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

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194 III,§2,2

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.

(2.8) 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

(2.13)

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. Formula (2.11) of §1 shows that theorem 3.1 of

II,§3 applies with

1=3,

h,=L,

h2=M,

go=S2+LT+MO,

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), `V + A + fL(L)(D + u) + FL(L) = 0;

(2.15)2

the result of the permutation of (L, T), (M, 0) in (2.15)2;

(2.15)3

L = const.,

= const.;

(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

L = L(s),

M = M(s),

(2.16)

L5(s)('Y + A) + MS(s)(G + u) + F,(s) = 0,

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

'Y + A[L, M, t] + MS 0

+ p [L, M, t]

1 Fs

E G(s),

271

27[

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

e G(s),

n 2n

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

MS _

G.C.D.(L M,, N,) = 1.

L, M,

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

t) =

pL2 + 2aLM + TM'

F(L

L(L2-M2)(pi-a2)

( )

is defined on V2\E", E" being the subset of V2, where

M2 LM L2

2°) If V2\E" is connected, then

( . )

mR° = 17r

1'I'

+ 1

arctan F(L, M, t) the integer part of )

on its universal covering space.

3°) The function

(2.19)

M -

- 2

20)

c[L,

is defined (that is, uniform) on V2.

III,§2,2

197

Remark 2.2. Assume a function

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,

V2\E" is connected.

Proof of 1°). 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

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

dR A w2 A w3 = G(L, M, t,'Y) dL A dM A dt

(2.22)

R sin 0

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",

198 III,§2,2

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,

where F is defined. by (2.17).

Calculation of mRa. Formulas (1.11) and (1.13) in §1 give

sin

0CO3 A wl

(2.23)

= 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),

dR A (03 A wl = Gy,(L, M, t, T) dL A dM A dt. (2.25)

k -sin 0

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

arctan F(L, M, t) mod 1.

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

III,§2,2

is defined. By (2.19), the function

MR.

- 1

`l, - 1

arctan F is defined on V2\E";

n n

199

(2.26)

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.;

by lemma 2.1, 1/f = 0 is equivalent to HQ = 0 and

(2.28)

H f <0

in a neighborhood of the points where HQ = 0; thus

arctan

+ arctan H4 is defined on V2\E";

now, from the orientation of I'[(2.9) of §1],

(2.29)

d on ['[L

is d

fi M]ct

+ 2 t

n H

30)

ne , .

ar a e

7zc[L

R M]

The formulas (2.26)-(2.30) prove 3°) for H generic; 3°) follows.

LEMMA 2.4.

1°) Define a constant No and a function r on Vl by the

relations

L1M-M1L=No, (2.31)

T + +Mlr=dD+µ-Llr=0.

(2.32)

The functions (r, s, t) are coordinates of the points of V1. A function

F : V1 - $ is defined by the formula

200

III,§2,2

_ N2

F(r, s, t)

L(L2 - M2)(pL1 + 2aL,M, + TM;)

(2.33)

when V, is generic.

2°) If V, is generic (No # 0; HL2L1 + 2HLML,M, + MM2M2

0 for

HQ = 0), then

MR° =

[!w

+ arctan F(r, s,

t)]([ ] is the integer part).

(2.34)t)]

3°) The function (2.20) is defined on V,.

Proof of 1°).

Formula (2.11)2, where F = 0 (remark 2.1), justifies the

definition (2.32) of r. By (2.2) and (2.10),

A,(pL2 + 2oL,M, + TM2) = 2n[L1NL2 + 2L,M,NLM + M2 NM2]

2 N

= 2n

ds2

= 0.

Proof of 2°). The apparent contour of V1.

Differentiating (2.32) gives

d'P+(pL1 +aM,)ds+M,dr=dI +(aL, +TM,)ds-L,dr

= 0 mod dR

by (2.10) and definition (2.1) of p, a, T. Whence by (2.21)

dR A w2 A cu3 = G(s, t, 'P)ds A dr n dt,

Rsin0

where

(2.35)

G(s,t,'F) =

HQ[L(pL2

+ 2aL,M, + TM1)sin'P + L2 N0M2cos'If

since dR =

RHQLLdt

mod(dL, dM), that is, modds. By 1°) and lemma 2.1,

the function G: V, - $ is defined and regular on V, Thus, by lemma I of

§1, the apparent contour ER° of V, is given by the equation

ER,,: tan 'P + F(r, s, t) = 0.

Calculation of mR°.

The same calculation used to deduce (2.35) from

(2.21) enables us to deduce the formula

dR A (a3 A 0), = Gyi(s, t,'P)ds A dr n dt

R sin 0

III,§2,2

201

from (2.24). Applying lemma 1 of §1 as is done in lemma 2.4, it follows that

mR° can

be expressed by (2.34).

Proof of 3°).

Suppose that V1 is generic. By (2.34),

mR°

- 1'f,

arctan F(r, s, t) is defined on V1.

(2.36)

From (2.33), where I M I < L by assumption (2.2) of §1, and from lemma 2.1,

F = 0 is equivalent to H. = 0, and

FHR

< 0

in a neighborhood of the points of V1 where H. = 0; thus

arctan F + arctanHQ

is defined on V1.

(2.37)

Formulas (2.30), (2.36), and (2.37) prove that the function (2.20) is defined

on generic V1, hence on every V1.

LEMMA Z.S.

1°) VZ satisfies Maslov's quantum condition if and only if, on

Vz, the functions L, M, and N are connected by a relation

L1CL-2+M1M+Nl(N+21 =f+N0,

(2.38)

where

L1, MI, N1, No E Z,

N1 : 0,

G.C.D.(L1, M1, N1) = 1. (2.39)

2°) V1 satisfies Maslov's quantum condition if and only if, on V1, the

functions L, M, and N are connected by the three relations

(L1, M1, N1) A

L - 2, M, N +

62I

= h(Lo, Mo, No),

(2.40)

where A denotes the vector product in E3 and

L1,M1,N1,Lo,MO,NoEZ,

Li + MI :g 0,

G.C.D.(L1i M1, N1) = 1,

LoLI + MOM1 + NON1 = 0.

Thus only two of the three relations (2.40) are independent.

(2.41)

202

III,§2,2

Preliminary to the proof.

A manifold V satisfies Maslov's quantum con-

dition (II,§2, definition 6.2) if and only if the function

21 (PR - 4 mR is defined mod 1 on V.

7rh

Then by (2.14) and (2.20), V, or V2 satisfies this condition if and only if

the function

(N j )-t

h +2)

2n+

h2n+(h

+2 c[L,M]

(2.42)

is defined mod 1 on V, or V2.

Proof of 1°).

By lemma 2.2,1°) the function (2.42) is defined on V2i

V2 = V2/2, where Z is generated by

(L1, M1, N1)

and acts on V2 according to (2.8). Thus Maslov's quantum condition is

(

+2)L,

+M, +

+')N,

EZ;

whence 11°). \

Proof of 2°).

By lemma 2.2,2°) the function (2.42) is defined on V,

V, = P, /22, where 22 is generated by

(0, M2N3, -M3N2),

(-L2N3, 0, L3N2),

(L2M3, -L3M2, 0),

L2, ... , N3 being defined by (1.12) and (1.13), and Z2 acts on V, according

to (2.8). Thus Maslov's quantum condition is

M2 N3 - (iNNi

2) M3 N2 E Z,

-(L +2)L2N3+(N+2)L3N2GZ,

(

)L2M3 -

L3M2 E z.

By (1.13), this quantum condition is equivalent to the condition (2.40)-

(2.41) supplemented by the following one: Lo, M0, and No are multiples of

L2, M2, and N2, respectively. Now

III,§2,2

203

L2 = G.C.D.(M,, N,), ... ;

thus this last condition is a result of (2.41); 2°) follows.

THEOREM 2.

Assume that the lagrangian equation

aU = 0 mod

(a the operator associated to H)

has a solution U that has a lagrangian amplitude > 0 and is defined mod (1/v)

on some compact lagrangian manifold V other than the tori V[L0, Mo]

studied in theorem 1; then the graph of the function

N: (L, M) -- N[L, M]

[see III,§1,(2.9)]

contains a rectilinear segment given by a pliickerian equation

(LI, MI, N,) A

M, N +

I = h(Lo, M0, NO)

(2.40)

such that

L1, M1, N1, Lo, Mo, No E Z, Li + M1

(2.41)

G.C.D.(L1, Ml, N1) = 1, LOLL + MOM, + NON1 = 0.

COROLLARY.

The condition

NoEZ

is necessary and sufficient for there to exist such a segment in a planar

domain belonging to the graph of N and satisfying an equation

L;(L-2) +M,M+Nf(N+hl=hNo,

(2.43)

where

'1 + 1 0 0,E,, Mi, Ni E Z, L

(2.44)

G.C.D.(L'I, M,, Ni) = 1,

No a R.

Remark 2.3.

This corollary makes it easier to apply the theorem (see

section 3).

Proof of the theorem.

By II,§3,theorem 6, V must satisfy Maslov's quan-

tum condition. Since V is not one of the tori V[L0, Mo], V must contain