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

Подождите немного. Документ загружается.

I1,§1,3

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

from the Taylor series of this function; arg Hess d) is given by I,§1, defini-

tion 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

Thus

e(x)P(x)e"O(x)dCx.

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. Let

0e 10, 11,

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,

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 modI (3.12)

V IV,

From this lemma we deduce the following.

LEMMA 3.5.

Let [p : X - R be infinitely differentiable with only one critical

point, which occurs at the origin and is nondegenerate, and critical value

zero:

(p (0) = 0,

(px(0) = 0, Hess (p(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 `O'(x)a(x)evO(x)d`x

mod (3.13)

j=0 j!

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]

where [

. ] denotes the integer part of . - that is, the greatest integer

Proof. By (3.12) and Taylor's formula,

2N-1 vlOJ

vm 2N

(1 -

e)2N-1

e= e

O e 0. (3.14)

J=1 J!

(2N - 1)!

Lemma 3.4 gives

v2N f

f1(1

0) 2N-1

0 mod N

x o v

(3.15)

since O2N vanishes to order 6N at the origin. When N tends to infinity,

(3.14) and (3.15) imply (3.13).

86 II,§1,3-II,§1,4

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

(Hess (D)-112 1

IvI

,1 r !j ! vr-i

where r, j e N. Denote

mod Ixt3, the condition

a I

[®'(x)a(x)]jmod 1

\ax/ x=0 v

(3.16)

the exponent of v by q = r - j. Since O(x) = 0

{ -

}x=o : 0 implies 3j

2r, that is j < 2q. Let

(')

q(o; a) _

O*q+j[Oj(x)a(x)1JI

i=0

(q + j)i j

x=0

in particular,

Io((p; a) = 00).

Now (3.16) can be written

ess)

1 2 gIq((p; a),

fX (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.

THEOREM 4.1.

Let Z be a 2-symplectic space, V a lagrangian manifold in

Z, R' a 2 -frame of Z, and

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)

(4.2)

nRl

1-14

) W(X )F X

(4 1)

. o( )

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.

Let

UR. =

11x,e°(0" E'0(V\ER U ER., R').

rEN

Then

UR = SUR. E .Fo(V\LR U ER,, R),

where R = SR',

is given at i by

x3r+1/2

II,§1,4

vroA(=)

J ('

(4 6)

R(V,

where

R,, ;

rEN

(dtX

x=RXZ,

x'=Rzz,

[dtx] 112 is defined in I,§3,corollary 3.

'IR.r is a linear function of the derivatives of the as (s <, r) of order < 2(r - s)

on V Its coefficients are infinitely differentiable functions of z E V\LR.;

these functions depend on V, R', and S.

JR.OIZ, x) = 00(2)

(4.8)

Formula (4.6) retains a meaning for all UR. E y (V \ER, ,

R') and i E

V\ER U ER' ; therefore the following theorem results from the preceding

one.

THEOREM 4.3. Formula (4.6) defines an isomorphism induced by S and

denoted S:

S:.F(V\ER U ER., R') , .F(V\ER u ER., R). (4.9)

The composition law of these isomorphisms is that of the group Sp2(l).

Proof of the preceding theorems. Let UR. E Ao(V\ER V ER., R') be such

that the restriction

: Supp UR. --+ X of RX : V X

is a diffeomorphism, with inverse denoted AX 1

The case S 0 Esp,. Then S = S,1 (I,§1,1). Let OR. be expressed by (4.5);

U' = rIR. Ua.

is given at x' by

u'(v, x') = E lxr(x')eVP'(x'),

(4.10)

rEN l

where

II,§1,4

a;(x') = a,(RX

1 x'),

cp'(x') = cpR,(RX 1 x') for x' e Supp u',

a; (x') = 0 for x' 0 Supp u' = RX Supp UR..

Since u' e po(X) c '(X), we have u = Su' e

'(X). By I,§1,(1.10), we

have the following expression for u at x :

IVI U1

2niA(A)

ar(x

)e"1n(x,X)+w'(x)]dlx'.

u(v' x) -

()2

X rEN

(4.11)

We apply theorem 3 to this asymptotic integral.

i. For each x e X let us find the critical points of the phase

X -3 x' -- A(x, x') + 1p' (x') e R

(4.12)

in Supp u'; that is, the x' e Supp u' such that

As.+9',,=0.

(4.13)

Let us write

i = AX 1x',

R'z = (x', p'); that is, p' = q . e X*.

(4.14)

Relation (4.13) becomes

p'+Ax,=0.

By I,§1,(1.11), this means that there exists p e X* such that

(x, P) = sA(x', P')

Thus (4.13) is equivalent to

x =

RxR'-1(x', p')

= RXR'

1(x', p').

By (4.14) this says that (4.13) is equivalent to

x=fix.

Therefore, on Supp u' the critical points of the phase (4.12) are the points

x' = I' i, where 1 e RX 1x n Supp UR..

(4.15)

This set is finite since Supp UR. is a compact set in V\ER and RX is a local

diffeomorphism in a neighborhood of this compact set.

ii. Let us find the critical values of the phase (4.12). At a critical point

x' defined by (4.15), let

II,§1,4

Iii

= (x, P),

= (X" p');

from the preceding formulas,

p= Ax,

p' = -Ax..

Hence, since A is a quadratic form in (x, x'),

A(x, x') = z <p, x> - i <p', x'>,

that is, in view of I,§3,(2.4),

A(x,

x') = cPR(z) - APR (z),

where cpR,(z) = cp'(x').

Therefore at the critical point x' defined by (4.15), the value of the phase

(4.12) is (PR(4

iii. The square root of the Hessian of this phase (4.12) is given by I,§3,

(3.7):

{Hess..[A(x, x') + (p'(X)]}l,Z =

J1/2.

(4.16)

While calculating Hess.-, x and x' are viewed as independent. Afterwards

we give x and x' the values x = RXi and x' = RXi so that x and x' are

local coordinates of the same point i e V\tR u

Taking these values

for x and x' in the right-hand side, it follows from (4.16) that

Hess.. [A (x, x') + p'(x')] : 0.

Consequently theorem 3 can be applied to the calculation of (4.11).

iv. Completion of the proof. This application of theorem 3 to (4.11) gives

U(V, X) _ Y_

UR(V, z),

zERx1X

where:

UR has the same support as OR. ;

defining x = RXi and x' = RXi, the value of OR at Z is

(4.17)

UR(v, i) = A(A)[Hess..(A + gy0')]_1/2 Y 1 I,(A + (p'; x') e"*-('). (4.18)

ti'

Now on one hand, (4.17) means

U = nRUR,

II,§1,4-II,§1,5 91

but on the other hand, by (4.16), (4.18) is equivalent to (4.6). Indeed, by

(3.3) Ir(A + (p'; a') is the value of a linear function of the derivatives of

the as(s < r) of order <2(r - s); the coefficients of this function are

rational functions of the derivatives of cp'; the common denominator of

these rational functions is [Hessx.(A + (p')]3i, whose value is given by

(4.16).

Thus theorems 4.1 and 4.2 hold for SA E Sp2(1)\Esp2

The case where S E Esp2. By I,§1,lemma 2.5, given S e Esp, there exist

SA and SA' E Sp2(1)\Esp2 such that

S = SA, SA.

Since the morphisms (4.1) and (4.2) are composed like the elements of

Sp2(l) that induce them, the compositions of the morphisms induced by

SA and SA, depend only on S = SA.SA; these are then morphisms induced

by S. Given UR. such that Supp UR c V\±R' U ESR', we can choose SA

sufficiently close to the identity so that tSAR, n Supp UR, = 0, whence

follow theorems 4.1 and 4.2, which imply theorem 4.3.

5. Norm and Scalar Product of Formal Functions with Compact Support

Let V and V' be two lagrangian manifolds in Z, V and V' their universal

covering spaces, R a 2-frame of Z, c°R and wR the phases of R V and R V,

0 and 0' the lagrangian phases of V and V. Let

UR = Y- laR reYWR E fo(V\iR, R),

reN V'

UR =

is aR sev'6

E Col V'\i' , R),

SEN

with projections

U = IIRUR,

u' = IIRU'R.

Definition 5.1.

Under assumption (5.1) the scalar product of UR and

UR is the asymptotic class

(UR I UR) = (u I u') E

(5.2)

where (u I u') is defined by the asymptotic integral (1.9). Thus

(URI UR) and (UR UR) are complex conjugates and the Schwarz inequality

applies:

II,§1,5

(UR I UR)I

(UR I UR)112(UR I UR)1f2.

The seminorm of UR is (UR UR)112 = iI u

0; this seminorm satisfies

the triangle inequality.

THEOREM 5.I.

The value of (UR I UR) depends only on the behavior

of UR and UR at the pairs of points of V and V' projecting onto the same

point of V n V'. This value is 0 if V n V' = 0.

2°) If V = V', which implies cpR = (pR, then

(UR Ua)

= Y

( .1)5

aR..(z)aR.s(z

a .°F,

(5.4)

.,s

zEV ;.Z

where:

x=Rxz,d'x>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.

Thus the phases of (UR I UU) are these periods c., of 0.

3°) If V and V' are transverse, then

vy2(OR I UR) a

(5.5)

If V and V are transverse and intersect at a single point z that is the pro-

jection of a single point i of Supp UR and a single point i of Supp UU, then

1/2

(UR I UR)

2li

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

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;

PO(aR, aR) = a R,o(Z)aR.o(fl

(5.7)

4°) [Invariance under Sp2(1)]. Let S E Sp2(1). Under the assumption

OR E .FO(V`tR U ±SR, R), U' c- FO(V'\ R V t'

SR, R),

which is stronger than (5.1), we have

(UR I UR) _ (SUR I SUR).

(5.8)

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;.,();

jEJ

rEN

u'(v, x)

= Y dk(v,

where vk(v, x) _

vk,r(X);

kEK

rEN V

J and K are finite sets. By part 1 of theorem 3,

(u I u') _ f u(v, x)u'(v, x) d'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).