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

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224

III,§3,3

Now by the equation (3.3) for F[Lo, Mo],

Q2 = K[R, Mo]

- Lo

hence the expression (3.6) for G, follows.

The preceding lemma permits a statement of the properties of problem

(2.3) to which problem (2.1), has been reduced by theorem 2.

Definition 3.

A function g: F[L0, Mo]\E - C is said to be even or odd

when

9(Q, R) = ±g(-Q, R)

V(± Q, R) E I'[Lo, Mo]\E.

THEOREM 3.1.

(Complement to theorem 2)

Assume (3.1).

1°) If problem (2.31), has a solution g such that go = 1, then it has a

unique solution such that

go = 1,

and

gs is real and has the parity of s (s < r),

99 = 1 -

R (9 dg

- g

do)

mod

2v dRdRvr+,.

Formula (3.10) means that, for 2 < 2s < r,

R 2s-1

(-1)5925 s95 =

Q S'o(-1)sg,'dR92s-1-5;

thus it expresses 92, using g1,

... , 92,-1

(3.10)

(3.11)

2°) If problem (2.31)2,_, has a solution, then problem (2.31)2, has a

solution.

Remark 3.

The formal function U" defined on F[Lo, Mo]\E by theorem

2,2°) is evidently given by

°oU"(v) = g

e .

)

( .

It satisfies

0 U"(v R) = R

Z {

K [R Mo] - Lo - 4v2 U (v, R),

(3.13)

where A is the laplacian, specifically (d 2 /dR 2) + (2/R)(d/dR).

III,§3,3

225

Proof of 1°).

Assume that 1°) has been proved when r - 1 is substituted

for r and that problem (2.3)r is solvable. Then, by theorem 2,2°) and

lemma 3, gr is defined up to a constant of integration by the conditions

dg,

= Ggr_1 on F[L0, Mo]\E,

Q3rgr is regular on F[L0, M0].

(3.14)r

Now, G changes parity. If r is odd, then a convenient choice of the constant

of integration makes gr real and odd. If r = 2s is even, then 92, is even and

can be chosen to be real.

We have

dg _

modvrl1,

Gg=G19+dR[G2dR]

By a trivial calculation analogous to the proof of (2.19), we deduce (3.10),

up to the addition of a formal number E,(c,/v`), which is canceled by

conveniently choosing the constants of integration of the g2,-

Proof of 2°). Assume that r = 2s is even and that problem (2.31)2s_1

is solvable. By theorem 2,3°), problem (3.14)2s has a solution g2s when t

is replaced by its universal covering space t. The proof of (3.11) remains

valid; (3.11) proves that 92s is defined on F[L0, Mo]\E; thus 92., is a

solution of problem (3.14)2..

Proof of remark 3.

By the expression (3.2) for a and theorem 2,2°),

{v2A

RZK[R'v

xzax

)

][U(v,x)U (vex)]

z 1

where U' is homogeneous in x of degree 0 and U" only depends on R.

This implies

A(U'. U") = U'- AU" + U" - AU' ;

hence (3.13) follows by (2.12)2.

Notation.

Let [R 1, R21 c R c C be the set of values taken by R on

F[L0, Mo]\E. Let o. be a simply connected neighborhood of the real

closed segment [R 1, R21 in the complex plane C; R E C.

THEOREM 3.2.

Assume that K is holomorphic in co. Then Q, defined by

(3.3), is holomorphic in (t)\[R1, R21.

226

III,§3,3-III,§3,4

1°) If problem (2.31)25 has a solution g, then, for all r _< 2s,

Q3i94, R) = (-Q)3'g.(-Q, R)

is the value of a junction of R that is holomorphic in w. In particular, 92,

is a function of R that is meromorphic in w, with poles R1 and R2.

2°) Problem (2.31)25+1 is solvable [and thus problem (2.31)25+2 also]

if and only if the primitive of (Gg2i)dR is defined (that is, uniform) in

w\[R1, R2].

Proof.

Suppose 1°) is true. [1°) is evident for s = 0]. Then, by (3.14),

the function g25+1: I'[L0, Mo]\t - C is obviously the restriction of the

primitive of (Gg2s)dR to the edges of the cut [R1, R2] in C, which is

defined on the universal covering space of w\[R1, R2]. Obviously, 925+1

is defined on I'[L0, Mo]\E if and only if this primitive is defined on

w\[R1, R2)

Assume that this condition is satisfied.

With a convenient choice of the constant of integration, this primitive

925+1 is an odd meromorphic function of Q in a neighborhood of R1

and R2. Like Q, it takes opposite values on the edges of the cut [RI, R2].

For R near R1 or R2i Q is near 0 and g25 is an even function of Q having

a pole of order 6s at Q = 0. From (3.3), (3.5), (3.6), and (3.14),

d 1 = G925 =

2-93-

n 925

RKR d 2s

where KR

dQ dQ

Q[ Q

Thus g25+1 is locally an odd function of Q having a pole of order 3(2s + 1)

at Q = 0.

Consequently Q3c2s+ng25+1 is holomorphic in w.

g25+2 is defined by (3.11). Then

Q3(2,+2

'92,+2 is holomorphic in a).

Consequently 1°) holds when s is replaced by s + 1.

4. The SchrSdinger-Klein-Gordon Case

In order to establish, for any r, the solvability of problem (2.1) which

we have reduced to problem (2.31) an appropriate assumption is clearly

necessary.

Assume that K is a second-degree polynomial:

K[R,M] = -R2A(M) + 2RB(M) - C(M).

III,§3,4

227

In other words, the expression of a in R0 is the Schrodinger-Klein-Gordon

operator (§1,4) and A, B, and C are affine functions of M.

Then in theorem 3.2, co = C. In the definition (3.5) of the operator G,

G3 is a polynomial in R of degree 3 and G1 and G2 are functions of R

holomorphic in C\[R 1, R21 and at infinity, where G1 vanishes to second

order. If g is holomorphic in C\[R1, R2] and at infinity, then so is the

primitive of (Gg)dR. Then all the g, exist and are holomorphic in

C\[R1i R21 and at infinity. Since g, is holomorphic at infinity and since

Q 3rg, is holomorphic in C, Q 3'g, is a polynomial in R of degree 3r. Thus we

have proven the following two theorems.

THEOREM 4.1.

(Existence and uniqueness)

Assume that the expression

of a in R0 is the Schrodinger-Klein-Gordon operator (example 4 of §1).

1°) The condition that the lagrangian system

aU = (aL2 - CL)U = (am - cM)U = 0,

(4.1)

where cL and cm are two formal numbers such that

C L

o-cM-M0=0mod

v2,

have a solution defined on a compact lagrangian manifold V is the following:

i. CL = Lo + (1/4v2), CM = MO ;

ii. V is one of the lagrangian tori V [L0, M0] = T(l, m, n) defined by

theorem 4.1 of §1.

2°) There exists a lagrangian solution U of (4.1) defined on such a torus

V having lagrangian amplitude

#0=1.

Any lagrangian solution of (4.1) defined on V is the product of U and a

formal number with vanishing phase.

Remark 4.1.

The projection of V on X is

Vx : R 1

x < Rz>

MoJxJ < Lo

xzl +

xzz,

where R1 and R2 are the two roots of the equation

AOR2-2B0R+C0+Lo=0

228

III,§3,4

(A0, Bo, and Co are the values of A, B, and Cat Mo.)

THEOREM 4.2. (Structure) There exists a unique solution U of (4.1)

defined on such a torus V having the following structure: Its expression

UR,, in the frame Ro (§1,1) has the form

U& M = U'(v) U"(v).

(4.3)

Using the local coordinates x e V, on V, U'(v) is a formal function of x

that is homogeneous of degree 0 and satisfies

U'( )) = M

1 U'(v

O v, x ,

, X

z i

(4.4)

U'(v, x).

(v' x)

Rz l L0 + 40

U"(v) is a formal function of R satisfying

V x)

= 1

{K[R, M0] - Lo _ 4vz} U"(v, x).

4.5)

U' and U" are defined by the formulas

U' v x

fv' t)ev(r.ovV+Mom)

( )' ,sin`' '

(4.6)

U"(v, x) =

g(v, Q, R)evo,

where T = cot `P and Q is the function of R defined by (2.8) of §1; arg sin 'Y

and arg Q have jumps of +n at the points 'P = 0 mod 7r on R and Q = 0

on ['[L0, M0], which are oriented in the directions dP > 0 and Q dR > 0.

Moreover,

f(v) _

rEN V

g(v) = I .rgr

rfN

are formal functions

with vanishing phase defined on R and on

I'[L0, M0]\E, respectively, by the following set of properties:

df = 1

Ff,

Gg,

(4.7)

III,§3,4

229

the differential operators F and G being given by

Ff = F1J + dT[Fzd-r

Gg G1g

+ dR

[G2]

where F1 and F2 are the even polynomials in T of degree 2 and 4 defined by

lemma 2.2,1°); QSG1 and QG2 are the polynomials in R of degree 3 and 1

defined by (3.6); fo = 1, go = 1; the functions f, and g, are real and have

the parity of r;

d f df

Jf =

)

- f

99 =

(

dR -

(4.9)

These formulas (4.9) give the even functions f2, and 92, in terms of fl,

f2,-1 and g1, ..

, 925-1.

The odd functions fzs+1 and 92,+1 are defined

by the quadratures

df2 +l = (Ffzs)dT,

dg25+1 = (Gg2,)dR.

f, is a polynomial in T of degree 3r. Q3rg, is a polynomial in R of degree 3r.

Remark

4.2. Thus

g25+1 - G2dg2 /dR

is the

odd

function

F[L0, M0]\E that is the primitive of the differential form G1g2,dR,

namely, of a form

II'(R)Q-65-sdR, where IT is

a polynomial of degree

6s + 3. By the preceding theorem, this primitive shall be

II(R)Q-6,-3

where fI is a polynomial of degree 6s + 3. Let us give a more direct proof

of this essential fact.

LEMMA 4.

For each s e N, the differentiation

fl(R)

fI'(R)

(4.10)

dR Qzs+,

Qzs+3

(Q2 a polynomial in R of degree 2, discriminant Q

0) defines an auto-

morphism fI " IT of the vector space of polynomials of degree 2s + 1.

Proof. Let Fl be a polynomial of degree 2s + 1; fI(R)Q-2s-1 is holo-

morphic at infinity; thus its derivative has a double zero at infinity.

Consequently (4.10) defines a polynomial fI' of degree 2s + 1. Hence the

mapping

II " rI'

230

III,§3,4-III,§4,O

is an endomorphism of a finite-dimensional vector space. Now it

evidently a monomorphism; thus it is an isomorphism.

Conclusion

This §3 is concerned with finding, for dim X = 3, a lagrangian system, with

one lagrangian unknown, having a solution defined on a compact lagrangian

manifold unique up to a multiplicative factor. In this §3, we have found a

system of this type: the system used in wave mechanics for studying atoms

(or ions) with a unique and spinless electron.

Remark 4.3.

Replacing the Schrodinger-Klein-Gordon hamiltonian by

the harmonic oscillator hamiltonian for E3,

H(x,p)2 [P2+AR2-2B],

that is, choosing

K[R, M] = -AR4 + 2BR2

where A and B are affine functions of M, one obtains another lagrangian

system (2.1) belonging to the same type.

This can be shown by computations similar to those performed above,

where R2 plays the role that R played above.

The energy levels are still those of wave mechanics.

Remark 4.4.

Of course, the study of the harmonic oscillator in R is

simpler; in E3, for A and B independent of M, it gives new lagrangian

solutions for the operator associated to the harmonic oscillator, but

again the same energy levels.

§4. The Schrodinger-Klein-Gordon Equation

0. Introduction

The following classical boundary-value problem will be called problem

(0.1): To find nonzero square-integrable functions

that have square-integrable gradients and that are solutions of the

differential equation

I1I,§4,0-I1I,§4,1

231

au=0,

(0.1)

where a is the differential operator associated to the hamiltonian

H[L, M, Q, R] = 21 P2 -

R2 K [R,

M]}

(0.2)

K being an affine function of M [compare (3.1) and (3.2) of §3]. This

operator is defined by substituting vo = i/h for v in the lagrangian opera-

tor associated to H [(4.5) of §1]. Hence it is

2V2

2RZKrR,

o(x1ax2

x2az1)]

In section 2, we assume (compare §3,4)

K[R,M] = -R2A(M) + 2RB(M) - C(M),

(0.4)

where A, B, and C are affine functions of M. Then a is the Schrodinger-

Klein-Gordon operator.

In §4, we briefly recall the solution of the classical problem (0.1) in order

to observe two facts:

i. There are formal analogies between its solution and that of the lag-

rangian problem (2.1) of §3, which is to solve

aU = (aL= - CL)U = (aM - c;,i)U = 0 on compact V.

ii. The condition for the existence of a solution to this lagrangian problem

[or to this problem mod(1/v2), namely, to (3.1) of §1] is the same as that

for the classical boundary-value problem (0.1) in the Schrodinger-Klein-

Gordon case, that is, under assumption (0.4).

This assumption is essential.

Remark 0. For a suitable choice of A, B, and C, the Schrodinger-

Klein-Gordon equation is the Schrodinger or Klein-Gordon equation

(4.21) or (4.22) in § 1, where the .

2 terms have been omitted. It is customary

to treat the terms of these equations that are linear in .' by "perturbation"

theory. We do not do this, but we rigorously solve problem (0.1) in order

to show that assertion (ii) is rigorous.

1. Study of Problem (0.1) without Assumption (0.4)

Review of the properties of spherical harmonics ui,m.

The set of poly-

nomials in x c- E' that are harmonic and homogeneous of degree 1 is a

232

III,§4,1

vector space over C of dimension 21 + 1. A basis consists of the poly-

nomials

R'ui,m,

I and m integers,

lml '< 1,

defined by the following system, where ul.m is homogeneous of degree 0

and thus satisfies R2 Au' = A0u' [see (2.4) and (2.24) of §3]:

Aui.m + R21(l + 1)u',, = 0,

I x1

axa - x2ax )u',m

= imui.m.

(1.1)

Let S2 be the unit sphere in E3 and let a be the usual measure on S2.

Then

J'S2

o ,

Ul,m

IU.ml2a

a = 1(1 + 1)

and for all (1,, ml)

(112, m2)

S/2

x Ul2'm2/ O =

fS2 U13,m1Ul2,"2Q = 0.

where < ,

> is the (sesquilinear) scalar product.

The restrictions of the ul,m to S2 form a complete system of functions

on S2 : every square-integrable function u : E3 -+ C with square-integrable

gradient has a unique expansion

ul,mul, m,

I'm

where the u;',,, are functions only of the variable R > 0, such that

R2luml2dR < o0

u12d3x

114",mlafo

JET

l,m S2

and

+ E

ml2a I R2

l.m

d3x =

I'm

Ul,m

2 + ac

a. J u,%ml2dR

Ul,m

dR<oc.

Resolution of problem (0.1).

By (0.3) and (1.1), the condition that u be a

solution of problem (0.1) is formulated as

II1,§4,1

233

d z

(d 2 + R dR)u`

m(R) +

Rz {

K[R, hm] - 1(1 + 1)

u';,m(R) = 0

(1.2)

for all 1, m. By writing Ru;,m = v, we may evidently state that result as

follows.

THEOREM 1.1.

Problem (0.1) is equivalent to problem (1.3), which is to

find integers 1, m and nonzero functions v:10, + oo I - R such that

I m l

<, 1,

dRZ + Ry

hzK[R,hm] - 1(1 +

1)}v = 0

and

+00 (I+' z)V2dR<

oo,

()2dR

< oo.

Remark 1.1. Thus problem (0.1) is solvable if and only if the following

problem is solvable: To find two integers 1, m and a nonzero square-

integrable function u: E3 -+ C with square-integrable gradient such that

lml<l

and

au = 0,

Dou + 1(1 + 1)u = 0,

(x, Y - x2 _ )

u = imu.

(1.5)

z i

Recall that system (1.5) plays an essential role in physics.

Remark 1.2.

Equations (1.1) and (1.2) and the system (1.5) are obtained

formally by replacing

U', U" U

uium, u

in equations (2.21) and (3.13) and in the system (2.1) of §3, where account

is taken of theorem 3 of §1 and theorem 2 of §3, which require