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

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II,§1,2

The set F° [respectively,

of elements of F [respectively,

.F(X )] with vanishing phase forms a subalgebra of.

[respectively, .

(X )].

Let Z denote a symplectic space with a 2-symplectic geometry, V a

lagrangian manifold in Z (I,§3,1 and 4), and V the universal covering of V.

A formal function UR on V consists of

i. a 2-frame R of Z

ii. a mapping

UR = UR (v) =

r- e"'PR : V -

rEN V r

where

4pR is the phase of R V (I,§3,2),

ar : V - C is infinitely differentiable.

Clearly,

Supp UR = U Supp ar

(2.6)

Given R and V, the set of these formal functions UR is a vector space

over 3F°, denoted F (V R); the set of those of its functions with compact

support is a subspace 3F°(V, R).

We give F(17, R) the topology defined by the following neighborhood

system /V(K, p, r, e) of the origin:

K is a compact set in V; p,reN,eE1(i+;

UR e .A'^(K, p, r, e) means that the derivatives of the as (s < r)

of orders < p have modulus < e on K.

ER denotes the apparent contour of V for the frame R (I,§3,2). ER

denotes the apparent contour of V, that is, the set of points of V that

project onto ER. We shall deduce the properties of -flV\ER, R) from

those of .f°(V\ER, R). The latter will be deduced from the properties of

.F°(X) using the morphism resulting from the composition of the two

morphisms that will be defined in theorems 2.1 and 2.2, respectively.

Notation.

If z e Z and Rz = (x, p), then we write x = RXz. The com-

position of the natural projection V - V and the restriction of RX to V

is denoted 1

: V -p X.

II,§1,2

THEOREM 2.1.

There exists a natural morphism

]ZR:

o(V\ER, R) - .9Fo(X) (2.7)

called the projection. It

is defined as follows. Let UR E .F o(V\ER, R);

u = FIR UR is given by

u(v, x) _ X UR(v, z).

(2.8)

ZERX'X

Remark 2.

IIR is a monomorphism if any period of cPR is non vanishing.

Proof.

Formula (2.8) makes sense because RXIx n Supp UR is a finite

set, since Supp UR is compact in V\ER and I

is a local homeomorphism

V\ER

. X.

THEOREM 2.2.

There exists a natural monomorphism of the algebra .F,(X)

into the algebra W (X) that allows the convention

.Fo(X) c 'e(X).

It is defined as follows. Let

e"`lj E J`o(X );

jEJ

put

(2.10)

T'-1 a

E Y-','ev4:I xX -C. (2.11)

jEJ r=O

ti.

1°) There exist functions f c P4(X) such that f - uN = 0 mod 1/vN VN.

2°) All these functions belong to the same asymptotic equivalence class

f e (X); the mapping u i-- f defines a natural morphism E--' r, (X).

3°) This morphism is a monomorphism.

Proof of 1°). It suffices to consider the case in which (2.10) reduces to

u(V, X) = I

a'(x) e°O(X). (2.12)

For all g e M (X) vanishing outside a compact set of X we use the norm

IgI, = Sup

(1 a } g ,

where v E I,

X E X, +sj < r. (2.13)

V, z. r \ V 8x

II,§1,2

We choose an increasing sequence of numbers

0<µo<µl<...

such that

2'I xre"' I r < µr, lim u, = oG,

(2.14)

and we choose a sequence of continuous functions go, E,

, ...: I -+

R+

such that

0 < E,(v) < 1,

Er(v) = 0

for IvI < µr,

(2.15)

Er(v) = 1

for 2µr < I i ' !

Now we define

V(p(x)

) _

(v)

-( )

2 16)e .

, x

( .

This series converges since, from (2.14)2 and (2.15)2, the number of non-

zero terms is finite on any bounded interval in the domain of definition

of v. In accordance with (2.11), we define

UN(V, X)

ar(X)evrp(x)

r=0

By (2.15)3, for 2µN < IV 1,

Er(V)

If(V) - uN+l(V)I N <

I I,

lore I

N+1

Now (2.14)1, (2.15)1, and (2.15)2 imply

E,(v)2'Ia,e"'I, < IvI.

Hence, for 2µN < I VI,

If(V) - UN+1(V)IN < E

2rVIr

2IVIN 2 - (1/IVI)

r=N+1

because 1

I vI ; thus

If - UNIN = 0 mod N.

(2.17)

(2.18)

11,§ 1,2

Let us prove a more general statement :

If-uNIM=0mod-

VM, N e N.

(2.19)

If M < N, (2.19) follows from (2.18) and the fact that

increases with

M. If N < M, (2.19) follows from the relations

If - U11

< If - UMI M + J UM - UNIM,

I f - uMI

M =

0 mod

vM,

and hence mod vN,

M-1a

I UM - UNI M = 0 mod

because UM - UN = Y- , e°v.

V IV

Since the supports of f and of the uN belong to Supp u = U Supp ar,

which is compact, (2.19) implies

x°(f - UN)IM = 0 mod

VM,N,geN;

hence

f e R(X), f-UN=0modI

Proof of 2°).

If f and f' c- .V(X) and satisfy

f - uN = f' - UN = 0 mod

VN,

then

f - f' = 0 mod N

VN.

Proof of 3°).

Assume

U - I = 0;

then

UN(V1 x) = 0 mod N

VN e N,

Vx e X.

II,§1,2

It has to be proved that

u(v, x) = 0

Vx E X.

It suffices to consider the following case: u is a formal number. Assume

that in expression (2.1)

aJ, = 0 for r < N

dj E J;

then we have

uN+1(V) _ y ae"i =

jQJ V

0 mod

VN+ 1'

whence

lim

I aJNe'lli = 0,

jEJ

which implies

aJN=0

Vj.

Theorem 2.2 follows.

COROLLARY 2.

Asymptotic integration, elements of Sp2(l), and differential

operators with polynomial coefficients are morphisms

0(X) - T, fFo(X) - `6(X),

.F0(X) -. .FO(X).

The end of this section describes the properties of these morphisms. We

use the following theorem in §2, 3 to study the norm of lagrangian functions.

THEOREM 2.3. 1°) Let

u =

x E 'F0

rcN

he a formal number with vanishing phase. The condition that it be real,

uERey°, is

a; is real, that is, i-'a, is real `dr.

The condition u > 0 is

II,§1,2

0 <

, that is, i`as > 0,

where s is the first r such that a.: 0. Thus every u E Re F ° satisfies

either u > 0 or u <, 0. Hence > defines an order relation on Re go.

2°) The condition u'12 E Re .° is equivalent to the following:

u c .f °;

u > 0; the integer s (defined above) is even.

Notation.

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,

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 = Re

Im u = E Im a ,

rEN

rEN V

Thus u c Re .y ° if and only if a,v-' is real for every r.

Let us study uSince

2 as 1

u = v2s

mod

V2s+1

the assumption u112 c Re

° implies:

s is even, 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

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<0.

II,§1,3

3. Integration of Elements of .po(X)

The essential properties of formal functions will be deduced from the

following theorem, which supplements some classical results (the method

of stationary phase). Its proof requires the use of lemmas 3.4 and 3.5 in

place of the lemma of Marston Morse [12], by which a function with a

nondegenerate critical point is transformed locally into a quadratic form

by a change of coordinates. We briefly recall the other lemmas needed

for the proof of this theorem.

By corollary 2, asymptotic integration is a morphism

:.F0(X) - W.

Clearly,

u(v, x) d`x e 9

when the phases of'u are all constant. Theorem 3 shows that

vy2 ju(v

x) d`x c- .F

n the important case considered in corollary 3.

THEOREM 3.

Let u E .FO(X).

1°) The value of f xu(v, x)d`x depends only on the restriction of u to an

arbitrarily small neighborhood of the set of critical points of the phases

of u; this value is 0 if this set is empty.

20) Let

u(v) =

e" ° e FO(X),

rrN v

where cp has a single nondegenerate critical point xc on Supp u. Let cpc =

cp(xc) be the critical value of p; arg i112 = nl/4 and v Hessc cp be the value

of ,/Hess at xc (I,§1,definition 2.3). Then

f u (v, x)

d`x =

(I2VI

nil11Z

(Hessc(P)112

rEN

(3.2)

where

II,§1,3

I0((p a) = ao(xc),

I,(cp, a) =

(3.3)0

r 2(r-s)

I I

1*r S+;

8 1

F0'(x)xs(x)1

S=O ;=0

(r - s + j)!j !

ax)

))x=x

(3.3),

4), dD*, and O have the following definitions.

Notation.

The second-order Taylor expansion of cp at xc is written

(p(x)=qc+ 1(x-xc)+O(x),

(3.4)

where O vanishes to third order at xc and b is a quadratic form X -+ R;

b* is its "dual" form, that is,

(D* (p) is the critical value of the function x H (D(x) + <p, x).

(3.5)

In other words, if M is the value of the matrix ((piX,) at xc, then

b(x) = z<Mx, x>,

*(p) _ -i<p, M-'p>,

where M = 'M: X - X*.

(3.6)

The following is evident.

COROLLARY 3.

Let u e .F0(X). If all the critical points of all the phases

of u are nondegenerate on Supp u, then

vq2 f

u(v,x)d'xe9;

the phases of 0 2 f

u d'x are the critical values of the phases of u on Supp U.

It clearly suffices to prove theorem 3 when u is replaced by a function

f given by

f(v, x) = a(x)e'1*),

(3.7)

where a e !2(X) (that is, x is infinitely differentiable and has compact

support). The theorem then results from the following lemmas, the first

of which proves part 1 of the theorem.

LEMMA 3.1. We denote by xc the critical points of (p belonging to Supp f

If they are all nondegenerate, then

a(0)e°1'(")d'x mod N

II,§1,3

is a linear function of the values of the derivatives of a of order <2N at

the points xc.

Proof.

This amounts to proving the following:

If a vanishes to order 2N at the points xc, then

f d`x = 0 modN.

(3.8)N

Now (3.8)o is obvious. Suppose (3.8)N holds. If a vanishes to order

2(N + 1) at the points xc, then there exist fi e 9 such that

8icpxi,

vanishes to order 2N + 1 at the points xc.

Hence by (3.8)N,

- 1ae"°d'x = - f v e"'d`x = 0 mod

Oxi

which proves (3.8)N+,

The next lemma is proved similarly.

LEMMA 3.2.

Let a e 9(X) (Schwartz space; see I,§1,1). Let (b be a non-

degenerate quadratic form X - R. Then

a(x)e"m's)d'x mod N

is a linear function of the values of the derivatives of a of order <2N at 0.

More explicit is the following.

LEMMA 3.3.

Let a e .9'(X). Let 4) be a nondegenerate quadratic form

X - R. Then

o (x)e"m(x)d'x

(2mi)h/2

[Hess(D]-1/2I rev'

(D*r

x(x)

modvl .

IVI

) I

(

rtN

Proof. We know (see I,§1, proof of lemma 2.2) that

e-'``1 ' dx

= 1p

1 '2 for larg pj

< 7t/2.

f-,

II,§1,3

Hence, by differentiating with respect to µ,

r°

(2µ)' r !

-l/z rexp?µd

xz.

/ `7`

L z=0

Moreover,

xzr+1

a-uX=/2

dx = 0

`dr

E N.

Thus for any I x I diagonal complex matrix M, with nonzero eigenvalues

whose arguments e I - n/2, n/2I, and for any polynomial P : X - C,

= 2

l/z (20f

-uX'/2d

x e

P(x)e-m`(X)dix

= (2ir)'/z[Hess cQ

-112[eY',(a/3X)P(x)]x=o,

(3.9)

where

(D,(x) =

i<Mx, x>, 'P (P) = i< P, MC 1P);

arg Hess (Dc is the sum of the arguments of the eigenvalues of ct . More

explicitly, we write

where v e ill, x[ and b+ and D are two quadratic forms X -+ R, in-

dependent 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;,

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

J(x)P(x)e v(x) &.(x)dix

\ - uz

= CHess

- (D+

[e' (era:)P(x)]s=o mod `1

; (3.10)

argHess((D - (1/v)(D.+) is the sum of the arguments of the eigenvalues