Leray J. Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index

Подождите немного. Документ загружается.

124

II,§2,5-II,§3,1

of U and U', a lagrangian function and a distribution both defined on a

lagrangian manifold V when Supp U n Supp U' is compact.

Definition 3.5 and theorem 3.2,2°) apply to the scalar product (U I U')

of U and U', a lagrangian function and a distribution both defined on a

lagrangian manifold V, when Supp U n Supp' U' is compact.

§3. Homogeneous Lagrangian Systems in One Unknown

0. Summary

The existence of an inverse a-' of a lagrangian operator a under the

assumption of theorem 1.1 of §2 shows that it is the homogeneous lagran-

gian systems in one (§3) or several (§4) unknowns whose study is non-

trivial. §3 and §4 begin this study, which will be concluded in chapters III

and IV with the special cases of the Schrodinger equation and the Dirac

equation, the latter of which is a system in four unknowns.

1. Lagrangian Manifolds on Which Lagrangian Solutions of a U = 0

Are Defined

Let a be the lagrangian operator associated to a formal function

a° : S2 .FO (S2 an open set in Z)

whose value at z is

a°(v, z) = E 1 ar(z).

,ENV

THEOREM 1.

The support of a solution U of the equation aU = 0 is a

lagrangian manifold V on which ao = 0.

Proof. Consider a point in V at which ao i4 0. Replace Q by a neighbor-

hood of this point on which ao

0. Then a-' exists on S2 and aU = 0

implies U = 0 on S2.

We shall assume that ao is real valued and that the subset of S2 where

ao = 0 is a regular hypersurface W given by the equation

W : H(z) = 0,

V is therefore a submanifold of W such that

dim V = l; d[z, dz] = 0 on V.

II,§3,1-II,§3,2

125

V is studied by applying E. Cartan's theory of differential forms to the

pfaffian form [z, dz].

2. Review of E. Cartan's Theory of Pfaffian Forms")

We first recall some well-known results that are set forth, in particular, in

the treatise of Y. Choquet-Bruhat [6].

Let co be a differential form defined on Z that in section 2 will be any

infinitely differentiable manifold of finite dimension. E. Cartan expresses

co locally using a minimal number of functions

The number n of these functions is called the rank of w. These functions

are the first integrals (independent and maximal in number) of a com-

pletely integrable system of pfaffian equations called the characteristic

system of co. E. Cartan explicitly describes this system.

In other words, this characteristic system is equivalent to

dit=...=dff=0;

there exists a differential form za such that

w(z, dz) = m(f (z), df (z)), where f = (fl,

... , f,)

Let co be a pfaffian form defined on Z (that is, a differential form of

degree 1 in the differentials). Let i, denote the interior product [6] with

a tangent vector of Z; the characteristic system of dw may be written

icdw(z, dz) = 0 V . (2.1)

The rank of dco is even; denote it by 2n. Let

f=(ft,...,f2.)

be 2n independent first integrals of (2.1) locally. Then there exists a dif-

ferential form m' such that

dw(z, dz) = m'(f (z), df (z)):

evidently dm' = 0. Then m' = dm locally. In other words, locally there

exist a pfaffian form m of 2n variables and a function fo such that

' See Cartan [5].

126

11,§3,2

w(z, dz) = dfo(z) + rm(f (z), df(z)), where f = (fl, ... , f2.)- (2.2)

The rank of w is evidently 2n or 2n + 1 depending on whether fo is a

composition with f = (f1,

. . .

fen) or not. In the first case it is possible

to choose w such that fo = 0.

Let us state some less familiar facts with which E. Cartan [5] (chapter

XII, "Les equations qui admettent un invariant integral relatif") has sup-

plemented this result; we recall their proofs.

LEMMA 2.

Let w be a pfaffian form; let 2n be the rank of dw. Locally there

exist 2n + 1 numerical functions fo,

. : .

f2,, such that

w(z, dz) = dfo(z) + I f2j-1(z)df2j(z);

(2.3)

j=1

fl, -

- - ,

f2, are 2n independent first integrals of the characteristic system

of dw.

Remark 2.1.

f1,

. . . ,

fen is not an arbitrary sequence of first integrals of

this characteristic system: see remark 2.2.

Proof.

Let f2 be a first integral of the characteristic system of do). The

restriction of dw to the hypersurfaces f2 = const. has even rank <2n, so

its rank is 52(n - 1). Let f4 be a first integral of the characteristic system

of this restricted form. Continuing in this fashion, the restriction of dw to

the manifolds

f2 = const., .. , f2; = const.

has rank < 2(n - j) and therefore vanishes for some value J of j such that

J <, n. In other words,

dw = U on the manifolds f2 = const.,

.. , f2, = const.

Then there exists a function fo such that

w = dfo mod(df2,

. , df2,);

that is,

w = dfo + I f2i-1 df2;.

J=1

But do) has rank 2n, so J = n and f1, ... , fen are 2n independent first

integrals of the characteristic system of dw.

II,§3,2

127

E. Cartan has supplemented this lemma by using the notion of the Poisson

bracket as follows.

Definition 2. Let co be a pfaffian form that has been put in the form (2.3).

Let g and g' be two first integrals of the characteristic system of dw. Each

is then a composition:

z"g(z) = G[f1(z), ...,f2.(z)],

z"g'(z) = G'[f1(z), ...,f2. (z)]

Then there exists a composition of the same type, called the Poisson

bracket of g and g' and denoted (g, g'), such that

n(dw)n-1 A dg n dg' = (g, g')(dw)".

(2.4)

This bracket

(g, g') _ -(g" g),

which is bilinear and independent of the choice of f = { f1, ... ,

may

be expressed as follows, using one such choice:

(g, g') (Z) _

3G'

8G' aG

(2.5)

af2;-1 Oft;

42j-1 8f2;

I=I(z)

Hence every triple g, g', g" of such first integrals satisfied the Jacobi

identity:

(g, (g', g")) + (g', (g", g)) + (g", (g, g')) = 0.

g and g' are said to be in involution when (g, g') = 0.

Remark 2.2.

By (2.4), the pairs fJ,fk (1 < j < k) in the expression (2.3)

for w are in involution with the exception of the pairs f2j_1, f2j; these

satisfy

(f2;-1, f2;) - 1.

Remark 2.3.

Let g1, . . ., g9

be independent first integrals of the charac-

teristic system of dw. The restriction of w to a manifold

91 = const.,

... , gq = const.

reduces the rank of dw by an even number that E. Cartan [5], chapter

XII, section 124, has made explicit: it is 2 (q - r) if all the Poisson brackets

(ga, gk) (j, k E 11,

... , q}) are compositions with g

1, ... , gq

and if the

128 II,§3,2

exterior form

Y- (9j.

A k

j, k

in the variables

, ... ,Q

has rank 2r on this manifold.

In particular (q = n, r = 0), we obtain the following theorem, which

will enable us to establish theorem 3.1. These theorems shed some light

on the beginning of the study of the Schrodinger equation.

THEOREM 2.

Let w be a pfaffian form; let 2n be the rank of d(o.

1°) Locally, the characteristic system of d(o has n-tuples { f2, f4,

f2ri} of independent first integrals that are pairwise in involution.

2°)) Given one {of these n-tuples, a quadrature locally gives n{+ (1 functions

f0,

f1, f3, ,

f2n-1 such that to is expressed by (2.3); f1,

f2,

f3, ,

f2'.

are independent first integrals of the characteristic system of d(O.

Proof of 1°). Lemma 2 and remark 2.2.

Proof of 2°).

By assumption,

df2 A df4 A ... A df2

(dw)..-1

A df2j A df2k = 0

yj, k.

In other words, there exist differential forms Oj such that

(dw)n

1 = Y Oj A df2 A ... df2j ... A df2,.

j=1

(^ suppresses the term that it covers). This relation expresses the existence

of pfaffian forms Oj such that

dw =

Oj n df2j,

j=1

or equivalently, the condition

dw = 0 mod(df2, .

. , df2,.),

that is, the existence of a function fo such that

to = dfo for f2 = const.,

... ,

f2n = const.,

or the existence of n + I functions fo, f1, f3, f2,.-1 satisfying (2.3).

II,§3,3

129

3. Lagrangian Manifolds in the Symplectic Space Z and in Its

Hypersurfaces

Let

w(z, dz) = z[z, dz],

(3.1)

that is, in a frame R

w(z, dz) = '

2 2

<p, dx>

- ' <dp, x>

= <p, dx> - zd <P, x>,

where Rz = (x, p); (3.2)

dw evidently has rank 21; its characteristic system is dz = 0.

Lagrangian manifolds V in Z are manifolds of dimension 1 on which

do) = 0. These are manifolds given by local equations

P = (PR,s(x)

(3.3)

outside the apparent contour ER of V, where (PR is the phase of V in R; coR

is an arbitrary function when V is arbitrary.

Given x e V, it is possible to choose R such that x

ER and use the local

equations (3.3) in a neighborhood of x.

But if we want to use a single frame, we must supplement the preceding

result as follows (x' and pp will denote dual coordinates of x and p).

LEMMA 3.1.

The lagrangian manifolds in Z that project under RX : z r+ x

to a manifold in X of dimension k S 1 given by equations

x'=F;(x',.-..,xk), k+1

j<1,

(3.4)

are the manifolds in V given by local equations (3.4) and

a(PR(xl, ... , xk)

OF.

1 i < k

(3 5)

ax`

j=k+1 ax

;

where coR is the phase of V in R and x'.... , xk, Pk+11 ... , p, are local

coordinates on V; cp, is an arbitrary function of x', ... , xk when V is

arbitrary.

If k = 1, the equations of V reduce to (3.3).

If k = 0, Visa plane:

x = const.,

p arbitrary,

cpR = const.

130

II,§3,3

Proof

1 d<

p, x> = < p, dx >

Pi +

P;J dx'.

2 +

\\\

1=k+I

Then dw = 0 on V if and only if there exists, on V, a function WR of

(x',

... ,

xk) satisfying (3.5). The condition dim V = ! is that

(x`, xk, pk+I,

pI) be independent on V. Obviously cpR is the phase

of V.

Notation.

Given a function f : Z

R and a frame R, fR denotes the

function Z(!)

R such that fR(x, p) = f (z) for Rz = (x, p); that is,

fROR = f. Put

JR,xk

afl,

fR,x = (fR.x', ...,fR,x,)

W is a hypersurface in Z given by the equation

W: H(z) = 0,

Hz # 0.

(3.6)

ww andfw are the restrictions of w and f to W.

LEMMA 3.2.

(E. Cartan) dww has rank 2! - 2; its characteristic system is

Hamilton's system

dx' _ _ dpk

on W dj, k e {1, ..., !},

(3.7)

HR,Pi(x, P)

HR,x°(x, P)

where the vector (HR,P, - HR.x) is evidently tangent to W.

Proof. The condition

(d(ow)'-' A dfw = 0 on W

(3.8)

may be written

(dw)'-' A df n dH

= O on

(dw)'

Then by (2.4) and (2.5), where we choose f2J_1 = p; and f2J = x' in order to

identify (2.3) and (3.2), condition (3.8) may be written

(fR,P,HR,xj - HR.P;fR.x;) = 0 for H. = 0.

(3.9)

J=1

II,§3,3

131

It does not hold identically; thus

(dcow)'_1 # 0. But (dcow)' = 0; thus the

rank of d( ow

2(I - 1). Hence the equivalent relations (3.8) and (3.9)

mean that fw is a first integral of the characteristic system of do)w. Hence

this system is Hamilton's system (3.7).

Remark 3.

As local coordinates on W we shall use 21 - 2 independent

first integrals of Hamilton's system (3.7) and a function t such that (3.7)

may be written

dx = HR,p(x, p)dt,

dp = -HR. x(x, p)dt. (3.10)

Definition 3.1.

Given a function g: Z - R, define g_ E Z at the point z by

dg = [dz, 9Z]

in other words,

(3.11)

R9z = (9R,P, -9R,x)

Hamilton's system (3.10) may then be written

(3.12)

dz = H_ dt.

(3.13)

By (2.5), where we choose f2J_1 = p3 and f2J = x' in order to identify (2.3)

and (3.2), the Poisson bracket of g and g' is

(9, g') = <9R.x, 9R,,> - <9R,x, 9x,,> = [9z, 9=].

(3.14)

The tangent vector K = H. of W is called the characteristic vector. The

curves in W tangent to this vector at each point of W, that is, the solutions

of Hamilton's system (3.7), are called characteristic curves.

The following classical theorem explicitly describes the lagrangian

manifolds in W, that is, the resolution of the nonlinear first-order equation

HR(x, (Px) = 0,

in which the unknown is a function (p.

THEOREM 3.1.

1°) Hamilton's system (3.7) has (I - 1)-tuples

of independent first integrals that are pairwise in involution, that is,

(dw)'-Z A dh; A dh,, = 0 on W

Vj, k e 11,

. . ., I - 1;

. (3.15)

132

II,§3,3

Given one of these (1 - 1)-tuples, a quadrature locally defines l functions

go,91, ,91-1

such that

1-1

w=dgo+

g3dh3 on W.

(3.16)

g1, ... y1_1, h1, ... , hi-1 constitute 2(l - 1) independent first integrals of

Hamilton's system (3.7). If t is defined by (3.10), then

t, h1, ...,h1-1,91, ...,9,-1

(3.17)

form local coordinates on W; go is a function of these coordinates.

2°) Lagrangian manifolds V in W are manifolds in Z that locally

i. are fibered by the characteristic curves

t arbitrary,

h 1 = const.,

... ,

h1 _ 1 = const.,

g, = const.,

... ,

g1 _ 1 = const.,

ii. have the lagrangian manifolds in the space Z(1 - 1) with the coordinates

x' = (h1, ... , h,-1),

P = (91, ... , 91-1)

(3.18)

as a base for this fibration.

3°) If the local coordinates (3.17) are used on W, we have the following

local equations of a lagrangian manifold V in W, after suitably permuting the

indices ( 1 , ... , 1 - 1), choosing k e (0, ... ,

l - 1} and choosing 1 - k

real numerical functions of k variables Fo, F,.,,,

... , F,_1:

h;=F;(h1,...,hk),

k+ 1 <,j,<l- 1

3Fo(h1, ... , hk)

1-1

OF.

(3.19)

91 =

g;,

ah1

;_k+1

ahi

t, h 1, ... , hk, 9k + 1 , . - . , g, -1 are

local coordinates on V. The lagrangian

phase

of V is given by

V/(t, h1, ..

, hk, 9k+1 .

, 91-1)

=9o(t,h,,...,hk,9k+1,...,g1-1)+Fo(h1,...,hk).

(3.20)

II,§3,3

133

If k = I - 1, the system (3.19) means

<i<l-1. (3.21)

Oh;

If k = 0, it means

It = const.,

g arbitrary,

F0 = const. (3.22)

Proof of 1°). Use theorem 2 and lemma 3.2.

Proof of 2°).

By (3.16), the condition that dw = 0 on V is equivalent to

the following: the mapping

V3zH(h1i...,h;-1;g1,---,g,-,)EZ(1-1) (3.23)

sends V onto a manifold B in Z(l - 1) satisfying Edh; A dg; = 0. Then

B has dimension <1 - 1. A necessary and sufficient condition for

dim V = I is that

i. dimB = I - 1, that is, B be lagrangian;

ii. (3.23) map a characteristic curve of V onto each point of B.

Proof of 3°). Lemma 3.1, where

Z, x, p,

dcpR = <p, dx>

on V,

are replaced by

:-1

Z(1 -

1), x', p' [see (3.18)], B,

dFo = Y_ g;dh; on B,

j=1

so that (3.16) gives

w=dgo+dFoon V.Now w=day.

Section 4 will use the following definition.

Definition 3.2.

Given a lagrangian manifold V in W, let us call a regular

measure q on V that is invariant under the characteristic vector K of W an

invariant measure on V ; in other words, an invariant measure on V is any

differential form q defined on V that is homogeneous of degree l in the

differentials and is annihilated by the Lie derivative Y. in the direction

of K:

Y.q = 0. (3.24)