR of U are not singularities of U; they may be described as apparent singularities. Their nature can be made explicit by making use of the Maslov index.
Historical account. V. P. Maslov [10] elucidated this essential characteristic of lagrangian analysis, even though he only studied the projections of lagrangian functions onto X without defining either lagrangian oper-
ators or lagrangian functions explicitly. He only used a subgroup of Sp 2(1), depending on a choice of coordinates of X : the subgroup generated
by Fourier transforms in a single coordinate.
§1. Formal Analysis 0. Summary In section 1 we define and study asymptotic equivalence classes of functions.
In section 2 we define formal functions; formal functions defined on X with compact support are asymptotic equivalence classes. Then we may integrate them (section 3) and transform them by Sp2(l) and by differential operators, whose definition can be generalized (sections 4 and 6). We deduce (sections 4 and 6) that Sp2(l) and formal differential operators act locally on formal functions defined on lagrangian manifolds V or their covering spaces V; these functions may be compactly supported or not. In section 5 we study their scalar product. Formal functions on V, which are no longer asymptotic equivalence classes, enable us to define lagrangian functions in §2. 1. The Algebra ' (X) of Asymptotic Equivalence Classes The algebra -4(X) Let I be the purely imaginary half-line
I = ill, 301EC. I(X) denotes the algebra of mappings
f:I x X-3 (v,x)F-- f(v,x)EC
69
II,§1,1
all of whose x-derivatives are continuous in (v, x) and satisfy 81
x9
Sup
8x J
(v,x)EI X X
dq,rEN`;
f(v, x) < oc
Sp2(l) and the differential operators with polynomial coefficients in (1/v, x),
a=
a+
(, x, -1-alOx/I=a_v, 1
v
vOX
V
x
act on V (X ).
If 1 = dim X = 0, 24(X) is denoted :#?: thus.1 is the algebra of bounded
continuous mappings I - C.
The algebra (X) Let N E R. Let -IN(X) be the set of f e .4(X) such that the mapping (v, x) - vNf(v, x)
still belongs to ,4(X). Clearly 24N(X) is an ideal in 2(X) that shrinks as is an ideal in 24(X). We define N grows; hence -4,)(X) = the algebra 16(X) as follows:
W(X) = 24(X)i2x(X)
(1.2)
l'(X) is an algebra over C; its elements are called asymptotic equivalence
classes. The condition that the elements f and f of .(X) belong to the same class is
f - f'E2N(X)
dNeR+.
Clearly, "eN(X) = -qN(X)114.(X)
is an ideal in 16(X) that shrinks as N grows:
f w,(X) = to}.
NeR,
Let f and f e :4(X) with classes f and f' e''(X ). To express the equivalent relations
f - f e 24N(X),
f-
' e'6,(X), where N 5 cc,
(1.5)
II,§1,1
70
we shall write one of the following:
f=f
modvN,
f = f' modf'E f modvN.
(1.6)
If I = dim X = 0, '(X) is denoted W. Clearly '(X) is a subalgebra of the algebra of functions X
W.
Thus an element f of '(X) has a restriction to any subset of X and a support
Supp(f)
X.
Remark 1.
Let F be an entire holomorphic function
(1.7)
F:Ci - C such that F(O) = 0;
for every fi c fi, the F(fl , ... , f,) are in the same class, denoted j). Thus d. f eW(X)(j = 1, ..., J), F(f,,...,.Tj) Ew(X). Let F be a Holder function F(.r,,...,
F : C' -p C such that F(O) = 0. We similarly define F(.Ti , ... , .fi) E W
bfi E W,
j = 1, ... , J.
For example, if f e ', then j f1, Re f, and Im f are defined. f c W is real if Imf = 0; then f+ (the positive part) and f_ are defined. f = 0 is written 0 1 and defines a partial ordering on Re W, the set of real elements of 1W: let f and E Re 16; f > 0 and g > 0 imply f + j > 0
0; f 0 (that is, -.r > 0) excludes f > 0 (that is, 0). However, f' may satisfy neither f < 0 nor f > 0.
and f g > 0; 12
0, 0 defined by
IIfll2
llf l + llg LEMMA 1.1.
I(f,g)l -
0 is
II,§1,2
0
0;
u c .f °; Notation.
the integer s (defined above) is even.
u'12 may be chosen in two ways; we take u'/2 > 0.
Proof. Let U = Z,EN(a,/V') E .F°; let s be the first r such that a, 0. If a,v-' is real valued for every r, the proof of theorem 2.2 constructs a real-valued function f c u, so that u c Re go. It follows that
Re u =
Im u = E Im a ,
Re '
rEN
rEN
V
Thus u c Re .y ° if and only if a,v-' is real for every r.
Let us study uSince as
2
mod
u = v2s
1
V2s+1
the assumption u112 c Re s is even,
° implies:
as/i' > 0, u e Re y ° (see remark 1).
For the converse, it evidently suffices to prove the following:
u1j2 e Re.y ° when a° > 0 and u e ReJF°. This is clear because the condition 2
.
a
V.
is equivalent to the condition f3 = a° and to certain conditions giving (i°/i, (for each r = 1, 2, ....) as a real linear function Of 0tr and Mr-1
(0 < t < r). Now u'12 E Re .F° implies u > 0 (see remark 1). Hence, if u e Re F
then ;v` > 0 implies u > 0. Thus u c Re _F' satisfies either u > 0 or
u,
'P (P) = i< P, MC 1P);
arg Hess (Dc is the sum of the arguments of the eigenvalues of ct explicitly, we write
.
More
where v e ill, x[ and b+ and D are two quadratic forms X -+ R, independent of v; (p+ is positive definite; we assume D is nondegenerate. The assumption that the matrix M, is diagonal is superfluous because a change of coordinates reduces two such forms to
+(x) = >x,
fi(x) _ > cix; i
i
,
where ci
0.
Let e c -9(X) be such that s(x) = 1 for x near 0. Lemma 3.2 and (3.9) give \1/2
C
2v
J(x)P(x)e v(x) &.(x)dix
= CHess
-
\ - uz (D+
[e' (era:)P(x)]s=o mod `1 ;
(3.10)
V
argHess((D - (1/v)(D.+) is the sum of the arguments of the eigenvalues
I1,§1,3
84
of b - (1/v)1+, which belong to 10, its. The argument of -v = Ivl/i is -n/2. The right-hand side is a function of 1/v, which is holomorphic for 1/v = 0. Hence, mod 1/v', it is an element of F whose limit, when 1 + vanishes, is
[Hess
mod
1
v
from the Taylor series of this function; arg Hess d) is given by I,§1, definition 2.3. Thus, mod 1/v°°, the left-hand side of (3.10) is an element of F; by lemma 3.2, its limit, when 1 + vanishes, is
f,
e(x)P(x)e"O(x)dCx.
Thus e(x)P(x)evo(x)dtx
=
(21ri
1/2
[Hess(D]-in[ecu")m'(a/Ox)P(x)]x=o modvI ;
(3.11)
IVI
lemma 3.3 follows by approximating a at the origin by its partial Taylor series P and applying lemma 3.2.
Lemma 3.3 proves part 2 of theorem 3 in the particular case where the phase cp is a quadratic form 1. The general case is deduced from this special case by applying the following lemma to the remainder of a partial Taylor series. The proof of the lemma is analogous to that of lemma 3.1. LEMMA 3.4.
0e 10, 11,
Let
xeX.
Let a: (0, X)HOC(o,x)eC
be infinitely differentiable with compact support, vanishing to order 2N when x vanishes. Let cp : (0, x) i-- cp (0, x) c- R
be infinitely differentiable and such that on Supp a, for every
0,
85
II,§1,3
(p: x r-+ 9(0, x) has only one critical point, which occurs at the origin and is nondegenerate. Then
JdOJc(Ox)evcP(0x)d1x = 0 mod
I
(3.12)
V IV,
x
o
From this lemma we deduce the following. Let [p : X - R be infinitely differentiable with only one critical point, which occurs at the origin and is nondegenerate, and critical value zero: LEMMA 3.5.
(px(0) = 0,
(p (0) = 0,
Hess (p(0)
0.
Let
cp(x) = I(x) + O(x) be its second-order Taylor expansion at the origin: 4) is a quadratic form; O vanishes to third order at the origin. Let a:X - C be infinitely differentiable, with support that is compact. Then
Y
Jx
`O'(x)a(x)evO(x)d`x mod
j=0 j! x
(3.13) v
This series converges in .`f since, by lemma 3.3, vi
f Oj(x)a(x)e''m(x) d'x = 0 mod v-('/2)-[(i+l)/2] x
where [ Proof.
.
] denotes the integer part of .
-
that is, the greatest integer
By (3.12) and Taylor's formula,
e= 2N-1 vlOJ J! v
e
vm
(1 -
+v 2N
J=1
0
e)2N-1
(2N - 1)!
O2Ne
0.
(3.14)
Lemma 3.4 gives v2N f xX
f1(1
-
0) 2N-1
o
0 mod N
(3.15)
v
since O2N vanishes to order 6N at the origin. When N tends to infinity, (3.14) and (3.15) imply (3.13).
II,§1,3-II,§1,4
86
Proof of theorem 3,2°). We may replace u by a function f given by (3.7); qp has a single critical point that is nondegenerate. We may assume that this critical point is the origin and the critical value is zero. Then by (3.13) and lemma 3.3, a(x)e,9(Y) drx_ 27ri
V2
(Hess (D)-112 1
,1 r !j ! vr-i
IvI
a
f
I
[®'(x)a(x)]jmod 1
\ax/
x=0
,
v
(3.16)
where r, j e N. Denote the exponent of v by q = r - j. Since O(x) = 0 mod Ixt3, the condition { }x=o : 0 implies 3j 2r, that is j < 2q. Let -
O*q+j[Oj(x)a(x)1JIlj Iq(o; a) _ 2 x=0 i=0 (q + j)i j z
1
i
(')
in particular,
Io((p; a) = 00). Now (3.16) can be written
ess)
= fX
gIq((p; a),
12
(FV
qeN
whence part 2 of the theorem follows. 4. Transformation of Formal Functions by Elements of Sp2(l)
Each S E Sp2(l) induces an automorphism S: ''(X) -+ W(X) (see section 1) whose restriction to
.FO(X) c '(X) (theorem 2.2) is a monomorphism:
S :.F0(X) - W(X). Theorems 4.1, 4.2, and 4.3 develop the properties of this monomorphism.
Let Z be a 2-symplectic space, V a lagrangian manifold in Z, R' a 2 -frame of Z, and THEOREM 4.1.
87
II,§1,4
S E Sp2(l);
R = SR' is then a 2 -frame of Z (I,§3,3). 1°) There exists an isomorphism, induced by S and denoted S,
u ER,, R') , o(V\ER u ER,, R),
(4.2)
characterized by the two following properties:
i. It is local. That is, if UR. E. '0(V\ER u ER', R'),
UR = SUR, E'F0(V\ER U ER', R),
then UR(v, 1), the value of UR at z e I, is a linear function of the values of UR. and its derivatives at the same point f. ii. The following diagram is commutative: y0(V\ER u ER,, R') - 'Fo(V\ER U Z`R., R) nRl
(4.2)
1-14
Fo( X)
S
) W(X )
.
(4 . 1)
It thus defines a morphism
S o IIR, = rIR o S:.FO(V\ER u ER,, R') - '0(X).
(4.3)
2°) The composition law of the morphisms (4.1) and (4.2) is that of Sp2(l). Remark 4.
The restriction of (4.1),
S:IIR. O(V\ER u ER,, R') - IIRAO(V\ER U ER,, R) - .FO(X),
(4.4)
is thus an isomorphism. While (4.2) is local, (4.4) is not local, but only pointwise: If U' E IIR,.,O(V\ER U
R'),
u = Su',
then the value u(v, x) at x e X is a linear function of the values of u' and its derivatives at the points of the finite set RXRX'x n Suppu'. The local character of the morphism (4.2) is made explicit as follows. THEOREM 4.2.
11x,e°(0" E'0(V\ER U ER., R').
UR. = rEN
Then
Let
V'
88
II,§1,4
where R = SR',
UR = SUR. E .Fo(V\LR U ER,, R),
is given at i by U
UR(V,
Z)
x3r+1/2 1
E (dtX = rEN
Vr
JR,, (' ; a) e
vroA(=)
(4 . 6)
where
x'=Rzz,
x=RXZ,
(4 . 7)
[dtx] 112 is defined in I,§3,corollary 3.
'IR.r is a linear function of the derivatives of the as (s 0; i and i are the points of Supp UR and Supp OR' projecting onto z. In (5.4), fi(i) - O(z) is one of the periods of 0: cY = 2
f [z, dz], where y is a cycle in V. r
Thus the phases of (UR I UU) are these periods c., of 0.
3°) If V and V' are transverse, then (5.5)
vy2(OR I UR) a
If V and V are transverse and intersect at a single point z that is the projection of a single point i of Supp UR and a single point i of Supp UU, then 1/2
2
li
(UR I UR)
Hess
'
3.-112P,
aR, aR
a°[0()-OV')]'
where
(p and (p' are defined near Rxz by
W(x) = (PR(RXIx n V), V(x) = (6(RX1x n Hess((p - (p) = {Hessx[(p(x) - (p'(x)]}x°R:z ;
II,§1,5
93
P, is a sesquilinear function of the values of the derivatives of OCR, . and YR,s (s < r) o f order < , 2(r - s) at 2 and z' ; the coefficients of this function depend on the behavior of V and V' at z; (5.7) PO(aR, aR) = a R,o(Z)aR.o(fl 4°) [Invariance under Sp2(1)]. Let S E Sp2(1). Under the assumption
U' c- FO(V'\ R V t'SR, R),
OR E .FO(V`tR U ±SR, R),
which is stronger than (5.1), we have (5.8)
(UR I UR) _ (SUR I SUR).
Proof of 1°), 2°), and 3°).
u and u' can be written in the form
u(v, x) = Y vv(v, x)e""AX) where vi(v, x) _ Y_ I, v;.,(); rEN '
jEJ
where vk(v, x) _
u'(v, x) = Y dk(v,
lr
vk,r(X);
rEN V
kEK
J and K are finite sets. By part 1 of theorem 3,
(u I u') _ f u(v, x)u'(v, x) d'x x
depends only on the behavior of the pairs vie"wJ, vke"c1 at the critical points of cps - wk, that is, at the points X E X such that ca,, = cpk,.
In other words, (OR I UR) depends only on the behavior of OR and UU at pairs of points (z, z') E V x V' such that
Al = Iii'
= (x, c'p
) = (x, 9"J
These pairs are the pairs of points of V x V' projecting onto a single point z of Vn V. 1°) and 2°) follow, as does 3°), by (3.2) and (3.3).
Proof of 4°). Lemma 1.1 and part 1 of theorem 4.1.
The invariance of (UR UR) under Sp2(1) stated in part 4 of theorem 5.1 raises the following problem: to give invariant expressions of the right-hand sides of (5.4) and (5.6). Theorems 5.2 and 5.3 will give such expressions mod(1/v).
94
II,§ 1, 5
Notation. define
Let , and n' be regular positive measures on V and V'; we
argil = argil' = 0. We assume_V and V' are both given 2-orientations: If z e V, x = Rxi, then d'(Rxz) = d'x '1
is a function V - R
n
vanishing on iR whose argument is given by I, §3, corollary 3: arg d'(Rxz) = rcmR(z) mod 4ic.
We define Qo : V - C by [d1(1x2)]h/2
f3 (2) =
aR,o(2), where 2 e
From formulas (4.6) and (4.8) of theorem 4.2, Qo is invariant when we replace R by SR without changing V ; $o is called the lagrangian amplitude of UR. This amplitude depends on the choice Definition 5.2.
of the measure n and the 2-orientation of V. Substituting the expression (5.9) for aR,o and ax the following.
o
into (5.4), we obtain
Suppose V = V', which implies ii = ii"; let Qo and Qo denote the lagrangian amplitudes of UR and U. Then
THEOREM 5.2.
(URIUR)= r
Qo(Z)Qo(fle"[V'(=)-PVC0]n
J ZEV Z.Z
mod 1,
(5.10)
v
where i and 2' are the points of Supp UR and Supp UU projecting onto z.
Notation. (Continued) Let 0 and 0' be the lagrangian phases of V and V'[1, §3,(1.7)]. We make the same assumptions as in part 3 of theorem
5.1. Let il, and , E A,(Z) be tangent to the 2-oriented manifolds V and V' at i and 2', respectively; let A and A' be their natural images in A(Z). Let 10 (and lo) be the measure on A (and %') that is translation invariant and equal to P1 (and rl') at i (and verse (as in part 3 of theorem 5.1); thus
V and V' are assumed trans-
II,§1,5
95
Z = A Q+ A';
110 A no is a measure on Z.
(5.11)
By (5.1) and part 1 of theorem 5.1, the two sides of (5.6) are zero unless we assume
z' E V'\R,
z E V\ER,
(5.12)
that is,
RA and RI' are transverse to X*. Substituting the expression (5.9) for aR.o and aR.o into (5.6) we obtain, under assumption (5.12), Cvl J112 (UR I UR)
[Hess( - Wr)] mod 1
112
[ d`
]112 r
U1
xz)
]1o2
xz ) (5.13)
.
V
Lemma 5 transforms this formula into the following.
If V and V' are each given a 2-orientation, are transverse, and intersect only at a point z that is the projection of a single point i of Supp UR and of a single point f of Supp UR, then THEOREM 5.3.
L VI
q Od
(UR I UR)
l2
112
mod
(5.14) V
where m is the Maslov index, defined mod 4 (I,§3,3) and rlo A 11o and d 21z are the measures on Z defined, respectively, by (5.11) and (5.15).
1°) The symplectic structure of Z defines a measure on Z that is invariant under translations and under Sp(Z):
LEMMA 5.
d21z
where
(dx A dp)1 = (-1)1(1 -1112d1x A d'p,
(5.15)
II,§1,5
96
dx n dp = > dxj n dpj ;
Rz = (x, p),
j=1
{xj} and { pj} are dual coordinates on X and X *. 2°) Under assumption (5.12), with the notation (5.9),
d'(Rxz) d'(Rxz )
Hess((p
d21z
(5.16)
no Ano
arg Hess(rp - (p')
d'(Axz) d'(Rz )I
L C
= nm(1.4i A4) mod 4n.
(5.17)
n'
n
The value of dp n dx on a pair of vectors z, z' is [z, z j
Proof of 1°).
Proof of (5.16). Since RA and RA' are lagrangian and transverse to X by (5.12), their equations have the form
RA:p = O(x),
R.Z':p = (D ,(x),
(5.18)
where 0 and 0' are two quadratic forms X -+ R. Since A and A' are transverse,
- (D')
Hess(
0.
By (5.11), each z E Z decomposes uniquely as the sum of a vector in A and a vector in A':
Rz = (x + x', is(x) + 'D
(x')) E Z(1),
where x and x' e X are unique. It follows by (5.15) that (-1)1(1-1Y2d'[x
d2'Z =
_
+ x'] A d'[''x(x) +' '.(x')]
(-1)1(1-1W2d'[x + x'] A d'[d'x(x)
(- l)t(t+1)/2 Hess((D
-
a)s(x)]
- cD')d'x A d'x'
_ (-1)tu+1U2 Hess((D - tb') d'(Rx()d'(Rx(') no A no, no
where
no
e A, ('e A'; hence (5.16) follows.
Proof of (5.17).
By definition [I,§3,corollary 3, (3.2) and (3.3)]
argd(RxZ) = nmR(z) = nm(X4, RA4) mod 41r;
I
II,§1,5-II,§1,6
97
thus
nm(A4, ,a) - arg d`(xz) + arg d`(Rxz ) n
n
= nm(14,,a) -
nm(X*a, RA4) + m(X a*, Rya)
= n Inert(X *, RA', R)) mod 4n
by I,§2,(8.2). Then, by the definition of arg Hess [I,§1,(2.1)], to prove (5.17) it suffices to show that
Inert(q) - tb') = Inert(X*, R)', R)). Proof of (5.19). RA:p = (D .,(x),
(5.19)
Let us write down the equations of R..., RA', and X*:
R)':p' _ Dx(x'),
X*:x" = 0.
The definition of Inert (I,§2,4) makes use of z' E RA',
z E R1.,
z" E X* such that z + z' + z" = 0.
Define z = (x, P),
z' = (x', P),
z , = W" P");
x"=0,
p+p'+p"=0,
hence
x+x'=0,
[z",z]=(P",x)=-(P,x>-(P',x>
_ - (cb (x), x> + (b' (x), x> = 2 b'(x) - 2 b(x). Consequently,
Inert(R), R)', X*) = Inert(O' - 0). (5.19) follows by I,§2,(4.4) and I,§1,definition 2.3, of the inertia of a form.
6. Formal Differential Operators Definition 6.1.
Let 0 be an open set in Z. Let
denote the set of
formal functions with vanishing phase
a° = 1 1r ar rEN v
(6.1)
98
II,§1,6
such that the a, : 0
C
are infinitely differentiable. We give the vector space y °(0) the topology defined by the following system of neighborhoods .N'(K, p, r, e) of the origin:
p, r e N,
K is compact in 0;
Ee
a° E A' (K, p, r, E) means that on K the derivatives of x5 (s < r) of order p have modulus <E. Let us say that a° is a polynomial if (v, z) i-- a°(v, z)
is a polynomial in 1/v and in the 21 coordinates of z. By the Weierstrass theorem the set of polynomials in.F °(Z) is dense in. °(Q). y °(S2(1)) is defined similarly by replacing 0 by a domain S2(l) of Z(1) _
XQ+X Every 1-frame R of Z (hence, a fortiori, every 2-frame R) induces an isomorphism .97'(Q) 3 a " aR e
°(SZ(1)), where Q(I) = RSZ,
defined by the formula a°(v, z) = aR(v, Rz)
Vv, z.
The formulas (3.4) I,§1, a,+, (v, x, P)
r
1
= CeXp 2v
08x' 8
8p
] a° R(v,.x, p),
(6.3)
aR (v, P, X) =L Cexp 2v - - `\8x a 8p a
I] ae(v, x, P)
define two automorphisms of y °(Z(l)) that are inverse to each other: o + aRHaR,
o aR"aR.
Definition 6.2 of formal differential operators rests on the following lemma.
II,§t,6
99
Let
LEMMA 6.1.
u = ae"°e.F(X),where a = E lrar. reN V
At a critical point x of the phase cp we have h
u(v, x) = 0 mod vlhl(z .
(v ax
Proof. It suffices to consider the case a = a°. Then at a critical point of q , h
a) u = vlh Phe"W (h is a multi-index), v where PP is a polynomial in the derivatives of a of order e 10, IhII and in the derivatives of vcp of order e 12, h] (since qpx = 0). Each monomial
in Pti is a product of derivatives whose orders sum to I hI. Hence the power of v in such a monomial is ] V
ax) `u(v,
=
e-tiaR
(V, X,
ax)U(V, X). V
Then, taking a+ (v, x, av/ax) = a- (v, av/ax, x) in the sense of section 1, we have (aRU)(v, x) = aR (V, X,
V
ax) U(v' x) = aR (v'
V ax'
X) U(V9 X).
Notation. Denote (aRU)(v, x) = aR (V, x,
v
-V, ax) u(v, x) = aR (
V aX,
X) U(V, X).
Let aR denote the unique local Definition 6.3. Let V c Q, a° E R) such that endomorphism of aRfIR = IIRaR.
(6.7)
aR, being local, extends to an endomorphism of.F(V\ER, R), which has the following properties.
11,§1,6
101
THEOREM 6. 1°) The formal differential operator aR is local (in the sense of part 1 of theorem 4.1); hence
Supp(aRUR) c Supp UR n Supp(a°). 2°) The mapping
`-V°(SZ) X .-(V\tR, R) 3 (aR, UR) r--+ aRUR E y (V\ER, R) is continuous. 3°) Each S E Sp2(1) transforms aR into an operator
aSR = SURS-
1
defined as follows: aSR(SUR) = S(aRUR) for all UR E
U 'SR, R).
aR and its transform aSR are associated to the same function a°, which satisfies
a°(v, z) = aR(v, Rz) = asR(v, SRz)
Vv, x.
(6.9)
4°) The formal differential operators aR and bR are adjoint, that is, (aRUR, UR) = (UR, bRUR)
'UR, UR,
if and only if b°(v, z) = aa(vz)z)
Vv, z.
5°) If aR and bR are two differential operators associated to elements a° and b° of 3 °(Z), then their composition cR = aR o bR is associated to the element c° of.F°(Z) defined by the formula
{[exp
c°(v, z)
=
- 2v (az' a?
)J [a°(v, z)b°(v, z')] }Z=: ;
(6.10)
there
Cazl
a
'N
x' tip'/-\ - \ex
p
(6.11)
where Rz = (x, p), Rz' = (x', p'), is clearly independent of R. Remark 6. co = a° o b° + (1/2v)(a°, b°) mod(1/v2), (,) being the Poisson bracket; see II,§3,(3.14).
II,§1,6-II,§1,7
102
Proof of 1°). The definition of aR. Proof of 2°). Use the definition of aR and the definitions of the topologies of .F°(Q) (section 6) and of R) (section 2). Proof of 3°.), 4°), and 5°).
Apply I,§1 (theorem 3.1, theorem 3.2, and
lemma 3.2) to the special case in which a° is a polynomial. This special case implies the general case, since (6.8) is continuous and the polynomials are everywhere dense in.&°(SZ).
Corollaries 3.1 (adjoint of an operator) and 3.2 (self-adjoint operator) of I,§1 clearly apply to formal differential operators. 7. Formal Distributions
In sections 1 and 2, it is not possible to replace .9(X) by .99'(X): the proof of theorem 2.2 breaks down because So'(X), unlike Y(X), does not have a countable fundamental system of neighborhoods of 0. The algebra
Definition 7.1.
R) is the set of functions
f:V\ER-.C that are infinitely differentiable and compactly supported. x = RXZ is used as a local coordinate on V\ER.
Given a compact set K of V\ER and a positive number b, for every 1-index r, we let
B(IC, {b,}) =
{f e
R)
Supp f c IC,
< b,f.
The subsets of these B(K, {b,}) are by definition the bounded sets of -9 U \tR, R). Definition 7.2.
f' :
The vector space
R) is the set of linear mappings
(V\R, R) - C
that are bounded on each bounded set of -9.
These f' are distributions; they have supports and derivatives of all orders with respect to x = RXZ ; these derivatives are distributions. We make the convention -9 c -9'; -9' is a vector space over the algebra -9.
II,§1,7
103
The value of f' E -9' at f e -9 is denoted
f'(z)f(f)d'x, where x = AXi. flP
Then
1a V ax
is defined on .9' by the requirement that it be self-adjoint:
f
[1f'()]JdI x =
IIV
f (Z)
ax f (z
ff,
d'x.
Given a bounded set B in -9(V\ER, R), we let
I FIB = Sup f7 f'(zf(Z)dix feB
< 00.
The topology of 9'(V\ER, R) is defined by the following fundamental system of neighborhoods of 0, each depending on a bounded set B of -9 and on a number e > 0: Al" (B, e) = { f' I If'IB , e}.
The f' with compact support form a subspace 2 of -9' that clearly is dense in -9'. It can be proved (as in L. Schwartz [13], chapter VI, §4, theorem IV, theorem XI and its comment) that -9 is dense in !'o, and thus R) is dense in Definition 7.3.
R).
A formal distribution UR on V consists of
i. a 2-frame R of Z, ii. a formal series
-
«r
a "),tR
reN V
where (PR is the phase of R V and a, e
Supp UR = U Supp a" c V.
R).
II,§1,7-II,§2,0
104
The set of formal distributions UR on V forms a vector space R). Its topology is defined by the following fundamental system of neighborhoods of 0, each depending on a bounded set B of R) and on two numbers c > 0, N > 0:
1''(B,s,N) _ {URIIa,IB < efor r < N}. Clearly .
-o(V\ER, R) is dense in
.
'(V\ER, R).
By (4.6), the isomorphism (4.9) defined by S = RR'-' E Sp2(l) extends to an isomorphism
S:.f '(V\ER V ER., R') - .y'(V\ER U ER., R) that is continuous and local. Similarly, by (6.6), the formal differential operator aR extends to an endomorphism
aR:R) -
R)
that is continuous and local and satisfies theorem 6. The scalar product of UR and UR, a formal function and distribution
on the same V\ER, is defined by formula (5.4) of theorem 5.1 when Supp UR n Supp UR is compact; it is invariant under Sp2(l) (part 4 of theorem 5.1) and satisfies theorem 5.2.
§2. Lagrangian Analysis 0. Summary
Synthesizing the properties of formal functions and formal differential operators from §1, we define and study lagrangian operators, lagrangian functions and distributions, and their scalar products in §2. These three notions make up the structure called lagrangian analysis. In section 1, we define lagrangian operators and study their inverses. In section 2, we define lagrangian functions on the covering space V of a lagrangian manifold V; their scalar product (' ) is defined when their supports are compact. This makes it possible to define lagrangian functions on V in section 3; their scalar product is defined when the intersection of their support is I
II,§2,0-II,§2,1
105
compact; section 3 requires the datum of a number v° e iJO, 001; it uses the "restriction" of v to the value v° in the expressions UR of a formal function U. This process of defining lagrangian functions on V has a theoretical justification, which is the possibility of defining their scalar product (I.), and a pragmatic justification, namely, the process of many "approximate calculations." An example is quantum physics, where 2ni/h plays the role of our variable v e ill, 001, h being Planck's constant; here h is taken to be infinitely small without comparing the orders of magnitude of the numerical values taken by the terms coming into play when h is given its physical value; h is given this physical value after completing the calculations in which h is assumed to be infinitely small. Another historical example is celestial mechanics; H. Poincare has elucidated how such a
process is an approximate calculation capable of predicting celestial phenomena with extreme precision using divergent series in which ultimately only the first terms have to be retained. V. P. Maslov wants to obtain "asymptotics" by the same process, that is, by making approximate calculations. In chapter III, we shall apply this process to cases in which it is certainly not an approximate calculation, and we shall recover numerical results that are in agreement with experimental results. Therefore, to make this process coherent is "a problem that poses itself and not a problem that one poses" in the sense that H. Poincare meant it. 1. Lagrangian Operators Let Z be a symplectic space and Q an open set in Z. Definition 1.1. a R= aR( v, x'
Let a° e ,y °(Q) (II,§I,defnition 6.1). Let
)
1 8 1_ _(v, 1 8 v ax J -
aR
V
X
x
be the formal differential operator associated to a° and to the 1-frame R (II,§l,definition 6.2). The lagrangian operator associated to a° is the collection a = {aR } of these formal differential operators aR (a° given; R arbitrary); aR is called the expression of a in the frame R. Theorem 2.3 will justify this terminology. We say that a is defined on Q. We define
Supp(a) = Supp(a°)
II,§2,
106
If
a(v,z) = ceC
Vv, Z'
then a is denoted
a=c. Formula (2.9) will justify the following definition. Definition 1.2.
Two lagrangian operators a and b associated to a° and
b° e F°(S2) are adjoint when b°(v, z) = a°(v, z)
Vv, z;
equivalently, by II,§1,theorem 6,4°), a and b are adjoint when aR and bR are adjoint for every R.
Notation.
The adjoint of a is denoted a*.
Formula (1.2) and the resulting formula (2.2) will justify the following definition. Definition 1.3.
The composition a° o b° of a° and b° e f °(S2) is given by
(a° o b°)(v, z) _ {(exp
where
- 2v[a_, 1
1
1
)I a°(v, z) b°(v,
Oz-
'
is defined by II,§1,(6.11).
° , [aZ 3z'
(1.1)
The composition a o b of the lagrangian operators a and b is the lagrangian operator associated to a° o b°. By part 5 of II,§1,theorem 6, (1.2)
(a o b)R = aR o bR.
Since composition of the aR is associative, composition of the a° and composition of the a are associative. It is easy to verify this directly by observing that (1.1) implies (a° o b° o c°)(v, z)
exp -
l
a
a
l
2v az' aZ' - 2v
a°(v, z)-b°(v,
a
a a a 1 LaZ' az" - 2v az ' 0z"
z")}z.z-=.
.
(1.3)
107
II,§2,1
This formula easily extends to the composition of any number of elements of Clearly
Supp(a o b) c Supp(a) n Supp(b),
(1.4)
(a° o b°)(v, z) = a°(v, z) b°(v, z) mod 1,
(1.5)
V
the right-hand side being the product in F ° of the values of a° and b°: (a° o a°)(v, z) = Fcosh 2v rLa L
z
l a°(v,
z). a°(v,
J
z')
(1.6) =-_
The set of lagrangian operators defined on S2 is thus a noncommutative algebra F °(S2) over the algebra 9' of formal numbers with vanishing phase; 3 °(S2) has an identity element. (Inverse of a lagrangian operator) The operator a has an inverse in the algebra of lagrangian operators defined on S2 if and only if THEOREM 1.1.
a°(v, z) : 0 mod
l
Vz E S2.
(1.7)
V
Recall that, by definition, a°(v, z) is a formal series: a°(v, z) = Y_
1,
,EN t'
a,(z)
(1.8)
Condition (1.7) means
a°(z) j4 0
Vz E S2.
Proof.
(1.7a)
The equivalent conditions (1.7) and (1.7a) are necessary by (1.5). Conversely, if these conditions hold, a right inverse a' of a is an operator associated to an element
a'(v, ') = Y v,a'(-) ,EN
of
°(S2) satisfying
{(exp that is.
d - 2vf ez-,c oz' )a°(v, z)a'0(v, z')}Z=Z. = 1, J 1 1
108
II,§2,1
ao(z)ao(z) = 1,/
((1.10)0
a0(z)ar(z) = Cr(z),
(1.10)r
where cr(z) is a linear combination of the Oz
)'a,(z)]
z
such that 0 < Isl = Isl = r - t - t', t' < r. These conditions determine a'. Therefore a has a right inverse and, similarly, a left inverse; these are identical since the composition of elements of F °(Q) is associative. The theorem follows. The definition of scalar products (sections 2 and 3) will use the following. Definition 1.4. A partition of unity is a collection (aj} of lagrangian operators defined on Z such that Uj Supp a3 = Z is a locally finite covering
of Z and
a, = 1 (identity operator).
(1.11)
This partition is said to be finer than an open covering Uk S2k = X of X when every Supp a3 belongs to at least one of the Stk.
(Partition of unity) 1°) There exist partitions of unity finer than a given open covering of X. 2°) These can be chosen so that the operators a3 are self-adjoint. 3°) More precisely, these partitions of unity can be chosen such that each a, has the form THEOREM 1.2.
a, = b* o b;,
Supp b; = Supp a3,
where b3 is a lagrangian operator and b# is its adjoint. Proof of 1 °) and 2°).
It suffices to prove the theorem when the operators
a; are replaced by C'-functions a;°:Z -+ R that are independent of v. It is thus a classical result.
Proof of 3°). Given a covering Uk S2k = X, we choose Cr-functions b°: Z -+ R such that
Y [b°(z)]2 = 1,
i
Suppb° c S2k for some k.
109
II,§2,1-II,§2,2
It follows by (1.6) that I
b° o bV = Y
2
where (3,:Z -+ R, flo = 1.
reNv
Let us try to find
c= o
I
Vzr Y.,
where Yr : Z - R,
yo = 1,
reN
such that
c°oc° = Yb°ob°, that is, cosh
1 2v
C
Yv az aZa, J ""
r v
zr
Pr'
This condition defines Yr as a function of Pr and of the
[()5
[J)s Y(Z)J
Y,
(z)]
such that
0 < I s l - Is l = 2(r - t - t'),
0 < t < r;
thus co exists, is unique, and has an inverse by theorem 1.1. Now
bi=b*,
c=c*,
Ybobi=coc; i
therefore the
ai = (bioc-1)*o(bioc-1) constitute a partition of unity finer than the given covering Uk S2k = X. 2. Lagrangian Functions on V
Let Z be a symplectic space provided with a 2-symplectic geometry. Let V be a lagrangian manifold in Z and let V be its universal covering space. Theorem 4.1 of §1 justifies the following definition. Definition 2.1.
A lagrangian.function 0 on V is defined by the datum of
110
II,§2,2
a formal function UR on V\ER for each 2-frame R, these formal functions satisfying
= RR'-'V,, on V\ER U ER.
UR
VR, R'.
UR is called the expression of U in the frame R. The support of U, Supp U, is the subset of V defined by the condition
Supp UR = (V\ER) n Supp U
VR.
This definition makes sense since S = RR'-' is local.
The set of lagrangian functions on V and the subset of those with compact support are vector spaces over F; they are denoted F(P) and .FF(V). Their dimension is infinite, as is proven by the following theorem. (Existence) Every UR. E F0(V\ ER., R') is the expression in R' of a lagrangian function U on V with compact support. THEOREM 2.1.
Proof.
Let R be any 2-frame. Let
S=
RR'-' E Sp2(l),
UR = SUR. ;
UR is a formal function defined on V\ER U ER, which is identically zero in a neighborhood of ER.. We extend its definition by making the following convention:
UR = 0 on ER,\R n ER.. Then OR becomes a formal function defined on V \ER ; U = { UR } is clearly a lagrangian function and Supp U = Supp UR.. II,§l,theorem 4.2 proves the following. (Structure) Let iG be the lagrangian phase of V, rl a positive regular measure on V, and x = 1 L E X, where i e V. We make the convention [rl] "2 > 0. We give V a 2-orientation; recall that [d'x] is defined by I,§3,corollary 3 using the Maslov index. THEOREM 2.2.
1112
We have URIV,
[' rEN
l3r+I/2 -d
/
1
(
yr YRr( )E
(2.1)
111
II,§2,2
on V\iR, where the YRr are infinitely differentiable functions V -+ C.
flRO is independent of R, is denoted f3o, and is called the lagrangian amplitude. Remark 2.1.
In (2.1), the exponent 3r + i cannot be decreased: see
III,§3,theorem 4.2, where the polynomial f, has degree 3r and the rational function g, poles of order 3r. II,§1,theorem 6 justifies the following definition. Let n be an open neighborhood of V. Let a be a lagrangian operator on n and U a lagrangian function on V. Let R and R' be 2-frames Definition 2.2.
and S = RR'-' E Sp2(l). Then aRUR = S(aR' UR,);
thus the formal functions aRUR form a lagrangian function, which we shall denote a V.
In other words THEOREM 2.3.
The algebra of lagrangian operators a, b defined on SZ
V
acts on the vector space f (V) [and .FO(V)] of lagrangian functions U [with compact support] defined on V; a(bU) = (a o b)U ;
(2.2)
Supp(aU) = Supp U n fl-' Supp a,
(2.3)
ft being the projection of V onto V c Z. Definition 2.3 of the scalar product (I ). Let U E .FO(V) and Let us first assume that the following condition holds:
There exists a 2-frame R such that
Supp U n ER = 0,
2.4)
Supp U' n Ex = 0'
This condition means
UR E 9'0(P \i,, R),
UR E
R).
We then define
(U U') _ (UR UR) E r6 for each R satisfying (2.4).
2.5)
II,§2,2
112
The right-hand side is an asymptotic class by II,§1,definition 5.1. It is independent of R by part 4 of II,§1,theorem 5.1.
Let U and U' no longer satisfy (2.4). Let Y ai = 1,
> aj. = 1
(2.6)
I
be two partitions of unity (definition 1.4) sufficiently fine (theorem 1.2) so that for every choice of (j, j') there correspond 2-frames Ri,i, satisfying the condition
Supp d, n ER,., = 0;
= 0,
Supp ai n ER;.;
thus (ai U I a;. U') is defined Vj, j'.
We define
(UI U') _ Y_ (aaUIaj.(Y)E`'. i.i'
The right-hand side is independent of the choice of partitions of unity (2.6); indeed, if
Ibk=1, k
>bk'=1 k'
is a second choice satisfying (2.7), then Y_ (aiUl a,, C') i1 j,
(ai OR,,
R;.r) =
(ai o bk UR;.r ai, o b,',, UR;.r)
i.f, k, k'
_ Y (bk U l bkXP), k,k'
since Ri,i. can be replaced by Rk,k' . THEOREM 2.4.
(Scalar product) 1°) The scalar product (- .) is a function I
of pairs U and U' of lagrangian functions on V and V' with compact support. It is sesquilinear over F ° and takes values in ' (asymptotic equivalence classes).
(UJU') and (U'IU) are complex conjugate and depend only on the
113
II,§2,2
behavior of U and U' at the pairs of points of V and V' projecting onto the same point of V n V.
If VnV'=0,then (UIU')=0. (aUI U') = (UJa* U') if the lagrangian operators a and a* are adjoint. (2.9)
2°) If V = V', which implies Vi = 0i', then
(UI U') E',
(2.10)
the phases of (U I U') being the periods of 0 (that is, the values of i f 7[z, dz] on the cycles y of V); Y(t')]g
flo(z)flo(z
(U I U') =
mod
(2.11) V
ZEJV z,zY
where the notation is that of theorem 2.2 and z and z are the points of
Supp U and Supp U' projecting onto z in V. We have
0 '< (U I U) E'.
(2.12)
The seminorm of U, U U)112 E
0
satisfies the triangle and Schwarz inequalities:
(Ui
(U U')I
U)112.(U'
U,)112 in W.
3°) If V and V are transverse (where V and V are given 2-orientations), then v1/2(U U') E
';
(2.13)
1riJt/2 (U I U') C2lvl
= I :Evn v,
no A no
d21z
1/2
Y
0
(z
mod 1
(2.14)
V
where
no A no and d
21z
are the measures on Z defined at the point z c- V n V'
by II,§1,(5.11) and II,§1,(5.15) respectively,
1 and 2' are the points of Supp U and Supp U' projecting onto z,
II,§2,2
114
m is the Maslov index defined mod 4 (I,§3,theorem 1),
A4 and , are the 2-oriented lagrangian planes tangent to f/ and fl' at i and i', respectively.
Proof of 1°). Use definition 2.3, part 1 of II,§1,theorem 5.1, part 4 of §1,theorem 6, and definition 1.2 of the adjoint of a lagrangian operator. Proof of (2.10).
Use §l,definition 2.3 and §1,(5.4).
Proof of (2,11). Use §1,theorem 5.2 and the fact that aU mod(1/v) is multiplication of U by the function ao. Proof of (2.12).
We use a partition of unity [part 3 of theorem 1.2]
Yb*obi = 1 sufficiently fine so that for every j there exists a frame Ri satisfying
Supp bi n IR, = 0. We have
(UI U) = >(bJUR I biUR.) > 0 (§1,definition 5.1).
Proof of the Schwarz inequality. By §l,definition 5.1, we have the triangle and Schwarz inequalities in R [§1,(5.3)] and in W (§I,remark 1); hence,
(UIU') _
(bi UR, I bi UR;)
(bi UR; bi QR ) (bi UR, I bi UR, )1'
2
(b, UR, I bi UR,
)1/2
)1112
[(bJUR,IbJURJ)] 11
[
(b, UR, I b, OR 1
J
U,)
The triangle inequality follows from the Schwarz inequality.
Proof of 3°).
Use §l,theorem 5.3.
115
II,§2,2-II,§2,3
Remark 2.2.
If V = V', (U U') is defined by the right-hand side of §1,
(5.4) under the assumption.
Supp U' n ER = 0-
Supp U n ER = 0,
If this assumption is not satisfied, the right-hand side is a divergent integral in view of theorem 2.2 (structure). Thus definition 2.3 of (- I ) gives a meaning to this divergent integral. 3. Lagrangian Functions on V
We are interested in functions on V rather than functions on V, that is, multiforms on V. The definition of lagrangian functions on V requires the datum of a number vo ; it will be convenient (see, for instance, Maslov's quantization, II,§3,6) to choose vo E i]0, oo[. The first homotopy group n, [ V ] acts on V: V is the quotient of V by
n1[V]. Clearly it,/[ V] leaves ER invariant /VR;
(YZ) = OM + cy!
(PR(YZ) = (PR(Z) + C
where c., = 1 J [z, dz]
dy e n 1 [ V];
that is,
q0 /-, = 0 - Cy!
(PR0Y-1
=(PR-CY.
Definition 3.1 of the group n, [V] of automorphisms of
OR = Y aRrr evv- E f (V\i,! R),
Let
Y E n, [ V].
(3.1)
V
Define the transform of OR by y not as UR
o 7-1 = e-vcy
y
aRr
Y
R)
V
but as yIIR = e(°-V,)C, UR o Y-1 = e-vocr Y aR.r 0 Y rEN
Clearly
evWR E
R).
(3.2)
II,§2,3
116
Y' (Y UR) = (Y"01R,
so n1 [V ] is a group of automorphisms of .y (V\ER, R). It clearly follows from definition (3.2) and the definition of the operator S E Spz(1) in II,§1 (lemma 1.1, part 1 of theorem 4.1, and theorem 4.3) that S and y commute: If
UR. E .f(V\±R. U :R, R') and R = SR', then
(3.3)
Y(SUR') = S(YUR,).
If UR are the expressions of a lagrangian function U on V, then yUR are the expressions of a lagrangian function on V; denote it by yU. Then (Y OR = yUR and n1[V] is a group of automorphisms of .F(). Clearly Vy e 7r, [ V], V O e
(V ), Va a lagrangian operator,
Supp(yU) = y Supp U;
(3.4)
y and a commute, that is, (3.5)
Y(aU) = a(yU).
A lagrangian function on V is a lagrangian function U on V having the following three properties, which are clearly pairwise equivalent and thus independent of the choice of the 2-frame R : Definition 3.2.
i. U is invariant under n1 [V], that is,
yU = U
Vy E n1 [V] (see definition 3.1);
ii. a "°`YaR,, o Y-1 is independent of y c- n1 [V] Vr E N;
iii. the function
UR =
URe(vo-v)WRI rEN
V
called the restriction of UR to vo, takes the same value at all points of V having the same projection onto V Therefore we may write: UR : V \ER - .y °.
The set of lagrangian functions on V[with compact support in V] is a Hence it is the set of points in V projecting onto a closed subset of V,
117
II,§2,3
which will be called the support of U in V and denoted Supp U. Then
Supp UR =Supp U\ER n Supp U,
t1R.
The set of lagrangian functions on V [with compact support in V] is a vector space over the algebra .F', It is denoted .y (V) [.y°(V )]. Clearly, by (3.5) the following theorem holds.
The algebra of lagrangian operators a, b defined on 0 acts on the space .y (V) of lagrangian functions on V; THEOREM 3.1.
V
Supp(aU) c Suppa n Supp U.
(a o b)U = a(bU),
The scalar product of lagrangian functions on V will be defined using the following lemma and definitions. LEMMA 3.1.
There exists a natural epimorphism
.y°(v)3CJ-+U
Y YUE.y°(V).
(3.7)
yER1[V]
Proof. Definition (3.7) makes sense: it defines U by a finite sum, since for every 2 e V, there are only finitely many y such that yz e Supp U. Clearly U is invariant under n1[V] and has compact support. Since y commutes with partitions of unity (definition 1.4, theorem 1.2), to prove (3.7) is an epimorphism it suffices to prove the following: Every U E FO (V) with support in a simply connected subset w of V is the image under (3.7) of an element U of .y°(17). This is proved as follows. The connected components of the subset of V projecting onto w are the transforms of any one of these components (b by the elements y of n1 [ V ]. Let U be the lagrangian function on V that vanishes outside th and equals U in th; then yU vanishes outside y(b and equals U = yU in y(b, hence
U = Y yU. Let us express the notion of restriction to v° more explicitly. Definition 3.3.
e°9 - e'ow
Let 5 be the ideal of F generated by the elements cp e R.
Each element of .5 may be written uniquely:
118
II,§2,3
I aJ(e`'PJ - ev0 j),
J finite,
ai E F °,
gyp; E R.
JEJ
.F is the direct sum of f and F °. Let us call the projection of F onto .F ° the restriction of F to vo: .FE) U = E c e"ip' F- 3 u ° = E a i e"O (P; E -IF' = -F/J. In particular, if U E flV), then the restriction of UR(v, 2) E to vo is the value UR (v, z) of the restriction UR of UR to vo at the point z, the projection of z onto V. Definition 3.4.
Any additive subgroup M of ' having the properties
.F c M c 16, M = 47-(complex conjugate) is called a submodule of 16. Multiplication in the algebra W makes .elf a module over the algebra .S. In particular,
ERimply ue"°E.11.
Let .' be the submodule of M whose elements are uJ(e°°i - ev00,),
J finite, u, E .i,
(p; E R.
JEJ
Let us call the quotient
..lf -+ -# / .N = ..lf °
the restriction of Al to vo; tf° is a module over the algebra F°. Since .N' = .N', conjugation in .,lf ° is induced by complex conjugation in M. Example 3.1.
If ..l' = .IF, then J(° _ -IFo
Example 3.2.
Let (1/IvI)s9,
...
denote the images of 9,
...
under the
multiplication 16 Z) J-+
1IsfE16,
sER+.
V
If .4Y = (1/Ivls).F, then tf° = (1/Ivl3),a-'°. Definition 3.5 of the scalar product ( I ). Let V and V' be two lagrangian manifolds in Z. Let.,# be a submodule of ' (definition 3.4) such that
II,§2,3
119
V U E .f°(V),
(UI U') E .,H,
U' E .F°(V')
(3.8)
For any U E .F°(V), U' E -FO (V'), there exist U E .`°(V), U' e .f° (V') such
that
U = E TO, YEidV]
U' =YEn1[V'I E y' U'
according to lemma 3.1. By definition, the scalar product of U and U' is (U I U') = (U I U')°, the restriction of (U I Cl') E .,H to v0.
(3.9)
Thus the scalar product has values in Mo. This definition makes sense by the following lemma. LEMMA 3.2.
Proof.
(UI U')° depends only on U and U'.
Let
Eb*obj=1 J
be a partition of unity (definition 1.4); we have
(UI U') _
(b;CI Ib;U');
it then suffices to prove that (bU I b;U')° depends only on
E yb;U = bjU and E y'b;U' = b;U'. YEn,[Vl
YeAdv ]
By theorem 1.2, it suffices to prove the lemma when the following two conditions hold: i. There exists a 2-frame R such that UR E 10'0(1 \'R, R), UR E
R).
ii. There exist two simply connected open sets, w in V and w' in V', that are projected injectively into X under RX and contain the supports of U
and U:
SuppUcwcV,
SuppU'cci cV.
The connected components of the subset of V projecting onto w are the open sets yw in V, where y e nl[V] (see the proof of lemma 3.1). Given x e RX Supp U, let
II,§2,3
120
z be the unique point of co such that Rxz = x, i be the unique point of to projecting onto z in V. By definition 2.3, §1,(1.9), and definition 5.1,
(fl UR)(V, X) - MR UR)(V, x)d'x.
(U I U') =
(3.10)
fX
By
definition (§1,theorem 2.1), if x E Supp(HRUR) = Rx Supp U, then
(nR UR) (v, x) = E UR(V, Y
' Z),
YE1,IV]
that is, by (3.2)
(nRUR)(v, x) = E e('0-Vk' YUR(v, Z)' YE1 11 VI
We substitute this expression for 11R UR and the analogous expression for
11RUR into (3.10). Using definition 3.4 of .N' and (3.7), we obtain the formula (U I U') =
UR(v, Z) - UR(v, Z )d'x
mod Al
(3.11)
J
under the assumptions (i) and (ii).
This formula (3.11) proves lemma 3.2 and is of use in proving the following theorem, part 1 of which is clear.
(Scalar product) 1°) Let V and V' be two lagrangian manifolds in Z. The scalar product (- I -) is a function of pairs U and U' of lagrangian functions on V and V' that is sesquilinear over F°.
THEOREM 3.2.
(U U') is defined when Supp U n Supp U' is compact. (U I U') e tl°, a module over .F' depending on V and V'. (U U') depends only on the behavior of U and U' in a neighborhood of Supp U n Supp U'.
(U I U') = 0 when Supp U n Supp U' = 0. (U I U') and (U' l U) are conjugate. (aU I U') = (U I a* U') when a and a* are adjoint lagrangian operators.
2°) Suppose V = V. Then (U I U') E .F° (algebra of formal numbers with vanishing phase):
0 0 nj=k+ Iwhen
0onW.
By part 1 of the theorem, n HR
[a(x)eP,(z)]
v. x, v
ax)
= 0 mod
(4.9)
i.
Let m(k) =
inf
(J + n j
jE{o,...,k}
if m(k) > k, we then have aR (v, x
1 (1
0 mod
V OX J
Vk+1'
But we assumed that a does not vanish identically on W. Therefore we can choose k such that
m(k) 5 k. We define m = m(k); let J be the collection of j such that
j + nj = m. By 1°), jbR (v, x, (1/v)a/ax) is mod(1/v) multiplication by the function V. Then, by (4.9), the De Paris decomposition gives aR Cv X, 1 S) [a(x)evv-(Y)] V ax -0
jY vj ib0(Z)[HR (V. X, v (x
m1 [2(x)e°'"(s)] mod
+i
139
II,§3,4-II,§3,5
where
1 <m s, it cannot be satisfied ; thus L, = 0. If r = s, it requires k = 0; (4.8) becomes III+IJOI=s
and (4.7) becomes 2-ICI
L3(X, p) _
I!
HR'z'pl (X, P),
the sum extending over the collection of 1-indices I such that III 0,
(6.2)
in chapter III because in quantum physics 2nh is chosen to be Planck's constant. Since 1/2
xR,o = flo
dx
,
where f3o 3 0,
the condition that U be lagrangian on V mod(1/v) amounts to )1,12
d`x
being defined (that is, uniform) on V. From the definitions of (d'x)112 and ,111/2 (I,§3,corollary 3 and II,§2,theorem 2.2), this condition is the following. Definition 6.2. A lagrangian manifold V satisfies Maslov's quantum condition when the function 2ni
APR
- 4mR,
where vo = i/h,
is defined mod 1 on V; this condition is independent of the choice of the frame R. Remark 6. If V is oriented, in the euclidean sense, MR is defined mod 2 on V. Then Maslov's quantum condition assigns to each period 1 f ,,[z, dz]
of 0, that is, to each period nh mod 2nh.
< p, dx> of r°R, one of the two values 0 or
If V has a 2-orientation, mR will be defined on V mod 4 and this condition requires that cpR and t be defined on V mod 2nh. This will not be the case in the applications that are given in chapter III. We have just proved the following theorem. 1 °) Let a be the operator associated to a hamiltonian H. A lagrangian solution U of the equation THEOREM 6.
I1,§3,6-II,§3,7
145
aU=0mod z V2 with lagrangian amplitude >-O can be defined mod(1/v) on a manifold V if and only if V simultaneously has the following three properties: i. V is a lagrangian manifold in the hypersurface W: H(z) = 0; ii. V has an invariant positive measure (invariant under the characteristic vector of H); iii. V satisfies Maslbv's quantum condition. 2°) The datum of V having these three properties defines U up to a constant
factor if and only if the invariant measure on V is unique, that is, if every function that is infinitely differentiable on V and constant on each of the characteristics generating V is constant on V. This theorem will be applied in III,§2. 7. Solution of Some Lagrangian Systems in One Unknown Theorems 7.1 and 7.2 supplement theorem 6. They are applied in 111,§1 and III,§3, respectively. THEOREM 7.1.
Let a(') (j = 1, . , l) be lagrangian operators associated to
l hamiltonians
H('): fl -. R (S2 an open subset of Z) that are independent and pairwise in involution; that is,
d1l"' A .. A dH(')
0, (dw)1-2 A dH(') A dH(k) = 0
Vj, k,
(7.1)
where (o = j[z, dz]. 1°) The system
a(')U=...=ac1'U=0
mod12
V
has a lagrangian solution U defined mod(1/v) on the connected manifold V if and only if V simultaneously satisfies the following two properties: i. V is a connected component of the manifold in Z given by the equations
V: H111(z) _ ... = H(1)(z) = 0, which imply that V is lagrangian.
(7.3)
II,§3,7
146
ii. V satisfies Maslov's quantum condition (definition 6.2). Then the lagrangian amplitude of U is constant when d21z
n
A dH(l
dHcl) A
(see remark 7)
(7.4)
is chosen as the measure on V. Thus, when V has been chosen, then U is defined mod(1/v) up to a constant numerical factor. 2°) The Lie derivatives £f 1 in the direction of the characteristic vectors i of the H111 commute. If V is compact, then V is a torus whose translation group is generated by the infinitesimal transformations IKj.
Recall that d21z is defined by §1,(5.15); (7.4) means that ry is the restriction to V of any form 1z on Z such that Remark 7.
dHc11 A ... A dH11) A ?1z = d21z;
rl is clearly independent of the choice of ?1z; rl is invariant under each K.
Proof of 1°). From theorem 1, the support of U belongs to one on the connected components V of the manifold given by the equations (7.3). Conversely, let V be one on these components. The rank of do) is 21
because its characteristic system is dz = 0. By theorem 2 (E. Cartan), locally there exist functions go,
. . .
, g, such that
1
= dgo + Y- gj dH c>> ; J=1
then the restriction a, of w to V satisfies dcoy = 0.
Now dim V = l; thus V is a lagrangian manifold.
Moreover this result could be deduced from (3.22) of theorem 3.1, 3°). Let R be a frame of Z; let (x, p) = Rz; the condition that H(') and H(k) be in involution is expressed all(j) OH (k)
ax'
that is,
OH (k)
3H1;i
ap/-Cax'-ap/-0'
II,§3,7
147
Y,,; H(k) = 0,
(7.5)
where $4, is the Lie derivative in the characteristic direction of H('): Kj = 8H(j) , 8p
-
OH(j) Ox
From (7.5) follows
5 K, d H(') = 0;
moreover the definition of the Lie derivative gives
d21z=0, hence, by the definition (7.4) of rj,
YKl7 =Q
VJ.
Then in view of theorem 4,2°), the condition that a lagrangian function U defined on V satisfy
a(i)U = .. = a(()U = 0
mod 12
is that its lagrangian amplitude /J satisfy YK'YO = 0
Vi;
since the H(') are independent, that is, since they satisfy (7.1)1, this condition can be expressed as /Jo is constant on V.
Then, by section 6, the condition that U be lagrangian on V mod(1/v) is equivalent to Maslov's quantum condition. Proof of 2°).
By §2,(1.1), the commutator of a(') and a(k),
ja(i), a(k)I = a(1) o a(k) - a(k) o a(1)
is the operator associated to the formal function
- 2 {sinh
2v
a] [-Oz-, az'
H(j)(z)H(k)(Z')j -Z
since .H(') and H(k) are in involution, this formal function vanishes
II,§3,7
148
mod(1/v2). It is an odd function of v, so it vanishes mod(1/v3), from which follows Jaci), a(k)]
= 0 mod 3
.
v
From theorem
4,2°),
aR) (aeYWrt) = V evWR yR2
xi (ayR 1/2) mod V2
da.
so that aci) a(k) ae°wR Re R]( )
= V12 eYwR yRlie
I -1/2) mod 1 . y /a, . XR V3
thus, by (7.6),
[yxh zJ = 0 Then, if V is compact, the infinitesimal transformations
, Yx'
generate an abelian group of homeomorphisms of V, whence 2°). We supplement theorem 7.1 as follows.
Let a(') (j = 1, , 1) be lagrangian operators that commute with each other and are equal mod(1/v2) to operators associated to l independent hamiltonians H(') [that is, satisfying (7.1)1]; these hamiltonians are THEOREM 7.2.
then in involution [that is, satisfy (7.1)2]. Let us study the problem of defining
mod(1/vr+'), on a connected manifold V, a lagrangian solution U of the system
a(1)U =
= a(')U = 0 mod
1 2,
r , 1.
(7.7)r
Assume that conditions i) and ii) of theorem 7.1 are satisfied; indeed they are necessary. Then this problem has a solution U if and only if, in addition, the following two conditions are satisfied: iii. A solution of (7.7)r_ 1 has to exist on V (we assume it is explicitly known).
iv. A function f -+ C that is defined by integration of a closed pfaffian form on V\ER, and by the condition that it have polar singularities on ER, has to
be a function V - C. (Knowledge of this function explicitly solves the problem.)
II,§3,7
149
When (i), (ii), (iii), and (iv) are satisfied, then the solution U of (7.7), is defined on V up to a factor that is a formal number with vanishing phase. Preliminary to the proof. Let R be a frame of Z; let V satisfy (i). We restate theorem4,1°),2°) as follows. Let /3: V\ER -+ C; then a(i)+(v x, 1 R
O
(x) 3(x)e'`°",
(Xx2
I
VOX/
;C'n/2(x)ev9-(s) rY
Mr (x' OX) /3(x),
V'
(7.8)
where Mi is a differential operator of order = 0.
This formula is supplemented by the following, where y is an arbitrary function V\ER -- C, and where d1dt is the derivative along the characteristic curves of W, that is, the Lie derivative PK : (x, (PR,X),
XR2
L1(x,
fl
d (Y CR 112)
[1'(x)f(x, (PR,.)] (1.7)
AX, R,x)Y(X).
Here J(x, p) is defined on W by the formula
J=z
[(g" bm)fm + bn (g", f,.) + 9"(bn, fm)] + m,n
in which
, m.n
gncnfm
(1.8)
denotes the Poisson bracket, defined by §3, (3.14).
J has the following properties, all obvious except for the first: Remark 1.1.
J only depends on b and c and the restrictions fw and gw
offandgtoW. Remark 1.2.
Multiplying f by h: Z -+ C\{0} multiplies gw by h-' and
adds d(log h)/dt to J. Remark 1.3.
If the matrix b is symmetric and the matrix c is antisymmetric,
then J = 0, since it is possible to choose f = g.
II,§4,1
154
Suppose the matrices b and is are self-adjoint. In particular, this is the case when the matrix a is self-adjoint, that is, when Remark 1.4.
a' = (an,,)*
dm, n.
Then the values of J are purely imaginary, since it is possible to choose 9 = f
Proof of the theorem.
Part 1 follows from §3, theorem 4. To prove part 2,
let
a = aR,
b' =
(P = (PR(X), b nz
b n ( x, 1Pz),
=
a6 m
na
x , p) ,
P=N. a9n(x, P)
9
Op
= w:
so that abm
ax`
I
n
b x, + E b P; (px,x;,
,
1
-'9; - 9Xt + L 9P,(Pxix" j=1
J=1
(1.9)
where x' and p; are the components of x and p (1, j = 1, ... , 1). By the definition of a+ in §1, (6.3)-(6.6), an m (V,
x,
e"'bn (x, Pz)am + eV
V ax
bnP,P;(Px'x' +
+ (2 mod
l
1 V
hence, by (1.9) and the definition (1.2) of L 1,
\g, L 1(X' ax) a) J,m,n
Qn( I
9nbnpl aXm + m.n
.
npj +
V
Thus, by choosing am = If,,, it follows from (1.5) that
b P' as 1 bn ;Pl + cn I or. 1J
11,§4, 1
155
=YHp+[2I ;n9nbp'-xm Y1 2im,n9nasi(bv;fm)+
+
E9ncnfnl ;
hence by (1.5) and the definition of d/dt on W,
= do + 2 E aa;xp;y + Jy,
(1.10)
l
where 1
J
E 9nbnp
2
;
Y axjbnp fm +
2
j, m, n
j, m, n
9nCnfm M' n
Now, by §3, (3.28), a
Y-
ax,
Hpj
dx1e.
1
_
XR
dt
hence (1.10) is equivalent to (1.7). Moreover, by (1.4),
axj
(Yg-b-)
I = 0;
a
hence (1.11) can be written _ bn m
9n
afm
ab
ag
axj
axj
axj
fm
J=
1
2j
Y-
mn
nm 9 Cn fm-
+ m,n
fm m p;
np;
9v,
By (1.9), this is -bm
fm
J=1
Y-
2j,mn
,nx'
f
mp
because
n
9n
m
nx' m
n p;
9xn '
+
nm
9 Cn fm, m,n
(1.12)
II,§4,1-II,§4,2
156
n
9 bnpi m
Jmpr
n
(Px'x' = 0,
gPl
r"PI
bnp
9;;
since the above determinant is an antisymmetric function of (i, j). Clearly, (1.12) is equivalent to (1.8).
Proof of remark 1.1.
J=
(1.8) can be written ('
9n(bf, Jm)] + E 9ncm fm -
[(9n,
2
11
m,n
m,n
Now, by (1.4) there exist regular functions F. such that
Ybnlm = HFn. m
Hence, on W, where H = 0,
J=
d9n
fm)J +
+ m,n
9ncmfm; m,n
thus J only depends on b, c, f, and the restriction of g to W. 2. Resolution of the Lagrangian System aU = 0 in Which the Zeros of det a°° Are Simple Zeros Section 5 of §3 is easily extended in this case.
Notation 2.
Notation 1 is kept. By theorem 1,1°), solutions of the system
t a' U. = 0
(2.1)
M=1
are defined on lagrangian manifolds V in the hypersurface W given by the equation W : H = 0, where H = det b (by assumption, H. Let UR =
1 V
reN
OfR,,e
N
0 on W).
(2.2)
157
II,§4,2
be the expression, in a frame R, of a solution
U=(U1,...,U") of the system (2.1), which will be written
aU = 0; the U. are lagrangian functions defined on V c W; the a,,r are functions V\ER -+ C". In view of theorem 1, the condition aU = 0 on V\ER may be written Lo(x)aR,r(x) + s= 1
L, \x' Oz)
Vr e N.
s(x) = 0
xR,
(2.3),
By (1.6), it implies 9(x, PR,x), Y- Ls S=1
x, )R.Sx))
= 0.
(2.4),
Let M 0 be one of the matrices V - C"x" such that the relation Lo(x)a(x) = a'(x), where 0,
dL A dR A cot/rw >1 0,
dRAco3Awl/w>, 0,
(1.19)
where the three left-hand-side functions do not vanish simultaneously. There exists a constant c such that, in a neighborhood of this point of V, the Maslov index MR. has the value
when dR A 2 A (D3/rv < 0 1 + c when dR A cot A (J)3/0 > 0.
`c MR°
jl
(1.20)
III,§1,1-III,§ 1,2
170
Proof of 1°).
Use expression (1.16) for d'x.
Proof of 2°). Let us calculate the jump of mR. across ERo by applying theorem 3.2 of I,§3. By (1.9), (1.15), and (1.18), the components d;x and
dap of dx and dp in the frame (J1, J2, J3) satisfy on V, at the points of 2:R0
dtp A d2x A d3x = R2[d(QR-') - LR-'w3] A w2 A w3
=RdQAw2Aw3, d1x A d2p A d3x = -RdR A [d(LR-') + QR-'w3] A w2
=dLAdRAw2, d1x A d2x A dap = (dR) A 0-)3 A [Lw1 - Qw2]
=LdRAw3Awl. The existence of w satisfying (1.19) and (1.20) follows by this theorem (theorem 3.2, of I,§3).
2. Choice of a Hamiltonian H Let
be an infinitely differentiable function in involution with L and M, that is, by (1.14), a composition with the functions L, M, Q, and R:
H(x, p) = H[L(x, p), M(x, p), Q(x, p), R(x)].
(2.1)
H[ ] is an infinitely differentiable function defined on an open subset 04 of the space R4 with the coordinates L, M, Q, R. Assume that, on 524
0 < R,
I MI < L,
(HQ, HR)
(0, 0) when H = 0.
(2.2)
HQ denotes the function aH[L, M, Q, R]/8Q. By (1.3), the three functions H, L, and M of (x, p) are pairwise in involution, so that the results of II,§3 can be applied explicitly to the lagrangian operator a associated to H. From theorem 2 (E. Cartan) of II,§3, a quadrature locally defines four real numerical functions on 526, 90,91+92,93,
171
111,§ 1,2
such that < p, dx> = dgo + g1 dL + 92 dM + g3 dH.
(2.3)
We shall use this formula when H = 0; it is made explicit by lemma 2, assuming
H = 0. According to (1.14), the quadrature is the definition of the function 0 by (2.8).
Let W be the hypersurface in 06 given by the equation
Notation.
W : H(x, p) = 0
(Compare II,§3). Let (Lo, M0) be a pair of real numbers such that IM01 < Lo; let V[L0, M0] denote any connected component of the subset of W given by the equations
L(x, p) = Lo,
V [LO, MO] : H(x, p) = 0,
M(x, p) = Mo.
(2.4)
W is the union of the V [La, M0]. By theorem 7.1, of II,§3, V [LO, M0] is a lagrangian manifold. It is the topological product of
the 2-dimensional torus with the coordinates D and T mod 27r, a connected curve I [Lo , M0] in the open half-plane with the coordinates
Q, R > 0. The equation of this curve is (2.5)
T [L0, MO]: H[L0, MO, Q, R] = 0.
By (2.2)3, this curve does not have any singular points. Let us define a real numerical function t of [L, M, Q, R] on W up to the addition of a function of (L, M) by the condition
dt = RHQ[L,IM,
RHR[L,dM,
Q, R]
Q, R]
on F [L, M];
(2.6)
when the curve T[L, M] is a closed curve, then t is defined modc[L, M], where c [L,
[L'
M] _
- fr[L,M] RHQ
_
dQ
.
-fr[L,M] RHR'
(2.7)
III,§1,2
172
t is monotone on F; (L, M, t, (D, `P) forms a system of local coordinates on W.
Let us define another real numerical function S2 on W up to the addition of a function of (L, M) by
A2 = Q
R on F[L, M];
(2.8)
when the curve F[L, M] is a closed curve, then
S2
is
defined
mod2nN[L, M] where
N[L , M] =
dR >0 1 f r[LM] R 1
Q
(2 . 9)
.
S2 is not monotone on F. Let its differential be denoted by
dc2[L, M, t] = Q R + 1 [L, M, t] dL + µ[L, M, t] dM.
(2.10)
The properties of the functions ) and µ thus defined will be specified in §2. Now we can give the explicit expression of the restriction of (2.3) to W. LEMMA 2.
1°) The restriction of the pfaffian form
w= to Wis
cow=d(S2+L`P+MD)-O +`P)dL-(µ+(D)dM.
(2.11)
2°) The characteristic system of dww is Hamilton's system: dx
dp
HH(x, P)
H,,(x, p) '
H(x, p) = 0.
(2.12)
Its first integrals are compositions with the functions
L, M,2+`P,µ+ b;
(2.13)
L and M are in involution; 2 + `P and p + (D are also in involution. 3°) On the solution curves of this system, that is, the characteristic curves of W, we have
_ --
dx
dt = H,(x,
p)
dp
HH(x, p)
_
dR RHQ[L, M, R, Q]
_ dP _
d(D
HL[' ]
HM[' ]
173
III,§1,2
dQ ,dL=dM=0.
(2.14)
RHR[-] 4°) On W d 3x A d 3
dH
_ dL n dM n dt n d(D n d`)'.
PI
(2.15)
w
Proof of 1°). Use the expression (1.14) for in Z and the expression (2.10) for Q dR/R on W. Proof of 2°). Lemma 3.2 of II,§3 (E. Cartan) proves that (2.12) is the characteristic system of dww and that dww has rank 4. Now, by (2.11),
dww, =dL n d(.1 +`Y)+dM n d(µ+(D); thus the four functions (2.13) are first integrals of this system (see E. Cartan's definition of a characteristic system, II,§3,2). The pair (L, M) and (2 + `F, p + 1) are in involution by remark 2.2 of II,§3. Proof of 3°). The same lemma (E. Cartan) and the expression (1.14) for prove that the characteristic system of dw,, is dR
d`P
RHQ[L,M,Q,R]
HL[
_
dD
dQ
HM[']
RHR[']
_
dL 0
_
dM 0 '
H = 0;
thus this system is equivalent to (2.12). Obviously dx
_
HP(x, p)
<x, dx> <x, HP(x, p)>
_
R dR <x, LP>HL + <x, MP>HM + <x, QP>HQ
Since x A (p + sx) is independent of the real variable s,
d(x A p) = 0 for dx = 0,
dp parallel to x;
therefore,
<x, LP> = <x, MP> = 0. Moreover,
<x, Q> = R2. The preceding five relations imply
III,§1,2-III,§1,3
174
dx Hp(x, p)
_
dR RHQ[L, M, Q, R]
_
d'Y
_ d) _ _
HL
dQ
RHR'
HM
dL = dM = 0
and thus (2.14), by (2.6) and (2.12).
Proof of 4°).
Formula (1.17) can be written
d 3x n d3 p= dL n dM n dH n dR n d(l) n d'Y, RHQ
where, for H = 0, dR RHQ
= dt mod(dL, dM)
by (2.14); hence (2.15) holds, the following meaning being given to its left member: d3x A dap
dH
1W
denotes the restriction ww to W of any form w of degree 5 such that dH A m = d 3 x A d 3 p ; ww is clearly independent of the choice of w.
3. The Quantized Tori T(1, m, n) Characterizing Solutions, Defined mod(1/v) on Compact Manifolds, of the Lagrangian System 2) aU=(aL=-LoU=(aM-Mo)U=O mod v
1
.1) (3.1)
a, aL2, and ay denote the lagrangian operators associated, respectively, to the hamiltonians H, L2, M,
which are in involution; Lo and MO are two real constants such that IM0I < Lo. From theorem 7.1 of II,§3, solutions of this system are lagrangianfunctions U with constant lagrangian amplitude, defined mod(1%v) on those
manifolds V[L0, M0] defined by (2.4), whether compact or not, that satisfy Maslov's quantum condition (II,§3, definition 6.2). V[L0, M0] is chosen to be connected. The measure nv on V, which is invariant under the characteristic vectors of H, L2 - Lo, and M - M0, and which serves to define the lagrangian amplitude, is
III,§1,3
175
nv = dt A d' A dW
(3.2)
from (2.15) and formula (7.4) of II,§3.
Recall the statement of Maslov's quantum condition: The function 1
Zh
cOR° - 1 MRo
(whereh =
i
is real
o
has to be uniform on V mod 1. By (2.11), where L = Lo and M = M0, the phase cpR, of V [L0, M0] (defined in I,§2,9 and I,§3,1) is
(pR0 = 0 + LOT + MO(D.
(3.4)
Let us calculate the Maslov index of V [L0, M0]. LEMMA 3.
1°) The apparent contour ER0 of V[L0, M0] is the union
E, u E2 of two surfaces in V[L0, M0] given by the equations E2 :HQ[L, M, Q, R] = 0.
E, :'Y = 0 mod 7r;
2°) The Maslov index m mR° _
of V[L0, M0] is
[i-p] -
[V [L0, M0] 77
up to the addition of an integer constant; [ Proof.
(3.5)
R
] denotes the integer part.
Apply lemma 1. By (1.11) and (1.13),
0)1 = - cos'I' sin O dW, L00)3 = L0 d'I' + M0 da)
0w2 = sin W sin O dcb,
on V[L0i M0]; hence, by (2.6),
dR A w2 A 0)3 = -RHQsin'I'sin0dt A d'I' A dD,
(3.6)
dQ A 032 A W3 = RHR sin W sin O dt A d'I' A d(D,
(3.7)
dR A w3 A 0), = -RHQcos'I'sinOdt A d'! A dcb,
(3.8)
where R sin O : 0 by (2.2), and (2.2)2 . By lemma 1,1°), ER° is given by the equation HQ sinW = 0, from which follows part 1 of lemma 3.
III,§1,3
176
In a neighborhood of a point of E1\E1 U E2, HQ cos `P # 0; in lemma 1,2°), we can choose
w = dR n w3 n w1;
dR n W2 n w3/w = tan `P,
whence by this lemma mRo =
[i: `Pl + const. n
J
By (2.2)3, in a neighborhood of a point of E2\E1 U E2' HR sin `P in lemma 1,2°), we can choose
0;
dR A 0)2 A m3/m = - HQ/HR,
w = dQ A wz A 0)3; where
MR,, = const. - C1 arctan L
(3.10)
].
HR
The global expression for mRo follows from the two local expressions (3.9) and (3.10).
By (3.4) and (3.5), Maslov's quantum condition may be formulated as follows: The function 1
h 2n
4n
arctan
HQ HR
+ Lo _ h
1
`P
Mo do
2
27c
it 2n
is defined mod 1 on V[L0, Mo].
If V[L0, Mo] is compact, that is, if the curve F[L0, Mo] is a closed curve, by (2.9) this condition is that
IN[Lo, M] + 1, 0 2
h
LO - 1, 2
1M
h°
be three integers. Denote them by
n-1,
1,
m
in order to recover the classical notation of quantum physics. Since N > 0 and IMOI < Lo, we have
Iml0,
L2O+Co>0,
Bo>-,/A0V/L+C
(4.7)
When it is satisfied, V [Lo, Mo] is unique, and by (4.3),
N [Lo, Mo] = 2n
f~ - AOR, + 2Bo R - Lo - Co R ,
where the integral is calculated along a cycle of C enclosing the cut JR1 i R21; taking residues at R = oo and R = 0 gives
N[Lo, Mo] =
BO
- v[L:o + Co > 0.
v' Ao
The statement of theorem 3 becomes the following.
Let a be the Schrodinger-Klein-Gordon operator, that is, the operator associated to the hamiltonian (4.6). Let aLl and am be the operators associated to the hamiltonians L 2 and M. Then the lagrangian THEOREM 4.1.
system
aU = (aL: - Lo) U = (am - MO)U = 0 mod
a V
has solutions defined on a compact manifold if and only if there exists a triple of integers (l, m, n) such that
Lo=h(l+Z), Iml
l < n,
Mo=hm,
(I + )z +
(4.10) Coh-z
> 0,
AO =Boh-z[n-l-I+
(l+1)2+Coh-z]
z
(4 . 11)
If this condition is satisfied, this compact manifold is unique: it is the torus T(l, m, n) given by the equations
III,§1,4
181
T(l, m, n): H(x, p) = L(x, p) - Lo = M(x, p) - Mo = 0.
(4.12)
We provide this torus with the invariant measure (3.16); then the solutions of (4.9) defined mod(l/v) on T(l, m, n) are the lagrangianfunctions with constant lagrangian amplitude.
Notation. (Related to physics and used only at the end of this section)
In E3, we choose an electric potential sago : E3 -+ R and a magnetic potential vector ('41, d2, a3): E3 - E3 satisfying the physical law
i=i axi
= 0;
(4.13)
they are time independent. The trajectories in E3 of the relativistic or nonrelativistic electron of
mass µ and electric charge -F < 0 are the solutions of Hamilton's system, defined by the hamiltonian given by
sii(z)12-E
H(x,p) = 1
2µi=1
CPi
+
- Fsdo(x)
c
(4.14)
J
in the nonrelativistic case, and by sii(z)12
H(x, p) = 2µ iY-i [pj +
- 2µc2 [E +
2 + iµc2
(4.15)
in the relativistic case. Here c is the speed of light, E is a constant, which is the energy of the electron whose position x and momentum p satisfies H(x, p) = 0 (energy including the rest mass µc2 in the relativistic case). By (4.113)
a'x,
ap) H(x, p) = 0.
Then by// I1,§1,(6.3) and II,§l,definition 6.2, in the nonrelativistic case the operator associated to H is the Schrodinger operator
a= 2µ
v2 0 + ce Ed'(x) v ax j + i
c2
E i
(x)1
-E
- Fdo(x),
(4.16)
182
III,§1,4
and in the relativistic case it is the Klein-Gordon operator
a=
+ 2z
2Lyz
I.i 1
+ E2 I d2]
0
-
2-t- [E + Ed0]2 +
µ2z (4.17)
Let us choose these hamiltonians and operators as follows: d0 is the electric potential of a nucleus with atomic number Z, placed at the origin; the magnetic field is constant, parallel to the third coordinate axis, and of intensity .* . In other words,
-Zrxz,
SIo(W ) = R ,
z(x) = Z. x1,
S13 = 0. (4.18)
Let us neglect the . Z terms, as is done in physics; the hamiltonians (4.14) and (4.15) of the electron become hamiltonians of the form (4.6)
H(x, p) = Z
[P2
µ
+ c .M - 2µE -
2µR2
Zl
(4.19)
in the nonrelativistic case, and
Hx ( ' P)
2µ '
I
P2+Ec M+ µz c 2
Ez_2e2ZE-e4Zz cz
c2R2
c2R
(4.20)
in the relativistic case. The Schrodinger operator becomes
a=2µ[v A+E
(xla3Z-xzOX
µz -2µE-2RZ(4.21)
the Klein-Gordon operator becomes
a=
1
1
2µ vz
+ ec xl aX2
xz
8x1
+ µzcz - Ez cz
2e2ZE e4Zz - czR - czR zl. (4.22)
Relation (4.11)3 defines E as a function of (l, m, n); in the course of this calculation, two constants, which are classical in physics, appear. z
he
-
137,
the dimensionless fine-structure constant,
183
III7§1,4
p=
,
2u
the Bohr magneton (j3.)(° has the dimensions of energy),
as well as the function given by
F(n, k) =
1
.
2
aZ
1 +
(4.23)
2
The statement of theorem 4.1 becomes the following. Let a be the Schrodinger operator (4.21) or the Klein-Gordon operator (4.22), H being (4.19) or (4.20). Let aL= and aM be the operators associated to L2 and M. For certain values of the constants E, Lo, and M0, the lagrangian system THEOREM 4.2.
aU = (aL2 - Lo )U = (am - Mo)U = 0 mod z
(4.9)
has solutions defined mod(1/v) on compact manifolds. These values of E, Lo, and MO are expressed as functions of a triple of integers (1, m, n) such that
ml < I < n and, in the Klein-Gordon case, l + z 3 aZ; these expressions for E, Lo, and MO are -ucZ
E_
azZz
2n2 +
j3.*'m in the Schrodinger case,
E2 = uc2(uc2 + 2f3.°m)F2(n, l + Z) in the Klein-Gordon case,
Lo = h(l + Z),
Ma = hm.
(4.24)
(4.25) (4.26)
These solutions of (4.9) are defined on the tori (4.12). Let us provide these
tori with the invariant measure (3.16); then these solutions are lagrangian functions, defined on these tori, with constant lagrangian amplitude. Remark 4.2.
±E
Physicists, neglecting j32 and j3a2, simplify (4.25) as follows:
uc2F(n, I + Z) + f.#°m.
The minus sign concerns antimatter (u < 0). The energy levels (4.24) and (4.25) are those given by the study of the partial differential equation Remark 4.3.
184
I I I,§ 1,4-III,§2,0
au = 0, whose unknown u:E3\{0} -+ C and its gradient have to be square integrable (III,§4).
§2. The Lagrangian Equation aU = 0 mod(1/vs) (a Associated to H, U Having Lagrangian Amplitude > 0 Defined on a Compact V) 0. Introduction
In §1, we studied problem (1) (introduction to chapter III), that is, a system of three lagrangian equations. In §2, we study the first of these three equations and, more precisely, problem (2) (introduction to chapter III). Each solution of problem (1) is a solution of problem (2), by theorem 3 of §1. In the exceptional case of a hamiltonian H independent of M (for Summary.
example, Schrodinger and Klein-Gordon without a magnetic field), a rotation in E3 acting simultaneously on x and p obviously transforms solutions of problem (1) into solutions of problem (2) that are no longer solutions of problem (1). Without considering this exceptional case, theorem 1 of §2 constructs solutions of problem (2) that are not solutions of problem (1): a solution of problem (2) defined on a torus T(l, m, n) (theorem 3 of §1) is not necessarily unique up to a constant multiplicative factor.
On the other hand, theorem 2 shows that even if H depends on some parameters, these tori T(l, m, n) are, in general, the only compact lagrangian
manifolds V on which solutions of problem (2) are defined. Moreover, theorem 3.1 specifies that in the Schrodinger and Klein-Gordon cases problems (1) and (2) define the same energy levels: the classical levels. Remark. Theorem 1 of §2 does not have any physical significance: it takes
into account the condition that some numbers, which in the Schrodinger and Klein-Gordon cases are measurements of physical quantities, take rational values. Let us call a problem that when posed for the SchrodingerKlein-Gordon equation has the essential features of the classical problem (4) (introduction to chapter III) a well-posed problem. Problem (1) is wellposed by theorem 4.2 of §1. From the preceding remark, the main result of CONCLUSION.
§2 is that problem (2) is not well-posed.
III,§2,1
185
1. Solutions of the Equation aU = 0 mod(1/v2) with Lagrangian Amplitude -> 0 Defined on the Tori V [L0 , Mo)
In section 2 of §1 we chose H and defined the manifolds
W:H=0,
V[L0,Mo]:H=L-Lo=M-Mo=0.
Any compact lagrangian manifold V in W is a union of characteristics K of H (whose paramter t varies from - cc to + oo); V is thus a union of compact closures K of such characteristics. Properties of a characteristic K of H with compact closure K. By (2.13) of §1, such a characteristic K stays on a torus V[L0, Mo] and is given by the equations
K:`Y-To+).[L0,M0,t]=(D -(o+p[L0,M0,t]=0 (0o, `1'o constants)
on this torus. Recall that the coordinates of a point of V[L0, M0], (`1', 4), t), are defined
(mod 2n, mod 2n, mod co), where co = c[L0, Mo].
More explicitly, let R3 be given the coordinates (`Y, D, t) and let Z3 be the additive group of triples of integers (S, q, () acting on R3 as follows: Z3 -3 ((1, q, C): (P, 4), t) i -+ (P + 2nd, 1) + 2n7, t + c0S);
the quotient of R3 by this group Z3 is V[L0, Mo]: there exists a natural mapping
R3 -' R3/Z3 = V[Lo, M0].
(1.2)
Given a function F: (L,
M,
Obviously
A,R=A,Q=0.
t
M, t].
III,§2,1
186
By the definitions (2.8) of 0 and (2.9) of N in §1,
O0[L, M, t] = 2nN[L, M]; then, by (1.3) and §1,definition (2.10) of
..
and
2ndN[L, M] = AA [L, M, t] dL + A,u[L, M, t] dM; that is, A,,l = 2nNL,
0,u = 2nNM.
Let
NL,, = NL[LO, MO],
NMo = NM[L,, MO]
We can now describe K. LEMMA 1.1.
NL° _
1°) Assume NL° and NM° are rational:
-L1 N,'
Ml,
NMo =
where (L,, M, N1) E Z3,
N,
G.C.D.(L,, M1, N,) = 1
(1.6)
(that is, L,, M,, and N, are integers with greatest common divisor 1). Then K = K is a closed curve given by the equations (1.1). More precisely, the equations (1.1) define an open curve R (that is, a curve homeomorphic to R) in R3. By (1.5) and (1.6), the subgroup 2 of Z3 generated by (L1, M,, N,) leaves ft invariant; we have
K=K=R/Z.
(1.7)
2°) Assume that NL° and N,M° are connected by a unique affine relation L, NL,, + M, NMO = N,
(1.8)
where (L,, M,, N,) E Z3, G.C.D.(L,, M,, N,) = 1. Then K is the 2-dimensional torus T 2 defined in V [L., M0] by the equation
L,{`P - To + 2[L0, M0, t]} + M,{0 - b0 + u[L0, M0, t]} = 0. (1.9)
More precisely, equation (1.9) defines a surface R2 in R3 homeomorphic to R2. By (1.5) and (1.8), the subgroup 22 of Z3 given by the equation
Z:LjI +M,q+N1 = 0
(1.10)
III,§2,1
187
acts on 1l2. We have
T2 = It2/22.
(1.11)
In order to describe the generators of Z2, let M2 = G.C.D.(L1, N1),
L2 = G.C.D.(M,, N1), N2 = G.C.D.(L,, M1).
(1.12)
M2 and N2 divide L1i G.C.D.(M2, N2) = 1 by (1.8)3; then there exists an integer L3 and similarly integers M3 and N3 such that L1 = L3 M2 N2,
Ni = L2 M2 N3.
M1 = L2 M3 N2 ,
(1.13)
Z2 is generated by its three elements (0, M2N3, -M3N2),
(-L2N3, 0, L3N2),
(L2M3, -L3M2, 0), (1.14)
which evidently are connected by the relation L3(0, MZN3, -M3N2) +
M3(-L2N3, 0, L3N2)
+ N3(L2M3, -L3M2, 0) = 0.
(1.15)
3°) Suppose there is no affine relation with integer coefficients connecting NL° and NMO. Then K is the torus V[L0, Mo].
Proof. The subset of R3 whose natural image in V[L0, Mo] is K is defined by the condition
C ` I ' - To +.[LO, MO, t]b - 0o + u[Lo, MO, t] 2n
2rc
JJ
E G,
where G is the image of Z 3 in the additive group R 2 under the morphism
Z33(
b)~'(S + NLOS,fl + NM00ER2.
Then K is the natural image in V[L0, Mo] of the closed subset of R3 defined by the condition C`I'
- To +2 [Lo, M0, t] (D - .bo + 2[Lo, Mo,
E
t] where G is the closure of G; G is a closed subgroup of R2. Three cases occur: (1°) G is discrete, (2°) dim 6 = 1, (3°) 6 = R2.
III,§2,1
188
1°) G is discrete, that is, C = G. This is the case if and only if NL° and NM° are rational (use the rapidity of convergence to an irrational number
of its rational approximations given by its continued fraction development), that is, if and only if (1.6) holds. Then K = K; the elements of the
subgroup Z of Z3 leaving invariant the curve R c R3 given by the equations (1.1) are the
N1 = L1C,
rl, l;) in Z3 such that
N,tl = M1 C-
By (1.6)4, N1 divides C. Thus Z is generated by (L1, M1, N1) E Z3. 2°) dim G = 1. Then G is the set of (0, t) e R2 satisfying a condition of the form
L10 + M1t E Z, where L1, M1 E R;
(1.17)
by the definition of G, G is the subgroup (1.17) of R2 if and only if (1.8) is satisfied and G is not discrete, that is, by (1°), if and only if NL° and NM° are connected by a unique affine relation with integer coefficients, which is (1.8).
Assuming this hypothesis, by definition (1.16) of k and definition (1.17) of G, K is the image under (1.2) of the manifold R2 in R3 given by equation
(1.9); obviously (1.10) defines the subgroup Z2 of Z3 that leaves A2 invariant; and (1.10) implies by (1.8)3 and (1.12) that
L2, M2, and N2 divide , ri, and t;, respectively.
(1.18)
By (1.13), the three elements (1.14) are contained in Z2. On one hand, (1.14)1 generates the subgroup of Z2 given by the equation
=0, because G.C.D.(M2N3, M3N2) = 1 by (1.12)1, (1.13)2, and (1.13)3. Thus
G.C.D.(N3, M3) = 1, which implies that the values taken by in the subgroup of 22 generated by the elements (1.14)2 and (1.14)3 are all the multiples of L2. Thus, by (1.18), the three elements (1.14) of Z2 generate
2.
3°) R2. By (1°) and (2°), G = R2 if and only if NL° and NM° are not connected by any affine relation with integer coefficients. If G = R2, then K is the image of R3 under (1.2); thus K = V[L0, M0].
Invariant measures on V[L0, M0]. Recall that V[L0, M0] has a measure > 0 that is invariant under the characteristic vector K of H [(3.2) of §1] :
189
III,§2,1
tlv = dt A dO A dW. Every measure on V [LO, M0] that is invariant under K is the product of
tlv and a function V[L0, Mo] -+ R that is invariant under K, that is, constant on the closures k of the characteristics K of H staying on V[Lo, Mo]. Then the following lemma is an obvious consequence of lemma 1.1. 1°) Suppose that NL° and NM,, are the rational numbers (1.6). Then the characteristics of H staying on V[L0, Mo] are the closed curves given by the equations LEMMA 1.2.
K(c1, c2): `P + %[Lo, Mo, t] = c1,
b + p[Lo, Mo, t] = c2;
el and c2 are constants defined mod 2n
M2
N,
27c
= L2N3,
2n
L2
Nl
=
2n
M2 N3,
respectively, where
L2 = G.C.D.(M1, N1), N3 = N1/L2M2 E Z.
M2 = G.C.D.(L1, N1),
The measures on V[L0, Mo] that are invariant under the characteristic vector K of H are given by
F(`f' + )[Lo, Mo, t], cD + p[Lo, Mo, t])r)v,
(1.19)
where
is an arbitrary function with periods 27r(M2/N1) and 27t(L2/N,) in the first and second arguments, respectively. 2°) Suppose that NL, and NM,, are connected by the unique affine relation
(1.8). Then the closures K of the characteristics K of H staying on V[L0, Mo] are the tori given by the equations
T2(co): L1 {`P + ).[Lo, Mo, t] + M1 {db + p[Lo, Mo, t]} = co,
(1.20)
where co is a constant defined mod 27t. The measures on V[L0, Mo] that are invariant under the characteristic vector K of H are given by
F[L1(`P + A) + M1(b + u)]'lv,
(1.21)
where F[ ] is an arbitrary function with period 2n. 3°) Suppose that there is no affine relation with integer coefficients connecting NL,, and NM,. Then the closure K of each characteristic K of H
190
111,§2,1-11 1,§2,2
staying on V[L0, Mo] is V[LO, Mo]. Every measure on V[L0, Mo] that is invariant under the characteristic vector x of H is given by (1.22)
const. g v .
The following theorem is an easy consequence. THEOREM 1.
1°) The tori V[L0, Mo] on which a lagrangian solution U of
the equation 12
aU = 0 mod
(a is the operator associated to H)
V
with lagrangian amplitude >,0 may be defined are defined by the condition (3.11), (3.12) of §1: there exist three integers 1,rn,n such that
Iml
1,0 Exist
We shall show that such manifolds V exist only exceptionally. The calculation of their Maslov index (lemmas 2.3 and 2.4) uses the following properties.
III,§2,2
191
Other properties of the characteristics K of H with compact closures. Differentiating the definition (2.10) (§1) of A and p gives
R dQ n dR + dJ n dL + dp n dM = 0, where L, M, Q, R are functions of (L, M, t) satisfying H[L, M, Q, R] = 0; then
HLdL+HMdM+HQdQ+HRdR=0, whence, by elimination of dQ:
[j_dR+d1]AdL+[!jy-dR+dp]AdM=o; RQ
thus there exist three real numerical functions p, a, and r of (L, M, t), defined when HQ * 0, such that
dA _ -
RddR + p dL + a dM, RHQ L
,u = -
dR + a dL + T dM.
RHQ By the expression (1.5) for A. and O,u and definition (2.6) of tin §1, these relations imply A,p = 21tNL1,
A,a = 21tNLM,
A,[L, M, t] _ -HL[L, M, Q, R],
O,t = 21tNMz,
(2.2)
p, _ -HM.
(2.3)
Let us describe explicitly the singular part of p, a, t, and PT - a2 when HQ = 0; by (2.1),
p = RLHL RHQ
T =
RMHM RHQ
+ L,
a = RMHL + M = RLHM + AL, RHQ
RHQ
+ AM;
thus
pt - a2 = RH [RLHLPM - RMHLPL - RLHMI.M + Q
+ i.L/M - MAL-
III,§2,2
192
Since H[L, M, Q, R] = 0,
HRRL+HL +HQQL=HRRM+HM+HQQM=O. When HR :j6 0, these relations make it possible to eliminate RL and RM from the preceding expressions for p, a, T, and pr - 62; by assumption (2.2) of §1,
HR # 0 when HQ = 0; hence: LEMMA 2.1.
The functions
2
P
+
HL
PT - a2 +
Q
RHQHR,
1
HLHM
+ RHQHR,
2
HM + RHQHR'
_
RHQ HR [Hi 12M - HLHM(uL + AM) + H2 AL]
are bounded in a neighborhood of the points where HQ = 0. Properties of compact lagrangian manifolds V in W.
V is generated by
characteristics of H with compact closure, which thus stay on tori V[L0, Mo]; then the functions
L, M, N = N [L, M],
c = c [L, M]
are defined on V. Let V2 be the open subset of V where dL A dM 96 0 if it is not empty. When dL A dM = 0 on V, let Vt be the open subset of V where (dL, dM) : 0 if it is not empty t3) Then Vt and V2 are lagrangian manifolds, not
necessarily compact, that are generated by characteristics K of H with compact closures k; they contain these closures. When V2 and Vt do not exist, then V is one of the tori V[L0, Mo]. By the following lemma, Vt and V2 only exist if the graph N : (L, M) i--+ N [L, M]
contains a rectilinear segment with rational direction.
Thus the consequence of theorem 2 stated in the introduction (§2,0) follows from this lemma. 'This notation means that dL and d.41 are not simultaneously zero. [Translator's note]
III,§2,2
193
LEMMA 2.2.
1 °) On V2, N is an aff ne function of (L, M). More precisely,
L1dL + M1dM + N1dN = 0, where (L1, M1, N1) E Z3,
(2.6)
G.C.D.(L1, M1, N1) = 1.
V2 is defined in W by the datum of a function F of two variables and by the equations
`P + I[L, M, t] + FL[L, M] _'V + µ[L, M, t] + Fr,[L, M] = 0.
(2.7)
More precisely, in the space R5 with the coordinates (L, M, `P, 'V, t), equations (2.7) define a 3-dimensional manifold P2. By (1.5), where
NM = -M,/N,,
NL = -L1/N1,
the elements q, c) of the subgroup Z of Z3 generated by (L1, M, , N,) E Z3 act on V2 as follows: (5, n, C): (L, M, T, (D, t) i-- (L, M, `P + 2irS,'V + 2nq, t + c [L, M] S ).
(2.8)
We have
V2 = V2/Z
(2.9)
V2 has a measure that is invariant under the characteristic vector of H, given by
qv = dL A dM A dt. 2°) On V1, L, M, and N are affine functions of a single variable s. More precisely,
dL -dM=dN=ds L1
M1
(2.10)
N1
where (L1, M1)
0,
(L1, M1, N1) E Z3,
G.C.D.(L1, M1, N1) = 1.
V, is defined in W by the datum of three functions of s-the affine functions
L and M satisfying (2.10) and F-and by the equations
L = L(s),
M = M(s),
L1{`P + %[L, M, t]} + M1{dV + u[L, M, t]} + F,(s) = 0.
(2.11)
More precisely, equations (2.11) define a 3-dimensional manifold V1 in R5
III,§2,2
194
on which the subgroup V2 of Z3 given by equation (1.10) acts according to (2.8). Recall that this subgroup is generated by its three elements (I.14). We have
V,=V,/Z2.
(2.12)
V, has a measure that is invariant under the characteristic vector of H, given by
qv = (M, d'1' - L, d(D) A ds A dt. 3°) The phases of V, and VZ in the frame Ro (section 1 of §1) are given by
(pR0 =Q+L''+MD+F. Remark 2.1.
(2.13)
(2.8) and (2.10) in §1 define
12 up to the addition of a function F of (L, M) , and µ up to the addition of its derivatives FL and FM.
Given V, or V,, it is possible to choose 0 such that
F=0 in (2.7), (2.11), and (2.13).
In order to quantize V, we shall use only the following consequence of (2.13) and the definitions (2.7) of c[L, M] and (2.9) of N in §1: choosing F to be zero, the function
NR° - 2nNc[Lt M]
- LT - MO
(2.14)
is defined on V, and on V2. Preliminary to the proof. II,§3 applies with
1=3,
h,=L,
go=S2+LT+MO,
Formula (2.11) of §1 shows that theorem 3.1 of
h2=M,
91 = -''-A,
9z= -(D -u,
the phase cpR, replacing the lagrangian phase. Hence any lagrangian manifold V in W is given locally by equations of one of the four following types:
III,§2,2
195
T +i +FL[L,M] =0+,u +F,,t=0;
(2.15)1
M = f(L),
(2.15)2
`V + A + fL(L)(D + u) + FL(L) = 0;
the result of the permutation of (L, T), (M, 0) in (2.15)2; L = const.,
M = const.;
(2.15)3 (2.15)4
F and f are functions of one or two variables. The phase (PR. of V is expressed by (2.13), where F = 0 in the fourth case. Proof of 1°).
Locally, V2 is given by equations of the form (2.15)1. Hence
V2 is generated by characteristics K staying on disjoint tori V[L0, Mo]. Their closures K are disjoint and are contained in V2 ; dim V2 = 3, so that
dim K = 1. Then by lemma 1.1 the values of NL and NM on V are rational numbers. Thus the functions NL and NM are constant on Vand satisfy (1.6), which expresses (2.6), whence 1°) and (2.13).
Proof of 2°). Locally, V1 is given by equations of the form (2.15)2 or (2.15)3, that is, equations of the form
M = M(s), L = L(s), L5(s)('Y + A) + MS(s)(G + u) + F,(s) = 0,
(2.16)
where (Lc, MS) -0 0. For each value of s, let T(s) be the manifold in W given by equations (2.16): dim T(s) = 2. Since V1 is generated by characteristics given by the equations
L = const.,
M = const.,
'Y + A = const.,
0 + µ = const.
and contains their closures, V1 contains the closures T(s) of the T(s). Let us determine the T(s). By expression (1.5) for i,A and D,µ and since L.,NL + M.,NM = Ns, T(s) is the image in W of the set of ('Y, 0, t) E R3 such that L,
'Y + A[L, M, t] + MS 0 + p [L, M, t] 2n
+ 1 Fs E G(s), 27[
271
where G(s) is the image of Z 3 in the additive group R under the morphism
Msrl + N5
.
196
III,§2,2
Then T(s) is the image in W of the set of ('F, (D, t) e R3 such that LS'I'
+ ..[L, M, t] +
Ms(D + µ[L, M, t]
+ IFS
2n
2n
n
e G(s),
where G(s) is the closure of G(s) and thus a closed subgroup of R. G(s) = R, and thus T(s) is the 3-dimensional torus V[L(s), M(s)] unless
G(s) is discrete, that is, unless there exists (L M,, N,) E Z3 such that
-
L' L,
MS
_ N.
M,
N,
G.C.D.(L M,, N,) = 1.
Since dim V, = 3, this must be the case for every s. The functions MS/LS and NS/LS have rational values; thus they are constants. The integers
L,, M, , and N, are independent of s; and s can be chosen so as to arrive at (2.10); thus 2°) and (2.13) hold. Quantization of V.
Let us impose Maslov's quantum condition on V. In order to express this condition, let us calculate the Maslov index MR,, of V2 and of V, using lemma 1 of §1. We make F = 0 in lemma 2.2 (see remark 2.1). LEMMA 2.3. 1 °) The functions p, a, r are defined on V2. Let R be the one-point compactification of R. Then the function F : V2 -. $ given by
F(L , M , t) =
pL2 + 2aLM + TM'
L(L2-M2)(pi-a2) E $
(2 . 17 )
is defined on V2\E", E" being the subset of V2, where
LM
M2
(2 . 18 )
L2
2°) If V2\E" is connected, then
1'I' + 1 arctan F(L, M, t) mR° = 17r n
the integer part of )
(2.19)
on its universal covering space. 3°) The function M&
-
n
- 2 c[L, M]
is defined (that is, uniform) on V2.
(2 . 20)
III,§2,2
197
Assume a function
Remark 2.2.
N: (L, M) i-4 N[L, M] satisfying (2.6)
is given. Choose H satisfying (2.9) of §1. If this choice is generic, then dim E" = 1, Proof of 1°).
V2\E" is connected.
Recall that (L, M, t) are local coordinates on V2. By (2.2)
and (2.6),
1 °) follows.
Proof of 2°). The apparent contour of V2. Formulas (1.11) and (1.13) in §1 give the relation L sin E)
W2 A
G)3
=Lsin`Ydb Ad`Y+L2 -M2(MdL-LdM)A(Ld`Y+Md(b) (2.21)
on W. Differentiating (2.7), where F = 0, gives
d'Y + pdL+adM =dD+adL+tdM =0 moddR on V2, from definition (2.1) of p, a, and T. By (2.6) of §1,
dR = RHQdt mod(dL,dM), hence L
R sin 0
dR A w2 A w3 = G(L, M, t,'Y) dL A dM A dt
(2.22)
where G denotes the function
G = -HQ[L(pt - a')sin`Y + pL2 L2aLM2 zM2cos'Yl, which is regular on V2 by lemma 2.1. Then by lemma 1 of §1, the apparent contour ER°of V2 is
ER°=E'UE",
III,§2,2
198
where E" is defined by (2.18) and E' is the surface in V2\E" given by the equation
E': tan 'P + F(L, M, t) = 0,
(2.23)
where F is defined. by (2.17). Calculation of mRa.
Formulas (1.11) and (1.13) in §1 give
L 0CO3 A wl sin
= Lcos`Pd(D A d'P - Ly-
2(MdL- LdM) A (Ld'P + Md(b) (2.24)
on W, from which, by the same calculations used to deduce (2.22) from (2.21),
L
k -sin 0
dR A (03 A wl = Gy,(L, M, t, T) dL A dM A dt.
(2.25)
Relations (2.22), (2.25) and lemma 1 of §1 give
mR = const. for G/Gy, < 0,
mR = 1 + const. for G/Gs, > 0 in a neighborhood of a point of E'. This result obviously holds for any function G vanishing to first order on E', for example, the function
G = 1'P + 1 arctan F(L, M, t) mod 1. n
n
Thus, on V2\E", m,0 is expressed locally by (2.19), up to an additive constant; 2°) follows.
Proof of 3°). First suppose that H and S2 are generic (remark 2.2). Then V2\E" is connected and
LHL + MHM : O for HQ = 0, which implies by lemma 2.1 that the function
f = pL 2 + 2QLM + TM 2: V2\E" - k
199
III,§2,2
is defined. By (2.19), the function MR.
-
1
n
`l, - 1 arctan F is defined on V2\E";
(2.26)
n
by the definition of F, where I M I < L from assumption (2.2) of §1, we have
F/f < 0 in a neighborhood of the points where F = 0; F = 0 is equivalent to f = 0, so that arctan F + arctan f is defined on V2\E";
(2.27)
we see that
arctan f + arctan f = const.;
(2.28)
by lemma 2.1, 1/f = 0 is equivalent to HQ = 0 and HQ H f 0 by assumption. Thus condition (4.11) of §1 is satisfied. Choose a to be the Klein-Gordon operator (4.22) of §1. Suppose the magnetic, field .$ 0 0. Then the tori T(l, m, n) defined by (4.12) of §1 are the only compact manifolds on which there exists a lagrangian solution U of the equation THEOREM 3.2.
a U = 0 mod i with lagrangian amplitude >,0.
207
III,§2,3-III,§3,1
Proof. The proof of theorem 3.1 shows that the necessary condition
stated in theorem 2 can not be satisfied when a is the Klein-Gordon operator (C 0 0) and ' 0 (A depends on M). Conclusion
We do not pursue this difficult study of the equation aU = 0 mod(1/v2). In particular, we do not describe the lagrangian manifolds other than the tori T(l, m, n) on which there exist solutions of the Klein-Gordon equation when .f = 0 or solutions of the Schrodinger equation.
§3. The Lagrangian System
aU = (am - const.) U = (aL - const.) U = 0 When a Is the Schrodinger-Klein-Gordon Operator 0. Introduction
In §3, we study the lagrangian system that was solved mod(l/v2) in §1. In section 1, we determine the condition under which theorem 7.2 of II, §3 applies.
In sections 2, 3, and 4, we apply this theorem under assumptions that become more and more strict. These assumptions finally amount to the assumption that a is the Schrodinger-Klein-Gordon operator. Existence theorem 4.1 is finally obtained. Remark 0.
A Voros orally pointed out that these properties of the
Schrodinger and Klein-Gordon equations extend to the case where the electric potential is any positive-valued function of the variable R, if the energy level E is not constrained to be a real number, and if it can be taken to be any formal number with vanishing phase. 1. Commutivity of the Operators a, aL2, and am Associated to the Hamiltonians H (§1, Section 2), L2, and M (§1, Section 1) We want to determine when theorem 7.2 of II,§3 applies to these operators, that is, when they commute. 10) am and a (thus, in particular, am and aL=) commute. 2°) aL, and a commute if and only if
LEMMA 1.
HM2Q = HM2R = 0
YL, M, Q, R.
(1.1)
III,§3,1
208
Proof. Let a and a' be the lagrangian operators associated to two hamiltonians H and H'. By formula (1.1) of II,§2, their commutator
a 0 a' - a'oa is associated to the formal function given by
-2 r
1
1
(2r + 1)! (2V) 2r+1
)]2,4 H(x, p)H'(x', P')
XxpX'ap
P =P
Suppose H and H' are in involution, which is the case for H(§1,2), L2, and M(§1,1) by (1.3) of §1. Then the first term of (1.2) is zero. If H' is a polynomial of degree 2 in (x', p'), then all of the other terms evidently are zero;
thus a and a' commute and part 1 of the lemma follows. Suppose that H' is a polynomial that is homogeneous of degree 4 in (x', p'). Then by (1.2), a o a' - a' o a is associated to H"/v3 where H" is the hamiltonian given by H"(x, p)
24
(ax /-)'
H(x, p)H'(x', p')
/-)]
x
Two applications of Taylor's formula show that the term of H'(x' + y, p' + q) that is homogeneous of degree 1 in (x', p') and degree 3 in (y, q) is
6 [(q, aP / + \Y, ax
)] 3H (x', P') _
[('aq) +
(X ay)] H'(y, q);
thus, if H' is homogeneous of degree 2 in each of its two variables, then
H"(x, p) =
4
[(p' HP (ap' ax)>
- \x' Hz, (ap' ax)!] H(x, p).
In particular, if H' = L2, that is, H'(x', p') = Ix' A p'I2, then H"(x, p)
1
/
a
=2PA --+XA P
a
a
Ox ' a P
A
a
\
ax/J
H(x, P).
In this formula, for each j the pair of operators
and Op n
r p n ca + x A z) P
;
z)
c
commutes.
(1.3)
209
III,§3,1
The operator
Cpna +xn x P is an infinitesimal rotation acting on x and p; thus it annihilates L, Q, and R. Obviously,. Cp
A a + X A Xl M(X, P) = x3P - P3x.
f
P
Suppose H is a composition with L, M, Q, and R [§1,(2.1)]. Then (1.3) becomes
H "(x, p) = 2 (a A X, HMX3P - HMP3X )
\P
Let
X F = P1
F,,
X2
X3
P2
P3
FX2
FT,
YF =
,
X1
X2
X3
P1
P2
P3
Fp2
Fp3
Fp,
for any function F of (x, p). The preceding expression for H" may be written H"(x, p) =
12 aax9/HM 3
1
- 2a
0 H"t.
P3
Now, the linear differential operators I and Y evidently annihilate P2, Q, and R 2, hence L2, by (1.2) of §1; moreover,
-M = -l-L2
1aL2
9M
219X3
2 eP3
Thus, letting !2F denote the functional determinant
1F
aL2 of
aL2 of
ax3 OP3
aP3
ax3,
the expression for H" becomes
H"(x, p) = 4'!HM2
210
III,§3,1-III,§3,2
Now, the linear differential operator .9 evidently annihilates L and M; moreover,
z9Q=P2x3-R2P3 and R -9R = Qxj - R2P3x3; thus the preceding expression for H" becomes i z 2 H"(x, p) = i (P2HM2Q + R HM2R) x3 - R HM2RP3x3 - 2R HM2QP3 (1.4)
By §1.1, p3/x3 is independent of (L, M, Q, R). Thus the condition
H"(x, p) = 0
Vx, p
is equivalent to (1.1), which proves part 2 of the lemma. 2. Case of an Operator a Commuting with aL2 and am
Suppose (1.1) holds. Lemma 1 proves that theorem 7.2 of II,§3 applies to the lagrangian system
aU = (aL2 - cL)U = (am - cM)U = 0 mod 1 2, v
r
where cL and cm are two formal numbers with vanishing phase such that CL
-Lo=cM-M°=Omod v2.
(2.2)
The lagrangian operators aL2 and am have the following expressions ail and am' in R° by I,§1,3 and the formula 1
O0x,
eXp2v
a
2
ap)] L
1
(x, p) =
C1 +
2v
(') a
a
1
+ 8v2
a
a
ap)2]
ax
.(p2R2 - Q2)
=PZRZ - QZ -
2Q- 2v2 3 V
namely aL2(v' x' vex
v (A0
2,J
(2.3)
211
III,§3,2
where (4) 2
a =R2A->xx-2zx. axe j, k aXi OXk k
i
is the spherical laplacian (2.24), which acts on the restrictions of functions
to spheres R = const.; =
+
aM (V,
x, v8X )
V
xl aX2 -
x2aX
which is an infinitesimal rotation. Let us require that the unknown U be defined mod(1/vr+') on a compact lagrangian manifold V. By theorem 3 of §1, i. V is necessarily one of the tori
V = V[L0, M0] = T(l, m, n); H = L2 - L2 = M - M0 = 0, which part 1) of this theorem defines. ii. The lagrangian amplitude /30 of U is necessarily constant. If (30 = 0, (2.1), reduces to (2.1)r_1: hence we impose the condition
J30 = const.: 0.
(2.5)
The problem defined by the system (2.1),, condition (2.5), and the condition that V be compact will be called problem (2.1),. Notation. The invariant measure is given by (3.16) of theorem 3 of §1; d3x by (1.16) and (3.6) of §1; arg d3x = nmRO by definition [(3.4) of I,§3]; MR. is given by (3.5) of §1; arg rl, = 0 by definition; hence the value of the
function x, defined and used in II,§3, and the value of its argument are, respectively,
x=
?IV
= [R3HQ[LOi M0, Q, R] sin 'Psin ®]-', where O = const.;
dr
arg x = - arg HQ - arg sin T, where arg H. = -n[(1/n) arctan(HQ/HR)], arg sin 'P = n['P/n]. (Recall that [ ] is the integer part.) The apparent contour of V is 'As usual: 0 = Ej(a/ax;)Z.
III,§3,2
212
ER°: HQ sin `I' = 0; thus X: V\ERO - R.
(2.7)
Let U be a lagrangian function on V with lagrangian amplitude
f30 = const.: 0. Its expression in Ro will be denoted UR0 (v) = fZ-#(v)e"R-, where fl(v) _ Y
#R
.
2.8)
Vs
By the structure theorem 2.2 and definition 3.2 of lagrangian functions in II,§2, the function X-3,135: V
C
is regular even on ERO
Let Dt,, DL, and DM be operators such that LaU]RO = V A e"-PR, DHl3, V
L(ae - Lo) U]R" =
V X eMR. DLQ, V
DH, DL, and DM commute since a, a,!, and am commute. Let us use the local coordinates (R, `P, (D) on V\ERO. LEMMA 2.1.
DM =
1°) With this choice of coordinates,
8
(2.10)
a(D.
2°) If U is a solution of the equation
(am - cM)U = 0 mod r1 2, v
where cm is a formal number with vanishing phase such that cm = MO modhv,
(2.11)
213
III,§3,2
then 1
cm = M. mod
r+21
$ depends, mod(1/v'+'), only on the coordinates (R, `I').
Proof of 1°). Let us calculate DM by using theorem 4 of II,§3: in this theorem, the hamiltonian M - Mo is substituted for H. By (3.18) of §1, the characteristics of this hamiltonian are given by the equations
dt=dT=0;that is,dR=dP=0. The parameter of these characteristics is b, which is substituted for tin this theorem. We obtain (2.10).
For r = 0, 2°) is obvious. Proceeding by induction on r, we can assume 2°) is true when r is replaced by r - 1. Then Proof of 2°).
cM=Mo+Mr+1
M,+1
C.
By (2.9) and (2.10), (2.11) is equivalent to
M,+i f o ,where J3o = const.: 0. Now, fi, is a function of (D having period 27r; thus
M,+1 =0,
00&
=0,
from which 2°) follows.
Notation. From now on, we assume that (i depends only on the variables (R, `I'). By (2.10), DH and DR° commute with a/a(D and thus act on functions
of(R,`I'). LEMMA 2.2.
DL $
= 2L o
1°) In the local coordinates (R, `P, (D), d-r
d`If
ra _ F1 1
aT
where T is the v ariable
T = cot F
v
J
_s X 4v
(2 . 12)
III,§3,2
214
and F is the operator with polynomial coefficients given by
FQ = F, /i +
OT
[F2
F, and F2 being the polynomials in T given by
5M2x2 Z LO +M0 8Lo(Lo Mo2)
F1 (z)
F-2 (T = (MO T2+ LO)(T2 +1). )
2Lo(L2O
-
M02)
2°) Let U be a lagrangian function that is defined on the torus Y [LO, M0],
has a lagrangian amplitude Qo j6 0, and is a solution of the system
(aL2 - cL) U = (am - MO) U = 0 mod,
(2.13)
where CL is a formal number, with vanishing phase, such that
cL = L p mod V2'
Let E be the set of points on the curve F[LO, M0] [(2.5) of §1], where
E:HQ=0. Then
cL = Lo - 4v2 mod v,+2
(2.14)
and fl(v, R, `F) = g(v, R)f(v, T) mod
vr+j,
(2.15)
g being some formal function with vanishing phase defined on
I'[L0, M0]\E and f being defined as follows. 3°) There exists a unique formal function f defined on R, of the form f (v, T) _
sf, (T), where r e V
such that
(2.16)
215
III,§3,2
df = 1 Ff' dT
(2.17)
V
fo = 1, fs is a real polynomial in z, of degree 3s, having the parity of s, (2.18) and
= 1 + vF2(IdT
- fd }
(2.19)
[I is the complex conjugate off ; (2.19) expresses f2s using f1,
1 f2s-11 .
Any solution of (2.17) mod(l/v') is, mod(1/v'), the product of f and a formal number with vanishing phase. Remark 2. Let U' be the formal function of x that is homogeneous of degree 0, is defined for
M0R < L0'/xi + x2, and is given by f(v ) ev(LOT+Mom)
Uvx=
('
\/sinq
)
(2.20) '
U' satisfies
(Xi-a V
2
- x2 as )U
vI2
0U' = R2 (Lo + 4v2) U'.
1
(2.21)
Proof of 1 °). Let us calculate DL using theorem 4 of II,§3 : in this theorem the hamiltonian L2 - Lo is substituted for H. By (3.17) of §1, the characteristics of this hamiltonian are given by the equations
dt = dD = 0; that is, dR = dQ> = 0. The parameter of these characteristics is `Y/2Lo, which is substituted for t in this theorem. The right-hand side of formula (4.5)2 in this theorem has to be replaced by
1/a a\l
exp2 C
2
r
ax' ap/ J L (x' P) = Ll +
1 /a a\ 2
x' ap
1/a a\2l + 8 ax ap/ J
(P2R2-Q2)=P2R2-Q2-2Q-32
216
III,§3,2
This theorem gives DLL = 2Lo a# + 1 [x_1/2A0x1/2)
- 2 fl],
(2.22)
where A° is defined by (2.4) in terms of the coordinates (x1, x2, x3). By (1.5) and (1.12) of §1, we have R
a OR
=
a
; thus R2 a22 J=1 'axJ 8R x
=
02
2: XJxk
Jk
8xJaxk
(2.23)
using the coordinates (R, T, (D) in the left-hand sides and the coordinates (x1i x2, x3) in the right-hand sides. Then definition (2.4) of 0° is formulated as
0
°
R2A-R2
az
2R-a.
Mk'
(2.24)
oR
Now, formulas (1.5) and (1.12) of §1 give
R = - sin O cos `P, where cos O =
MO
by §1, (1.11);
0
hence, on V, KRX, '1'
J = 0,
R20'P = cot
R2
(1
\T
`
,
W
)= /
1
+
Cot 2 O
`P '
- cot201 sin2'P
The expression for R 2 t acting on functions of (R, `P) follows. Substituted into (2.24), it gives Cot
0° =
C1 +
2
0)
2
T a.j + cot `P1 - sm2 P)
(2.25)
on these functions; hence, from (2.6), X-112Ao(NX112) = ,/sin `P Do (1V
I sT.
-2L°dTFfl
+ a13,
by a trivial calculation. Thus (2.12) follows from (2.22).
(2.26)
III,§3,2
217
Proof of 2°). By lemma 2.1, /3 locally depends only on the variables (R, `P). By this lemma and (2.12), fi° depends only on R; by (2.5), )3o is not identically zero.
For r = 0, 2°) is evident. Proceeding by induction on r, we can assume that P°, ... , Pr_, are polynomials in r whose coefficients are functions of R and that 2°) is true when r is replaced by r - 1. Then CL
2L
5
L°2
v0Lr+1 modv,l 2,
4v2 +
where L,+1 E C;
thus equation (2.13)1 may be written
DC
Vc# +
4i,2
- 2Lr+1r+1 1 33 = 0 mode,+2 o
(2.27)
By the induction hypothesis, this equation holds mod(1/v'+ 1); thus, by (2.12), it may be written duir = (Fflr_ ) dT + L,+ 1 fo dT for R = const.
Then, since F is an operator with polynomial coefficients, fl, is the sum of a polynomial in T = cot W and the function Lr+ 11 P, where f o # 0. But 13r is a function of IF with period 21r. Hence Lr+ 1 = 0,
Fir
is a polynomial in T;
thus (2.27) may be written df3 =
(FfJ) A mod v,l for R = const. 1
If 3°) isv true, (2.15) follows.
Proof of 3°). written
fo = const.,
The condition that (2.16) satisfy (2.17) mod(1/v') may be df, dT
= Ff,-1 for s = 1, ... , r - 1.
Thus f, is a polynomial of degree 3s in r containing an arbitrary constant of integration. Then f is well defined up to multiplication by a formal number with vanishing phase. Let us choose the constants of integration to be real. Then the f, are real.
III,§3,2
218
There exists a unique choice of the constants of integration of the f 2,- 1 such that the f, have the parity of r. Proof of (2.19). Let f be the complex conjugate off. Since v is purely imaginary and F is real, (2.17) implies dT
v
dT d
FI,
(f) = v (IFf - fFf
that is, by the definition of F,
d(ff)= dT
v
dT
(F2L) dT
-f
1[WdT
AdCF2df dT
v
F2Cf4I
dT
- fdlll. dT JJ
thus
if -
1
F22
v
T-
f df dT
c
=
,
where co = 1,
SseN ENV
that is, since the f, are real, for all s e N c2s+1 = 0, 2s-1
2s
Z
d
2F2 Y (-1)sf,'dTf2s-1-s' + C2, ,'=o
s =o
If s > 0, this formula expresses f2s using f1, ... , f2s-1 and c2s, which is cancelled by an appropriate choice of the constant of integration of f2,. Proof of (2.21)2.
DJ
By (2.12) and (2.17),
4vf'
that is, by the definition (2.9) of DL,
[a2(vx,
- L+
1(f fe^)
0
or, bythe expression (2.3) for at= and the expression (2,6) for x, since Do acts
on the restrictions of functions to spheres R = const. and since cpR can be expressed by (3.4) of §1, where Q only depends on R, CV2
Do - Lo -
4v2} U'(v, x) = 0
219
III,§3,2
where U' is defined by (2.20); U' is homogeneous of degree 0 in x; (2.21)2 follows from this by the definition (2.24) of A0. Proof of (2.21)1. From the definition (2.16) off and the expression (2.10) for DM it follows that
DMf = 0, or, by the definition (2.8)-(2.9) of DM,
(a+ - Mo)(f
.fe"w"")
= 0.
In the beginning of section 2, we obtained
-1( xl x2
+ am
V
-
X2
-8x° J / 1
Thus a;, annihilates functions of the single variable R. Then by the expression (2.6) for X and the expression (3.4) of §1 for (PR., the preceding relation is equivalent to (2.21)1.
There exists an operator D, acting on formal functions g
LEMMA 2.3.
defined on ['[LO, Mo]\E, such that DH[f(v, `P)g(v, R)] _ .f (v, `V)Dg(v, R).
(2.28)
Locally,
D=
D.' R, d M .N y` 1
,
(2.29)
where DS(R, d/dR) is a differential operator defined on ['[L0, M0]\E and Do = RHQ
Proof.
Since DH commutes with DL, which is expressed by (2.12),
d'I' Ca = DH
(2.30)
dR
V
Fl DH [ f (v,'I')g(v, R)]
[dz
d`I' O-T
- 1 F .f (v, `')g(v, R)0. v
Thus by lemma 2.2,3°), DH[ fg] is multiplication of f by a formal function, which is defined on ['[L0, M0]\E and denoted Dg.
220
111,§3, 2
Theorem 4 of II,§3 proves that D has the form DH = sGN vS
DH,., (R,
T, 8R , 8`I')
where DH S is a differential operator. It follows that D has the form (2.29). Since the characteristics of H satisfy (2.14) of §1, the same theorem proves
that
D1(fg)dt = d(fg) mod I for dR = RHQdt, dP = HLdt. But
f=1mod!; V
thus
DNg = RHQ dR mode . This relation is equivalent to (2.30).
10) Problem (2.1)r, defined at the beginning of section 2, is solvable if and only if THEOREM 2.
CL
=Lo -4v2,
cm =Mo.
2°) Problem (2.1), is equivalent to problem (2.31)r, which is stated as follows: To define a formal function mod(1/vr+1) on F[Lo, Mo]\E, g(v) _
1
sEN V
gs, where go = 1, gs: F[Lo, Mo]\E -+ C,
such that
Dg = 0
mod(1/vr+'),
(2.31)r
(HQ)3rg is regular, mod(1/vr+1), on F[Lo, Mo]. Any formal function satisfying condition (2.31)r is, mod(1/vr+l), the product of g and a formal number with vanishing phase. The condition that g be a solution to (2.31)r evidently is equivalent to the following :
III,§3,2-III,§3,3
221
g is a solution of problem (2.3
RHQdg,/dR + E;=1 D,g,-, = 0 on r[L0, M0]\E,
(2.32),
(HQ)3rg, is regular on r[Lo, M0].
On r[L0, M0]\E, define the formal function e"n
U"(v) = /R3HQ, g where S2 is defined by (2.8) of §1. Define U' by remark 2 and lemma 2.2. Then
U'(v)U"(v) is the expression URo of a solution U of problem (2.1)r. Every solution of the system (2.1), defined on V[LO, M0] is, mod(1/vr+l), the product of that solution and a formal number. 3°) Suppose that g is a solution of problem (2.31)r_1. Then there exists a
function gr: r[L0, MO] \i -* C satisfying (2.32) where r is replaced by its universal covering space 1'". The function g, is defined up to an additive constant; moreover, gr is defined on F[L0, Mo]\E if and only if the equivalent problems (2.1), and (2.31), are solvable.
Proof of 1°).
Use lemmas 2.1,2°) and 2.2,2°).
Proof of 2°).
Use (2.6), (2.8), lemmas 2.1,2°), 2.2,2°), and 2.3, and theorems 5 and 7.2 of II,§3.
Proof of 3°).
Use the above theorems.
3. A Special Case
In order to supplement theorem 2, choose
H[L, M, Q, R] = 2 {P2
=
R2
K[R, M] 5
2RZ {L2 + Q2 - K [R, M] }
as in §1,4, and choose K to be an affine function of M. Then condition (1.1) is satisfied. By (4.5) of §1, the expression in Ro of the operator associated to
His a
2v2A - 2RZKLR,v(x18x\\
2
x2i?x)j.
1
(3.2)
III,§3,3
222
Moreover,
Lo + Q2 - K[R, Mo] = 0 on the curve I,[L0, Mo].
(3.3)
E is the set of points on this curve where Q = 0. LEMMA 3.
D
R
We have
[dR
V
G]
where G acts on functions g:11[Lo, Mo]\E -+ C and is given by
Gg=G19+dR[G2dR] G1 and G2 being functions defined on I'[L0, Mo]\E given by Gl(R, Q) = GQR),
G2(R, Q) _
-2 Q ,
(3.6)
where 32R(KR)2 + g(K - L2)(KR + RKR2).
G3(R) =
Proof. The operator D may be described as follows with the aid of II,§3, theorem 4: in formula (4.5) of this theorem, s = 2 and H(2)(X'
2 L eap
ox ' ap
P) =
[
2
=
2 P2;
(3.7)
the equations (2.14) of §1 giving the characteristics imply that
dR =
dt,
d'1' = L0 dt;
R then, by the definition (2.9) of D11, when /3 only depends on the coordinates (R, Yf) of V, this theorem gives DH/3dt = d/3 + -1 X-1r2A(fX112)dt
I1I,§3,3
223
for dt, dR, and d`' satisfying (3.8); in other words, /3 Q
__
/ _LO
2vX '120(QX'n
OR R + 8YJ RZ +
DHR
By (2.6)
O = const.;
x = [QR sin IF sin O]
therefore, by the definition (2.24) of Do, which acts on the restrictions of functions to spheres R = const., X
112 0(flX112)
sin
= RZ
A. is-in -IF
R
R(8RZ
+
OR)(/QR).
Replacing Do by its expression (2.26) and substituting the result into (3.9) yields DH
02 QR (8R2 2v
l - QoQ R OR +
LzdT(8 +R d'Y
8r
8 2
Q
+R
1 )1 2 vF+8vR
Choose
)(v, R,'1') = f(v,'Y)g(v, R), where f is the formal function of 'Y defined by lemma 2.2 and g is a formal function defined on I'[L0, Mo]\E. In accordance with lemma 2.3, we obtain
DH(fg) = fDg with Dg _ _ Q
9
R IdR +
2v IQ (RZdRZ + 2R dR/ (
QR) + 8v QR1.
By a trivial calculation, we deduce the expression (3.4) for D, with G expressed by (3.5) and G1 and G2 given by G1
_1d RdQ 4 dR [QZ
1R dQz
dR + 8 Q3 (dR)
GZ
__1R 2Q
224
III,§3,3
Now by the equation (3.3) for F[Lo, Mo],
Q2 = K[R, Mo] - Lo hence the expression (3.6) for G, follows.
The preceding lemma permits a statement of the properties of problem (2.3) to which problem (2.1), has been reduced by theorem 2.
A function g: F[L0, Mo]\E - C is said to be even or odd
Definition 3. when
9(Q, R) = ±g(-Q, R)
V(± Q, R) E I'[Lo, Mo]\E.
(Complement to theorem 2) Assume (3.1). 1°) If problem (2.31), has a solution g such that go = 1, then it has a
THEOREM 3.1.
unique solution such that
gs is real and has the parity of s (s < r),
go = 1, and
99 = 1 -
I
2v
do) mod dRdRvr+,. -g
R (9 dg
1
(3.10)
Q
Formula (3.10) means that, for 2 < 2s < r, 2s R 2s-1 d
sE
(-1)5925 s95 =
Q
S'o(-1)sg,'dR92s-1-5;
(3.11)
thus it expresses 92, using g1, ... , 92,-1
2°) If problem (2.31)2,_, has a solution, then problem (2.31)2, has a solution.
Remark 3. The formal function U" defined on F[Lo, Mo]\E by theorem 2,2°) is evidently given by
U"(v) = g
QR
e °o .
( 3 . 12)
It satisfies
0 U"(v R) = R Z { K [R Mo] - Lo - 4v2 U (v, R), where A is the laplacian, specifically (d 2 /dR 2) + (2/R)(d/dR).
(3.13)
III,§3,3
225
Proof of 1°). Assume that 1°) has been proved when r - 1 is substituted for r and that problem (2.3)r is solvable. Then, by theorem 2,2°) and lemma 3, gr is defined up to a constant of integration by the conditions dg,
Q3rgr is regular on F[L0, M0].
= Ggr_1 on F[L0, Mo]\E,
(3.14)r
Now, G changes parity. If r is odd, then a convenient choice of the constant of integration makes gr real and odd. If r = 2s is even, then 92, is even and can be chosen to be real. We have dg
_
dR
modvrl1,
Gg=G19+dR[G2dR]
1
By a trivial calculation analogous to the proof of (2.19), we deduce (3.10),
up to the addition of a formal number E,(c,/v`), which is canceled by conveniently choosing the constants of integration of the g2,-
Proof of 2°). Assume that r = 2s is even and that problem (2.31)2s_1 is solvable. By theorem 2,3°), problem (3.14)2s has a solution g2s when t is replaced by its universal covering space t. The proof of (3.11) remains valid; (3.11) proves that 92s is defined on F[L0, Mo]\E; thus 92., is a solution of problem (3.14)2..
Proof of remark 3. By the expression (3.2) for a and theorem 2,2°), {v2A
RZK[R'v
z
xzax
)
][U(v,x)U (vex)]
0,
1
where U' is homogeneous in x of degree 0 and U" only depends on R. This implies
A(U'. U") = U'- AU" + U" - AU' ; hence (3.13) follows by (2.12)2.
Let [R 1, R21 c R c C be the set of values taken by R on F[L0, Mo]\E. Let o. be a simply connected neighborhood of the real Notation.
closed segment [R 1, R21 in the complex plane C; R E C. THEOREM 3.2.
Assume that K is holomorphic in co. Then Q, defined by
(3.3), is holomorphic in (t)\[R1, R21.
III,§3,3-III,§3,4
226
1°) If problem (2.31)25 has a solution g, then, for all r _< 2s,
Q3i94, R) = (-Q)3'g.(-Q, R) is the value of a junction of R that is holomorphic in w. In particular, 92, is a function of R that is meromorphic in w, with poles R1 and R2. 2°) Problem (2.31)25+1 is solvable [and thus problem (2.31)25+2 also] if and only if the primitive of (Gg2i)dR is defined (that is, uniform) in w\[R1, R2].
Suppose 1°) is true. [1°) is evident for s = 0]. Then, by (3.14), the function g25+1: I'[L0, Mo]\t - C is obviously the restriction of the primitive of (Gg2s)dR to the edges of the cut [R1, R2] in C, which is Proof.
defined on the universal covering space of w\[R1, R2]. Obviously, 925+1
is defined on I'[L0, Mo]\E if and only if this primitive is defined on w\[R1, R2) Assume that this condition is satisfied. With a convenient choice of the constant of integration, this primitive 925+1 is an odd meromorphic function of Q in a neighborhood of R1 and R2. Like Q, it takes opposite values on the edges of the cut [RI, R2]. For R near R1 or R2i Q is near 0 and g25 is an even function of Q having a pole of order 6s at Q = 0. From (3.3), (3.5), (3.6), and (3.14),
ddQ 1 = dQ G925 = 2-93-n 925 la - 4dQ[ RKR Q ddQ2s
,
where KR
0.
Thus g25+1 is locally an odd function of Q having a pole of order 3(2s + 1)
at Q = 0. Consequently Q3c2s+ng25+1 is holomorphic in w.
g25+2 is defined by (3.11). Then Q3(2,+2 '92,+2 is holomorphic in a). Consequently 1°) holds when s is replaced by s + 1. 4. The SchrSdinger-Klein-Gordon Case
In order to establish, for any r, the solvability of problem (2.1) which we have reduced to problem (2.31) an appropriate assumption is clearly necessary. Assume that K is a second-degree polynomial:
K[R,M] = -R2A(M) + 2RB(M) - C(M).
227
III,§3,4
In other words, the expression of a in R0 is the Schrodinger-Klein-Gordon operator (§1,4) and A, B, and C are affine functions of M. Then in theorem 3.2, co = C. In the definition (3.5) of the operator G,
G3 is a polynomial in R of degree 3 and G1 and G2 are functions of R holomorphic in C\[R 1, R21 and at infinity, where G1 vanishes to second
order. If g is holomorphic in C\[R1, R2] and at infinity, then so is the
primitive of (Gg)dR. Then all the g, exist and are holomorphic in C\[R1i R21 and at infinity. Since g, is holomorphic at infinity and since Q 3rg, is holomorphic in C, Q 3'g, is a polynomial in R of degree 3r. Thus we
have proven the following two theorems. (Existence and uniqueness) Assume that the expression of a in R0 is the Schrodinger-Klein-Gordon operator (example 4 of §1). 1°) The condition that the lagrangian system THEOREM 4.1.
aU = (aL2 - CL)U = (am - cM)U = 0,
(4.1)
where cL and cm are two formal numbers such that
= v2, o-cM-M0=0mod 2
CL
1
have a solution defined on a compact lagrangian manifold V is the following:
i. CL = Lo + (1/4v2), CM = MO ;
ii. V is one of the lagrangian tori V [L0, M0] = T(l, m, n) defined by theorem 4.1 of §1.
2°) There exists a lagrangian solution U of (4.1) defined on such a torus V having lagrangian amplitude
#0=1. Any lagrangian solution of (4.1) defined on V is the product of U and a formal number with vanishing phase. Remark 4.1. Vx : R 1
The projection of V on X is
x < Rz>
MoJxJ < Lo
xzl +
xzz,
where R1 and R2 are the two roots of the equation
AOR2-2B0R+C0+Lo=0
III,§3,4
228
(A0, Bo, and Co are the values of A, B, and Cat Mo.) (Structure) There exists a unique solution U of (4.1) defined on such a torus V having the following structure: Its expression THEOREM 4.2.
UR,, in the frame Ro (§1,1) has the form (4.3)
U& M = U'(v) U"(v).
Using the local coordinates x e V, on V, U'(v) is a formal function of x that is homogeneous of degree 0 and satisfies
V
I x1
a z
- xz as
1 U'(v , X )
= M U'( v, x ) , O
i (4.4)
V2
AU
Rz l L0 + 40 J
(v' x)
U'(v, x).
U"(v) is a formal function of R satisfying x)
V
=1
{K[R,
M0] - Lo _ 4vz} U"(v, x).
4.5)
U' and U" are defined by the formulas
U' v x (
'
fv' t)ev(r.ovV+Mom)
)
,sin`'
'
(4.6)
U"(v, x) = g(v, Q, R)evo, QR
where T = cot `P and Q is the function of R defined by (2.8) of §1; arg sin 'Y
and arg Q have jumps of +n at the points 'P = 0 mod 7r on R and Q = 0 on ['[L0, M0], which are oriented in the directions dP > 0 and Q dR > 0. Moreover,
f(v) _
?r rEN V
g(v) =rfNI .rgr
fr
are formal functions with vanishing phase defined on R and on I'[L0, M0]\E, respectively, by the following set of properties:
df = 1 Ff, dT
v
dg
dR
=
1
v
Gg,
(4.7)
III,§3,4
229
the differential operators F and G being given by Gg
Ff = F1J + dT[Fzd-r
G1g
+ dR
[G2]
where F1 and F2 are the even polynomials in T of degree 2 and 4 defined by lemma 2.2,1°); QSG1 and QG2 are the polynomials in R of degree 3 and 1
defined by (3.6); fo = 1, go = 1; the functions f, and g, are real and have the parity of r;
Jf =
1
+
1
Fz
v
f d f - f df ) C
dT
dT
99 =
1
+
1
v
G2
(
dg
dg 9
dR -
9
dR (4.9)
These formulas (4.9) give the even functions f2, and 92, in terms of fl, , f2,-1 and g1, .. , 925-1. The odd functions fzs+1 and 92,+1 are defined by the quadratures df2 +l = (Ffzs)dT,
dg25+1 = (Gg2,)dR.
f, is a polynomial in T of degree 3r. Q3rg, is a polynomial in R of degree 3r. Remark 4.2.
Thus g25+1 - G2dg2 /dR
is
the odd function
on
F[L0, M0]\E that is the primitive of the differential form G1g2,dR, namely, of a form II'(R)Q-65-sdR, where IT is a polynomial of degree II(R)Q-6,-3 6s + 3. By the preceding theorem, this primitive shall be where fI is a polynomial of degree 6s + 3. Let us give a more direct proof of this essential fact. LEMMA 4.
d fl(R) dR Qzs+,
For each s e N, the differentiation
- fI'(R) Qzs+3
(4.10)
(Q2 a polynomial in R of degree 2, discriminant Q 0) defines an automorphism fI " IT of the vector space of polynomials of degree 2s + 1.
Let Fl be a polynomial of degree 2s + 1; fI(R)Q-2s-1 is holomorphic at infinity; thus its derivative has a double zero at infinity. Proof.
Consequently (4.10) defines a polynomial fI' of degree 2s + 1. Hence the mapping
II " rI'
230
III,§3,4-III,§4,O
is an endomorphism of a finite-dimensional vector space. Now it
is
evidently a monomorphism; thus it is an isomorphism. Conclusion
This §3 is concerned with finding, for dim X = 3, a lagrangian system, with one lagrangian unknown, having a solution defined on a compact lagrangian manifold unique up to a multiplicative factor. In this §3, we have found a system of this type: the system used in wave mechanics for studying atoms (or ions) with a unique and spinless electron.
Replacing the Schrodinger-Klein-Gordon hamiltonian by the harmonic oscillator hamiltonian for E3, Remark 4.3.
H(x,p)2 [P2+AR2-2B], u
that is, choosing
K[R, M] = -AR4 + 2BR2 where A and B are affine functions of M, one obtains another lagrangian system (2.1) belonging to the same type. This can be shown by computations similar to those performed above, where R2 plays the role that R played above. The energy levels are still those of wave mechanics. Remark 4.4.
Of course, the study of the harmonic oscillator in R is
simpler; in E3, for A and B independent of M, it gives new lagrangian
solutions for the operator associated to the harmonic oscillator, but again the same energy levels.
§4. The Schrodinger-Klein-Gordon Equation 0. Introduction
The following classical boundary-value problem will be called problem (0.1): To find nonzero square-integrable functions
that have square-integrable gradients and that are solutions of the differential equation
I1I,§4,0-I1I,§4,1
231
au=0,
(0.1)
where a is the differential operator associated to the hamiltonian
H[L, M, Q, R] = 21 P2 - R2 K [R, M]}
(0.2)
,
K being an affine function of M [compare (3.1) and (3.2) of §3]. This operator is defined by substituting vo = i/h for v in the lagrangian operator associated to H [(4.5) of §1]. Hence it is a
2V2
-
2RZKrR,
o(x1ax2
- x2az1)]
In section 2, we assume (compare §3,4)
K[R,M] = -R2A(M) + 2RB(M) - C(M),
(0.4)
where A, B, and C are affine functions of M. Then a is the SchrodingerKlein-Gordon operator. In §4, we briefly recall the solution of the classical problem (0.1) in order to observe two facts:
i. There are formal analogies between its solution and that of the lagrangian problem (2.1) of §3, which is to solve
aU = (aL= - CL)U = (aM - c;,i)U = 0 on compact V. ii. The condition for the existence of a solution to this lagrangian problem [or to this problem mod(1/v2), namely, to (3.1) of §1] is the same as that for the classical boundary-value problem (0.1) in the Schrodinger-KleinGordon case, that is, under assumption (0.4). This assumption is essential. Remark 0.
For a suitable choice of A, B, and C, the Schrodinger-
Klein-Gordon equation is the Schrodinger or Klein-Gordon equation (4.21) or (4.22) in § 1, where the
.
2 terms have been omitted. It is customary
to treat the terms of these equations that are linear in .' by "perturbation" theory. We do not do this, but we rigorously solve problem (0.1) in order to show that assertion (ii) is rigorous. 1. Study of Problem (0.1) without Assumption (0.4) Review of the properties of spherical harmonics ui,m.
The set of poly-
nomials in x c- E' that are harmonic and homogeneous of degree 1 is a
232
III,§4,1
vector space over C of dimension 21 + 1. A basis consists of the polynomials I and m integers,
R'ui,m,
lml '< 1,
defined by the following system, where ul.m is homogeneous of degree 0 and thus satisfies R2 Au' = A0u' [see (2.4) and (2.24) of §3]:
Aui.m + R21(l + 1)u',, = 0,
I x1
- x2ax )u',m
axa
\
= imui.m.
i
z
(1.1)
Let S2 be the unit sphere in E3 and let a be the usual measure on S2. Then o J'S2
2
, Ul,m
OX
f
a = 1(1 + 1)
IU.ml2a
S2
and for all (1,, ml)
(112, m2)
S/2
J
x Ul2'm2/ O = fS2 U13,m1Ul2,"2Q = 0.
a
, > is the (sesquilinear) scalar product. The restrictions of the ul,m to S2 form a complete system of functions on S2 : every square-integrable function u : E3 -+ C with square-integrable gradient has a unique expansion
where
0, such that u12d3x = Y
114",mlafo
R2luml2dR < o0
l,mJ. S2
JET
and a
L
ax
a,
2
u
d3x =
J +E
I'm
l.m
2
OX
Ul,m
2
a. J
ml2a
I
I
U! 2
Jo
+ ac
u,%ml2dR
R2
2
d
dR
Ul,m
dR 0 for s near + oc. Let E E 10, a[. From (2.5) we have
scs > 2ec5_ 1 for s near + oe ;
thus ESc,R' > c'e2,1 for R near + oc, where c' = const. > 0, and the integral (1.7)1 diverges.
III,§4,2
236
Case when az < 0: The classical integral expression for v gives an asymptotic expression for v which is also classical: for R near + oc, v(R) = ceaRR-Pla + ce-aRRP1a + .
.
where c and care constants and a is purely imaginary. Hence the integral (1.7)1 diverges. (See Whittaker and Watson, [19], chapter XVI, "The Confluent Hypergeometric Function: Asymptotic Expansion.") Case when a
= 0, f < 0:
v(R) = Ry' 1/2
By (2.5),
c,R5, where cs > 0
Vs;
sEN
thus the integral (1.7)1 diverges.
Case when a = 0, $ > 0: R-r-'/zV satisfies a differential equation with linear coefficients. Laplace's expression for its solution gives
v(R) _v k JT I t-2y-'e`2 R(-t-')dt where T is the boundary of a half-strip in C containing the cut I - oo, 0[.
By replacing T by a path on which Re(t - t-') -< 0 (=0 only for t = i) and assuming that R is near + oo, we obtain the asymptotic value
v(R) =
R+ ce-2i,/2%1R] + ...
Thus integral (1.7)1 diverges.
Theorem 1.2 and lemma 2, where a and $ are defined by (2.1) and y by (1.6), prove the following.
When a is the Schrodinger-Klein-Gordon operator, then, for each of the following problems, namely, THEOREM 2.
1°) the classical problem (0.1) that we have just reviewed, 2°) the lagrangian problem (2.1) in §3,
aU = (aL: - cL)U = (am - cM)U = 0 on compact V, 3°) the same problem mod(1 jv2), that is, problem (3.1) in 41,
the condition that there exist a solution is the same: the existence of a triple of integers (l, m, n) satisfying condition (4.11) of §1, theorem 4. Remark 2.
A comparison of theorem 1.2 and theorem 3.1, 1°) of §1
III,§4,2-1I I,§4,Conclusion
237
proves that the preceding theorem does not hold for every operator a associated to a hamiltonian H of the form (0.2). Conclusion
Although the classical boundary-value problems and the lagrangian problems are completely independent, they define the same energy levels for the Schrodinger and Klein-Gordon equations. As a matter of fact the experimental values of the energy levels agree with those of the Dirac equation, which is studied in the following chapter.
I
Dirac Equation with the Zeeman Effect
Introduction In §1, we solve, mod(1/v2), a homogeneous lagrangian problem in several
unknowns. That is one of the simplest problems to which theorem 3 of II,§4 can be applied. The resolution of this problem introduces the quadruple of quantum numbers that arises in the study of the Dirac equation. In §2, we use lagrangian analysis to reduce the Dirac equation mod(1/v2)
to the simpler system solved in §1. This reduction is analogous to the reduction theorem 2.2 of II,§4. Thus we prove that the lagrangian solutions
of the Dirac equation (one-electron atoms in a magnetic field) defined mod(l/v2) on compact manifolds have energy levels that are exactly those for which the classical solution of this equation exists (taking into account the Zeeman effect and even the Paschen-Back effect).
§1. A Lagrangian Problem in Two Unknowns In this §1, we give an application of II,§4,theorem 3. 1. Choice of Operators Commuting mod(1/v3) As in III,§1,1, we let
X = X* = E3, R(x) = Ix
x,pcE3,
P(p) _ Cpl,
Q(x, p) = ,
L(x, p) = Ix A Ph
so that
L2 + Q2 = P2R2.
(1.2)
Let (M1, M2, M3 = M) be the components of x A p. By III,§l,(1.3), the following formulas define three functions H"1, H« and H13j that are pairwise in involution (see [1I,§l,2): H")(x, p) = H[L(x, p), M(x, p), Q(x, p), R(x)], (1.3) H13 (x, P) = L2 (x, p), P) = M(x, P) j121, To use theorem 3 of I1,§4, we must choose three it x it matrices and J(3) that are functions of (x, p) c E3 O+ W. These must be such that the matrices of lagrangian operators associated to the three matrices
H'k)E +
J'k)
(E the p x p identity matrix)
(1.4)
IV,§ 1,1
239
commute mod (1 /0). Since the H(k) are in involution, it follows by remark 3 of II,§4 that this commutivity condition is equivalent to the condition that for all i, k,
(H(h J(k)) - (H(k) J( )) + J(n J(k)
- j(k) j(i) = 0,
(1.5)j k
where (, ) is the Poisson bracket. We choose p = 2. Then the values of the J(k) are matrices acting on vectors in the space C2, which we provide with a hermitian structure. We choose the iJ(k" to be self-adjoint. (The vectors in C2 are called spinors.)
Notation.
Let ao = 1 be the 2 x 2 identity matrix. Every 2 x 2 self-
adjoint matrix has the form x0c0 + or, where xo is a real number and a is a self-adjoint matrix with zero trace:
x1 - ix2
x3
Q = Q[x1, X2, x3]
x1 + 1x2
-x3
= x1Q1 + x262 + x363,
(1.6)
where (x1, x2, x3) e E3 and
Q1 =
(01
10),
62 =
(0
i
1
00,
Q3 = (0
--
)
are the Pauli matrices, which satisfy the relations
Qk = 1,
Cork = -ak(ri,
Q1Q2Q3 = i.
(1.7)
It follows that
Q2[x] _
N2.
(1.8)
Remark 1.1. Recall the consequences of (1.8): Let u2 be a measurepreserving automorphism of C2, that is, u2 E SU(2); u2Q[x]u2 1 is self-
adjoint and has zero trace; thus it is of the form a[y]. Let y = ux. Then
Q[ux] = u2Q[x]uj By (1.8), u preserves Ixl2. More precisely, u is a rotation in E3, that is, u E SO(3), whence we obtain a morphism SU(2) 3 a2 H u e SO(3),
IV,§1,1-IV,§1,2
240
which proves that SU(2) is the covering group of SO(3) of order 2. It is the universal covering group of SO(3) (see Steenrod [17], p. 115). LEMMA 1.
Assume
J(1j = if a3 + iga[x A p], is a datum, where
f(x, p) = f [L(x, p), M(x, p), Q(x, p), R(x)],
g = g[L, M, Q, R].
Then (1.5) is satisfied by choosing J(2) = 0,
P) = - 2Q3.
Remark 1.2.
Relations (1.5) evidently remain valid when arbitrary com-
(1.10)
plex numbers (that is, their products with the identity matrix denoted by co = 1) are added to the H)`) and P'`). (H12!, J(1)) = 0 because, by III,§1,1, L is in involution with Q, R, and M = M3, and similarly with M1 and M2. Proof of (1.5)1.2.
Proof of (1.5)1,3. (H)3), J(1))
Since M is in involution with L, Q, and R,
= ig(M, a[x A P]) = iga[-M2, M1, 0]
(1.11)
because an immediate and classical calculation gives
(M. M1) = -M2,
(M, M2) = M1.
Moreover, (1.7) implies
a3a[x A p] - o[x A p]a3 = 2icr[-M2, M1, 0]; whence, by the definitions (1.9) and (1.10) of Jj1) and J(3), J(3)J(l)
_ j(l)j(3) _ -iga[-M2, Ml, 0];
(1.5)1,3 results from (1.11) and (1.12). The proof of (1.5)2,3 is evident.
2. Resolution of a Lagrangian Problem in Two Unknowns
Notation. Let
af.s'
aL2,
am
(1.12)
IV,§1,2
241
be the 2 x 2 matrices of lagrangian operators associated to the 2 x 2 matrices
H[L, M, Q, R] + v f [L, M, Q, R]a3 +'g[L, M, Q, R]a[x A P], L2,
M.
(2.1)
Let us study the solutions U = (U1, U2) of the system
at,gU = (aLl - Lo =
(a,M
2i Lol') J
V
U
- Mo - 'm'+ 2v a3) U = 0 mod
z,
V
where Ul and U2 are two lagrangian functions defined on a compact lagrangian manifold V, l' and m' are real numbers near 0 whose squares are negligible, and Lo and Mo are two real numbers. (The interest of this system is shown by lemma 1 and remark 1.2.) As in 111,§ 1, let V [Lo, Mo] denote the lagrangian manifold in E3 Q E3 given by the equations
V [Lo, Mo] : H [Lo, Mo, Q, R] = L(x, P) - Lo = M(x, P) - Mo = 0. (2.3)
When this manifold is compact, it is a torus whose points have the coordinates
t modc[L0,M0],
'P mod 27r,
(D mod 27r.
F[Lo, Mo], t, and c[Lo, Mo] are defined in III,§1 by (2.5), (2.6), and (2.7), respectively. Assume that on V [Lo, Mo]
(f - ZHN)c[L, M] and Lgc[L, M] are near zero and have negligible squares. For V[L0, Mo], and hence I'[L0, M0], compact, let .fo =7 0, 0
(2.4)
IV,§1,2
242
where IMOI < L0. We shall see that f0, g0i and c have negligible squares. THEOREM 2. The compact manifolds V in E3 Q+ E3 on which solutions U of the system (2.2) are defined are the tori V [L0, M0] such that
Lo=h(l+i-I'), h(l +
i) +
N [h(1
M0=h(m-rn'),
+ 4), hm] = fin ± hs[h(1 + ' ), hm],
(2.5)
where 1, m - i, and n are three integers satisfying the inequalities Iml
l+z,
1 0); then, by (2.11), condition (2.10) that U be lagrangian on V may be formulated as follows, since e-"3 = -1: There exists e e R such that
y(t + c[L0, M0]) = e2" y(t),
(2.18)
N[L0,M0]+1'NL+m'NM+Z+e=Lo+l'-Z
= M0+m'-z=0mod 1. (2.19)
IV,§1,2
245
The search for solutions y of (2.16) satisfying (2.18) is a classical problem:
(f - 4Hm)r3 + Log(Q1 sin 0 + o3 cos O) is a self-adjoint matrix with zero trace; it is a periodic function of t; the period is c[L0, Mo]; then the 2 x 2 matrix u(t) defined by
du + i[(f - IHM)Q3 + Log(Q1 sin 0 + o3 cos O)]u dt = 0, u(0) = 1,
(2.20)
is a unitary matrix with determinant 1, that is, u c- SU(2); it satisfies
u(t + c[L0, Mo]) = u(t)u(c[L0, MO]); the general solution of (2.16) is
y(t) = u(t)d, where 6 E C2;
y satisfies (2.18) if and only if 6 is one of the eigenvectors of the matrix u(c[L0, Mo]) E SU(2). Let e±2aie[Lo.Mo]
(0 -< e[Lo, Mo]
i)
(2.21)
be the eigenvalues of u(c[L0, M0]). Then, in (2.18),
e = +e[L0, Mo]. Thus, there exists a lagrangian solution U of (2.2) defined on the torus V[L0, Mo] if and only if there exist three integers 1, m - Z, and n such that
ILo+1'+1N[Lo, MO]+1'NL+m'NM=n+e[Lo,MO], Lo + I' - 2 = 1,
Mo + m' = m, where IMo < Lo.
(2.22)
This assertion does not assume that 1', m', (f - HM)c[L, M], and Lgc[L, M] have negligible squares on V[L0, Mo]. z U and /l will be denoted by U± and #± depending on whether (2.22) holds with ±e[L0, Mo]. By (2.15), (2.17), and (2.18), where e = +e[L0, Mo], fl± satisfies (2.8), where the c±k have the values (2.9). Assume that the squares of 1' and m' and the squares of the derivatives of he are negligible. Then (2.22) reduces to (2.5). The integers 1, m - Z, and n must satisfy condition (2.5)3, which is independent of l' and m'. Since 1 and n are integers, since a is near 0, and since N > 0, (2.5)3 implies
IV,§1,2
246
I < n. Since 1' and m' are near 0, (2.5)1 and (2.5)2 imply Iml < 1 + Z, which completes the proof of (2.6).
It remains to prove that (2.4) is an approximate expression for e[Lo, Mo]. Since we have assumed that, on V[Lo, M0],
(f - 1HM)c[L, M] and Lgc[L, M] are near zero and have negligible squares, the solution of (2.20), modulo these squares, is
u = 1 - i [ r (f + Z pt) dt a3 + L0
g dt (a, sin 0 + a3 cos O) fo
0
by (2.14)2. In other words, choosing the right-hand side to be in SU(2), like u,
u = e-i[fo(J+j,.
2)dta3+L0S0gdt(a,sin0+a,cos®)
Thus, by (2.17), definition (2.21) of e[Lo, Mo], and definition (2.4) of fo and go, the eigenvalues +2rce[Lo, Mo] are approximately the eigenvalues of the self-adjoint matrix with zero trace given by for3 + go(al sin 0 + a3 cos 0). Now, by (1.7) and (2.13), its square is
(fo + go cos 0)2 + go sine 0 = f o + 2 Mo fogo + go ; 0
hence the approximate expression (2.4)2 for e[Lo, Mo] follows.
The-quantum numbers
Remark 2.2. 1,
m,
n,
±1
become those used in the classical solutions of the Dirac equation by imposing a condition stricter than (2.6) less strict than the condition that, on account of (2.7), would result from
the choice 1' = m' = 0, namely,
lmI
