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

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204

III,§2,2-III,§2,3

a lagrangian manifold V2 or V1 satisfying this quantum condition. Then

by lemma 2.5 the graph of the function N contains

either a rectilinear segment with the equations (2.40)-(2.41) or

a planar domain with the- equation (2.38)-(2.39) and hence such a

segment.

Proof of the corollary.

The condition No E Z evidently emplies that such

a segment exists. Conversely, assume that in the space R3 with coordinates

(L, M, N), the plane with the equations (2.43)-(2.44) contains a line with

the equations (2.40)-(2.41). This assumption is expressed by the four

relations

L1L'1 + M1Mi + N1Ni = 0,

N0 =

MiNo

NiMo

Nj'L0

- LiNo

LIMO

MiL0

L1 M, N1

(two of the last three result from the other two). These relations imply

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

3. Example: The Schrodinger-Klein-Gordon Operator

Let us choose a to be the operator associated to the hamiltonian H defined

by (4.6) of §1, where A is assumed to be an affine function of M, B and C are

assumed to be constant, and B > 0.

In §1, we studied the system

aU = (aL: - L')U = (am= - M0)U = 0 mod Z

and recovered the classical energy levels. In studying the single equation

aU = 0 mod

we shall find again the same condition for existence, thus the same classical

energy levels.

THEOREM 3.1.

The lagrangian equation

aU = 0 modv

(3.1)

III,§2,3

205

has a solution U with

lagrangian amplitude >,O defined mod(1/v) on a

compact manifold V if and only if there exists a triple of integers (1, m, n)

satisfying condition (4.11) of §l,theorem 4.1.

Remark 3.

Under this condition, neither the unicity of V nor the unicity

of U up to a constant factor is assured.

Proof.

Under condition (4.11) of §1, the existence of such a solution of

(3.1) is assured by theorem I and also by theorem 4.1 of §1.

By theorem 1, there exists such a solution of (3.1) defined on a torus

V[L0, MO] only if this condition is assumed.

By theorem 2 and formula (4.8) of §1, which gives the value of the func-

tion N as

N[L, M]

.11/1 M

- + C,

the existence of such a solution on a manifold V other than a torus

V[L0, Mo] requires that the graph of N contains a line segment. Thus it

requires that one of the three following cases occur.

First case:

A(M) = Aa is independent of M; C = 0.

Then by (3.2) the graph of N is the plane with the equation

L+N=

,r^_o

Theorem 2 and its corollary require the existence of an integer n such that

hn.

Then condition (4.11) of §1,theorem 4.1 is satisfied since it is independent

of I for C = 0 and independent of m for A(M) independent of M.

Second case: C = 0; A depends on M.

By (3.2), the only lines contained in the graph of N are given by the

equations

M = Mo, N + L = Ao,

where MO = const.,

Ao = A(M0). (3.3)

Their pluckerian equations are then

206 III,§2,3

(1,0,-1) A

(L-2,M,N+21=(MO,-

,1l'10).

Theorem 2 requires the existence of integers m and n such that

MO = Aim,

f= hn.

(3.4)

From (3.3) and (3.4) it follows that

n > I m I because N > 0 and L > I MI by assumption [see (2.2) of §1].

Thus condition (4.11) of §1 is satisfied.

Third case:

A = Ao is independent of M; C 0 0.

By (3.2), the only lines contained in the graph of N are given by the

equations

L=Lo, N= -v'Lo+C,

where Lo = const., Lo + C > 0.

Their pliickerian equations are

(0,1,0)A(L-2,M,N+2

=I =

"[LO

C+2,0,2-Lol.

`f o

Theorem 2 requires the existence of integers 1 and n such that

+Lo-\/Lo+C=hn,Lo=h(l+

VIAo

0 < 1 because 0 < Lo; I < n because N > 0 by assumption.

Thus condition (4.11) of §1 is satisfied.

THEOREM 3.2.

Choose a to be the Klein-Gordon operator (4.22) of §1. Sup-

pose the magnetic, field .$ 0 0. Then the tori T(l, m, n) defined by (4.12) of

§1 are the only compact manifolds on which there exists a lagrangian solution

U of the equation

a U = 0 mod i

with lagrangian amplitude >,0.

III,§2,3-III,§3,1

207

Proof.

The proof of theorem 3.1 shows that the necessary condition

stated in theorem 2 can not be satisfied when a is the Klein-Gordon

operator (C 0 0) and '

0 (A depends on M).

Conclusion

We do not pursue this difficult study of the equation aU = 0 mod(1/v2).

In particular, we do not describe the lagrangian manifolds other than the

tori T(l, m, n) on which there exist solutions of the Klein-Gordon equation

when .f = 0 or solutions of the Schrodinger equation.

§3. The Lagrangian System

aU = (am - const.) U = (aL - const.) U = 0

When a Is the Schrodinger-Klein-Gordon Operator

0. Introduction

In §3, we study the lagrangian system that was solved mod(l/v2) in §1.

In section 1, we determine the condition under which theorem 7.2 of II,

§3 applies.

In sections 2, 3, and 4, we apply this theorem under assumptions that

become more and more strict. These assumptions finally amount to the

assumption that a is the Schrodinger-Klein-Gordon operator. Existence

theorem 4.1 is finally obtained.

Remark 0.

A Voros orally pointed out that these properties of the

Schrodinger and Klein-Gordon equations extend to the case where the

electric potential is any positive-valued function of the variable R, if the

energy level E is not constrained to be a real number, and if it can be taken

to be any formal number with vanishing phase.

1. Commutivity of the Operators a, aL2, and am Associated to the

Hamiltonians H (§1, Section 2), L2, and M (§1, Section 1)

We want to determine when theorem 7.2 of II,§3 applies to these operators,

that is, when they commute.

LEMMA 1.

10) am and a (thus, in particular, am and aL=) commute.

2°) aL, and a commute if and only if

HM2Q = HM2R = 0

YL, M, Q, R. (1.1)

208

III,§3,1

Proof. Let a and a' be the lagrangian operators associated to two hamil-

tonians H and H'. By formula (1.1) of II,§2, their commutator

a 0 a' - a'oa

is associated to the formal function given by

1 1

-2

(2r + 1)! (2V) 2r+1

XxpX'ap

)]2,4

H(x, p)H'(x', P')

P =P

Suppose H and H' are in involution, which is the case for H(§1,2), L2, and

M(§1,1) by (1.3) of §1. Then the first term of (1.2) is zero. If H' is a poly-

nomial of degree 2 in (x', p'), then all of the other terms evidently are zero;

thus a and a' commute and part 1 of the lemma follows. Suppose that

H' is a polynomial that is homogeneous of degree 4 in (x', p'). Then by

(1.2), a o a' - a' o a is associated to H"/v3 where H" is the hamiltonian

given by

H"(x, p)

24 (ax /-)'

/-)]

H(x, p)H'(x', p')

Two applications of Taylor's formula show that the term of H'(x' + y,

p' + q) that is homogeneous of degree 1 in (x', p') and degree 3 in (y, q) is

6 [(q, aP

/ + \Y, ax

)] 3H (x', P') _

[('aq)

(X ay)] H'(y, q);

thus, if H' is homogeneous of degree 2 in each of its two variables, then

H"(x, p) =

[(p' HP (ap' ax)>

\x' Hz, (ap' ax)!] H(x, p).

In particular, if H' = L2, that is, H'(x', p') = Ix' A p'I2, then

a \

H"(x, p)

= 2 P A

- + X A A

' a

ax/J

H(x, P).

(1.3)

In this formula, for each j the pair of operators

commutes.

r p n ca + x A z)

and Op n

P ;

III,§3,1

209

The operator

Cpna +xn

is an infinitesimal rotation acting on x and p; thus it annihilates L, Q, and

R. Obviously,.

Cp A a + X A

M(X, P) = x3P

- P3x.

P f

Suppose H is a composition with L, M, Q, and R [§1,(2.1)]. Then (1.3)

becomes

H "(x, p) = 2 (a A X, HMX3P - HMP3X )

\ P

Let

X2 X3

X F =

YF =

X1 X2 X3

P2 P3

F,,

FX2 FT,

P1 P2

Fp,

Fp2

Fp3

for any function F of (x, p). The preceding expression for H" may be

written

H"(x, p) =

12 aax9/HM

2 a

H"t.

Now, the linear differential operators I and Y evidently annihilate P2, Q,

and R 2, hence L2, by (1.2) of §1; moreover,

-M =

-l-L2

1aL2

219X3

2 eP3

Thus, letting !2F denote the functional determinant

aL2 of

ax3 OP3

aP3

ax3,

the expression for H" becomes

H"(x, p) = 4'!HM2

210

III,§3,1-III,§3,2

Now, the linear differential operator .9 evidently annihilates L and M;

moreover,

z9Q=P2x3-R2P3

and R -9R = Qxj - R2P3x3;

thus the preceding expression for H" becomes

H"(x, p) = i (P2HM2Q + RHM2R)

x3 - R HM2RP3x3

i z

- 2R HM2QP3

(1.4)

By §1.1, p3/x3 is independent of (L, M, Q, R). Thus the condition

H"(x, p) = 0

Vx, p

is equivalent to (1.1), which proves part 2 of the lemma.

2. Case of an Operator a Commuting with aL2 and am

Suppose (1.1) holds. Lemma 1 proves that theorem 7.2 of II,§3 applies to

the lagrangian system

aU = (aL2 - cL)U = (am

- cM)U =

0 mod 1

where cL and cm are two formal numbers with vanishing phase such that

-Lo=cM-M°=Omod v2.

(2.2)

The lagrangian operators aL2 and am have the following expressions ail

and am' in R° by I,§1,3 and the formula

1 a 2

eXp2v

O0x,

ap)]

(x, p) =

C1 +

(')

+ 8v2

ap)2]

.(p2R2 - Q2)

=PZRZ - QZ -

2Q- 3

2v2

namely

aL2(v' x' vex

(A0

(2.3)

III,§3,2

where (4)

=R2A->xx-

-2zx.

j, k

aXi OXk

axe

211

is the spherical laplacian (2.24), which acts on the restrictions of functions

to spheres R = const.;

(V,

x, v8X )

aX2 -

x2aX

which is an infinitesimal rotation.

Let us require that the unknown U be defined mod(1/vr+') on a compact

lagrangian manifold V. By theorem 3 of §1,

i. V is necessarily one of the tori

V = V[L0, M0] = T(l, m, n); H = L2 - L2 = M - M0 = 0,

which part 1) of this theorem defines.

ii. The lagrangian amplitude /30 of U is necessarily constant.

If (30 = 0, (2.1), reduces to (2.1)r_1: hence we impose the condition

J30 = const.: 0.

(2.5)

The problem defined by the system (2.1),, condition (2.5), and the

condition that V be compact will be called problem (2.1),.

Notation. The invariant measure is given by (3.16) of theorem 3 of §1;

d3x by (1.16) and (3.6) of §1; arg d3x = nmRO by definition [(3.4) of I,§3];

MR. is given by (3.5) of §1; arg rl, = 0 by definition; hence the value of the

function x, defined and used in II,§3, and the value of its argument are,

respectively,

?IV

x =

= [R3HQ[LOi M0, Q, R] sin 'Psin ®]-', where O = const.;

arg x = - arg HQ - arg sin T,

where arg H. = -n[(1/n) arctan(HQ/HR)], arg sin 'P = n['P/n].

(Recall that [ ] is the integer part.)

The apparent contour of V is

'As usual: 0 = Ej(a/ax;)Z.

212

III,§3,2

ER°: HQ sin `I' = 0; thus X: V\ERO - R.

(2.7)

Let U be a lagrangian function on V with lagrangian amplitude

f30 = const.: 0.

Its expression in Ro will be denoted

UR0 (v) = fZ-#(v)e"R-, where fl(v) _ Y

2.8)

By the structure theorem 2.2 and definition 3.2 of lagrangian functions in

II,§2, the function

X-3,135: V

regular

even on ERO

Let Dt,, DL, and DM be operators such that

LaU]RO = V A e"-PR, DHl3,

L(ae - Lo) U]R" =

V X

eMR. DLQ,

DH, DL, and DM commute since a, a,!, and am commute.

Let us use the local coordinates (R, `P, (D) on V\ERO.

LEMMA 2.1.

1°) With this choice of coordinates,

DM = a(D.

2°) If U is a solution of the equation

(am - cM)U = 0 mod

r12,

where cm is a formal number with vanishing phase such that

(2.10)

(2.11)

cm = MO

modhv,

III,§3,2

213

then

cm = M. mod

r+21

$ depends, mod(1/v'+'), only on the coordinates (R, `I').

Proof of 1°). Let us calculate DM by using theorem 4 of II,§3: in this

theorem, the hamiltonian M - Mo is substituted for H. By (3.18) of §1, the

characteristics of this hamiltonian are given by the equations

dt=dT=0;that is,dR=dP=0.

The parameter of these characteristics is b, which is substituted for tin this

theorem. We obtain (2.10).

Proof of 2°).

For r = 0, 2°) is obvious. Proceeding by induction on r, we

can assume 2°) is true when r is replaced by r - 1. Then

cM=Mo+Mr+1

M,+1 C.

By (2.9) and (2.10), (2.11) is equivalent to

M,+i f

o ,where

J3o = const.: 0.

Now, fi, is a function of (D having period 27r; thus

M,+1 =0,

00&

=0,

from which 2°) follows.

Notation.

From now on, we assume that (i depends only on the variables

(R, `I'). By (2.10), DH and DR° commute with a/a(D and thus act on functions

of(R,`I').

LEMMA 2.2.

1°) In the local coordinates (R, `P, (D),

d-r

$ = 2LD

_ 1 1

12)(2

L o

d`If

where T is the v

aT v J

ariable

T = cot F