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

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94 II,§ 1, 5

Notation.

Let , and n' be regular positive measures on V and V'; we

define

argil = argil' = 0.

We assume_V and V' are both given 2-orientations: If z e V, x = Rxi,

then

d'(Rxz)

d'x

is a function V - R

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

f3 (2) =

[d1(1x2)]h/2

aR,o(2), where 2 e

Definition 5.2.

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

of the measure n and the 2-orientation of V.

Substituting the expression (5.9) for aR,o and ax

into (5.4), we obtain

the following.

THEOREM 5.2.

Suppose V = V', which implies ii = ii"; let Qo and Qo denote

the lagrangian amplitudes of UR and U. Then

(URIUR)=

Qo(Z)Qo(fle"[V'(=)-PVC0]n

mod 1,

J ZEV Z.Z

(5.10)

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

V and V' are assumed trans-

verse (as in part 3 of theorem 5.1); thus

II,§1,5

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\ER,

z' E V'\R,

(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)

[ `

xz)

]112 r

xz )

]1o2

[Hess(

- Wr)]

112

mod 1.

(5.13)

Lemma 5 transforms this formula into the following.

THEOREM 5.3.

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

L VI

(UR I UR)

q Od

112

mod

(5.14)

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).

LEMMA 5.

1°) The symplectic structure of Z defines a measure on Z that

is invariant under translations and under Sp(Z):

d21z

(dx A dp)1 = (-1)1(1 -1112d1x A d'p,

(5.15)

where

II,§1,5

Rz = (x, p),

dx n dp = > dxj n dpj ;

j=1

{xj} and { pj} are dual coordinates on X and X *.

2°) Under assumption (5.12), with the notation (5.9),

Hess((p

d'(Rxz) d'(Rxz ) d21z

no Ano

(5.16)

arg Hess(rp - (p')

d'(Axz) d'(Rz )I

= nm(1.4i A4) mod 4n.

(5.17)

Proof of 1°).

The value of dp n dx on a pair of vectors z, z' is [z, z j

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 trans-

verse,

Hess(

- (D')

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

d2'Z =

(-1)1(1-1Y2d'[x

+ x'] A d'[''x(x) +' '.(x')]

(-1)1(1-1W2d'[x + x'] A d'[d'x(x)

a)s(x)]

(- l)t(t+1)/2 Hess((D

- cD')d'x A d'x'

_ (-1)tu+1U2 Hess((D - tb')

d'(Rx()d'(Rx(')

no A no,

no no

where

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;

II,§1,5-II,§1,6

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)). (5.19)

Proof of (5.19).

Let us write down the equations of R..., RA', and X*:

RA:p = (D .,(x),

R)':p' _ Dx(x'), X*:x" = 0.

The definition of Inert (I,§2,4) makes use of

z E R1.,

z' E RA',

z" E X* such that z + z' + z" = 0.

Define

z = (x, P), z' = (x', P),

z , = W" P");

hence

x+x'=0, x"=0, p+p'+p"=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

(6.1)

rEN v

98 II,§1,6

such that the

a, : 0

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:

K is compact in 0;

p, r e N,

E e

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)

= CeXp

08x' 8

] a° R(v,.x, p),

(6.3)

aR (v, P, X) = Cexp - - ` a a

]

ae(v, x, P)

L 2v \8x 8p

define two automorphisms of y °(Z(l)) that are inverse to each other:

o -

aRHaR, aR"aR.

Definition 6.2 of formal differential operators rests on the following

lemma.

II,§t,6

LEMMA 6.1.

Let

u = ae"°e.F(X),where a = E lrar.

reN V

At a critical point x of the phase cp we have

u(v, x) = 0 mod

vlhl(z .

Proof.

It suffices to consider the case a = a°. Then at a critical point

of q ,

a) u = vlh Phe"W (h is a multi-index),

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 <, Ih1/2. The lemma

follows.

Definition 6.2.

Let

(y, x) = qq(x) - qq(y) - <9,,(y), x - y>

(6.5)

denote the remainder of the first-order Taylor series of the phase cP of

(6.4) at the point y. To an element a° of .F°(SZ) and to a frame R, let us

associate the local endomorphism aR of .F(X) and .F°(X), called a formal

differential operator, defined by the two equivalent formulas

(ax u) (v, y)

(

h l

Ja'lhPhR

(V, X, P) l

)h [a(v,

x)e°o(Y,xl] lx=Y

e"w(Yl

P=Wr(y)

l 1 a

h [IhIc2_

vo(y.x

)

"W(Y)

2Ph

, p, x)a(v, x)e

(6.6)

x=v

P W,(y)

By lemma 6.1, each of these two formulas makes sense when

(y, (py) e SZ(l): we say that aR is defined on S1(l) = R.

Proof of the equivalence of these two formulas. Leibniz's formula gives

(compare I,§l,proof of lemma 3.1)

100

h ! (V

ax)h[aIhla_(v

p, x)(

p)h

v(v,

x)]

a1zj+klaR(v, p, x)

1 a

uv x

= f j!k!

(ap)'+k(v axy

v ax

)

1 alk'aR (v, x,

p) (1

akll(V, X)

= : k!

(ap)

v OX)

because

II,§1,6

a a

j ! (ap)'(v ax)'

[exp

v ax ap

]

aR (v, p, x.

= aR (v, x, p).

Justification of definition 6.2.

Let us show that it is equivalent to the defi-

nition in section 1 when a° is a polynomial, that is, when aR is a classical

differential operator with polynomial coefficients in (1/v, x). In this case

(6.6) means

(aRU)(v, X)

{a+

v' x, P + V

aX)

`U(v' x)e

v<P,x-Y>]}Y

IP=(O,

Now

(V, X, P +

V ax)

`u(v,

x)e-v(P.x-Y>]

e-ti<P,s-Y>aR

(V, X,

ax)U(V, X).

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,

V ax)

U(v' x) = aR (v'

V ax'

X) U(V9 X).

Notation. Denote

-V,

V aX,

U(V, X).

(aRU)(v, x) = aR

(V,

ax) u(v, x) = aR (

Definition 6.3.

Let V c Q, a° E

Let aR denote the unique local

endomorphism of

R) such that

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-

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

[a°(v, z)b°(v, z')] }Z=: ;

(6.10)

c°(v, z)

{[exp

- 2v

(az'

there

Cazl a'Nx' 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).

102 II,§1,6-II,§1,7

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.

Definition 7.1.

The algebra

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

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.

The vector space R) is the set of linear mappings

f' : (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

flP

f'(z)f(f)d'x,

where x

= AXi.

Then

V ax

is defined on .9' by the requirement that it be self-adjoint:

[1f'()]JdI

x =

ff,

f (Z)

IIV

f (z

d'x.

Given a bounded set B in -9(V\ER, R), we let

I FIB = Sup

feB

f7 f'(zf(Z)dix

< 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 R).

Definition 7.3.

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 R).

Supp UR = U Supp a" c V.