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

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114

II,§2,2

m is the Maslov index defined mod 4 (I,§3,theorem 1),

A4 and , are the 2-oriented lagrangian planes tangent to f/ and fl' at

i and i', respectively.

Proof of 1°).

Use definition 2.3, part 1 of II,§1,theorem 5.1, part 4 of

§1,theorem 6, and definition 1.2 of the adjoint of a lagrangian operator.

Proof of (2.10).

Use §l,definition 2.3 and §1,(5.4).

Proof of (2,11).

Use §1,theorem 5.2 and the fact that aU mod(1/v) is

multiplication of U by the function ao.

Proof of (2.12).

We use a partition of unity [part 3 of theorem 1.2]

Yb*obi = 1

sufficiently fine so that for every j there exists a frame Ri satisfying

Supp bi n IR, = 0.

We have

(UI U) = >(bJUR I biUR.) > 0 (§1,definition 5.1).

Proof of the Schwarz inequality.

By §l,definition 5.1, we have the tri-

angle and Schwarz inequalities in R [§1,(5.3)] and in W (§I,remark 1);

hence,

(UIU') _

(bi UR, I bi UR;)

(bi UR; bi QR )

(bi UR, I bi UR, )1'

(b, UR, I bi UR,

)1/2

[(bJUR,IbJURJ)] 11

[

(b, UR, I b, OR

)1112

U , )

The triangle inequality follows from the Schwarz inequality.

Proof of 3°).

Use §l,theorem 5.3.

II,§2,2-II,§2,3

115

Remark 2.2.

If V = V', (U U') is defined by the right-hand side of §1,

(5.4) under the assumption.

Supp U n ER = 0,

Supp U' n ER = 0-

If this assumption is not satisfied, the right-hand side is a divergent

integral in view of theorem 2.2 (structure).

Thus definition 2.3 of (- I ) gives a meaning to this divergent integral.

3. Lagrangian Functions on V

We are interested in functions on V rather than functions on V, that is,

multiforms on V. The definition of lagrangian functions on V requires

the datum of a number vo ; it will be convenient (see, for instance, Maslov's

quantization, II,§3,6) to choose vo E i]0, oo[.

The first homotopy group n, [ V ] acts on V: V is the quotient of V by

n1[V]. Clearly

it,/[ V] leaves ER invariant /VR;

(YZ) = OM + cy!

(PR(YZ) = (PR(Z) + C

where c., = 1 J [z, dz]

dy e n 1 [ V];

that is,

q0 /-, = 0 - Cy!

(PR0Y-1

=(PR-CY.

Definition 3.1 of the group n, [V] of automorphisms of

Let

OR = Y aRrr evv- E f (V\i,! R),

Y E n, [ V].

(3.1)

Define the transform of OR by y not as

7-1 = e-vcy

aRr Y

but as

yIIR = e(°-V,)C, UR o Y-1 = e-vocr Y

aR.r 0 Y

evWR E

R). (3.2)

rEN

Clearly

116

II,§2,3

Y' (Y UR) = (Y"01R,

so n1 [V ] is a group of automorphisms of .y (V\ER, R).

It clearly follows from definition (3.2) and the definition of the operator

S E Spz(1) in II,§1 (lemma 1.1, part 1 of theorem 4.1, and theorem 4.3)

that S and y commute: If

UR. E .f(V\±R. U :R, R') and R = SR',

then

Y(SUR') = S(YUR,).

(3.3)

If UR are the expressions of a lagrangian function U on V, then yUR

are the expressions of a lagrangian function on V; denote it by yU. Then

(Y OR = yUR and n1[V] is a group of automorphisms of .F().

Clearly Vy e 7r, [ V], V O e

(V ), Va a lagrangian operator,

Supp(yU) = y Supp U;

(3.4)

y and a commute, that is,

Y(aU) = a(yU).

(3.5)

Definition 3.2. A lagrangian function on V is a lagrangian function U

on V having the following three properties, which are clearly pairwise

equivalent and thus independent of the choice of the 2-frame R :

i. U is invariant under n1 [V], that is,

yU = U

Vy E n1 [V] (see definition 3.1);

ii. a "°`YaR,, o Y-1 is independent of y c- n1 [V] Vr E N;

iii. the function

UR =

URe(vo-v)WRI

rEN

called the restriction of UR to vo, takes the same value at all points of

V having the same projection onto V Therefore we may write:

UR : V \ER - .y °.

The set of lagrangian functions on V[with compact support in V] is a

Hence it is the set of points in V projecting onto a closed subset of V,

II,§2,3

117

which will be called the support of U in V and denoted Supp U. Then

Supp UR =Supp U\ER n Supp U, t1R.

The set of lagrangian functions on V [with compact support in V] is a

vector space over the algebra .F', It is denoted .y (V) [.y°(V )].

Clearly, by (3.5) the following theorem holds.

THEOREM 3.1.

The algebra of lagrangian operators a, b defined on 0

acts on the space .y (V) of lagrangian functions on V;

(a o b)U = a(bU),

Supp(aU) c Suppa n Supp U.

The scalar product of lagrangian functions on V will be defined using

the following lemma and definitions.

LEMMA 3.1. There exists a natural epimorphism

.y°(v)3CJ-+U

Y YUE.y°(V).

(3.7)

yER1[V]

Proof.

Definition (3.7) makes sense: it defines U by a finite sum, since for

every 2 e V, there are only finitely many y such that yz e Supp U.

Clearly U is invariant under n1[V] and has compact support.

Since y commutes with partitions of unity (definition 1.4, theorem 1.2),

to prove (3.7) is an epimorphism it suffices to prove the following: Every

U E FO (V) with support in a simply connected subset w of V is the image

under (3.7) of an element U of .y°(17). This is proved as follows.

The connected components of the subset of V projecting onto w are the

transforms of any one of these components (b by the elements y of n1 [ V ].

Let U be the lagrangian function on V that vanishes outside th and equals

U in th; then yU vanishes outside y(b and equals U = yU in y(b, hence

U = Y yU.

Let us express the notion of restriction to v° more explicitly.

Definition 3.3.

Let 5 be the ideal of F generated by the elements

e°9 - e'ow cp e R.

Each element of .5 may be written uniquely:

118 II,§2,3

I aJ(e`'PJ - ev0 j),

J finite,

ai E F °, gyp; E R.

JEJ

.F is the direct sum of f and F °. Let us call the projection of F onto .F °

the restriction of F to vo:

.FE) U = E c

e"ip' F- 3 u ° = E a

i e"O

(P;

E -IF' = -F/J.

In particular, if U E flV), then the restriction of UR(v, 2) E to vo is the

value UR (v, z) of the restriction UR of UR to vo at the point z, the projection

of z onto V.

Definition 3.4. Any additive subgroup M of

' having the properties

.F c M c 16, M = 47-(complex conjugate)

is called a submodule of 16. Multiplication in the algebra W makes .elf a

module over the algebra .S.

In particular,

ERimply ue"°E.11.

Let .' be the submodule of M whose elements are

uJ(e°°i - ev00,), J finite, u, E .i, (p; E R.

JEJ

Let us call the quotient

..lf -+ -# / .N = ..lf °

the restriction of Al to vo; tf° is a module over the algebra F°. Since

.N' = .N', conjugation in .,lf ° is induced by complex conjugation in M.

Example 3.1. If ..l' = .IF, then J(° _ -IFo

Example 3.2.

Let (1/IvI)s9,

... denote the images of 9,

...

under the

multiplication

16 Z) J-+

1IsfE16,

sER+.

If .4Y = (1/Ivls).F, then tf° = (1/Ivl3),a-'°.

Definition 3.5 of the scalar product ( I ). Let V and V' be two lagrangian

manifolds in Z. Let.,# be a submodule of ' (definition 3.4) such that

II,§2,3

119

(UI U') E .,H,

V U E .f°(V), U' E .F°(V')

(3.8)

For any U E .F°(V), U' E -FO (V'), there exist U E .`°(V), U' e .f° (V') such

that

U = E TO, U' = E y' U'

YEidV] YEn1[V'I

according to lemma 3.1. By definition, the scalar product of U and U' is

(U I U') = (U I U')°, the restriction of (U I Cl') E .,H to v0.

(3.9)

Thus the scalar product has values in Mo.

This definition makes sense by the following lemma.

LEMMA 3.2.

(UI U')° depends only on U and U'.

Proof. Let

Eb*obj=1

be a partition of unity (definition 1.4); we have

(UI U') _

(b;CI Ib;U');

it then suffices to prove that (bU I b;U')° depends only on

E yb;U = bjU and E y'b;U' = b;U'.

YEn,[Vl

YeAdv ]

By theorem 1.2, it suffices to prove the lemma when the following two

conditions hold:

i. There exists a 2-frame R such that

UR E 10'0(1 \'R, R), UR E R).

ii. There exist two simply connected open sets, w in V and w' in V', that

are projected injectively into X under RX and contain the supports of U

and U:

SuppUcwcV,

SuppU'cci cV.

The connected components of the subset of V projecting onto w are the

open sets yw in V, where y e nl[V] (see the proof of lemma 3.1). Given

x e RX Supp U, let

120

II,§2,3

z be the unique point of co such that Rxz = x,

i be the unique point of to projecting onto z in V.

By definition 2.3, §1,(1.9), and definition 5.1,

(U I U') =

(fl UR)(V, X) - MR UR)(V, x)d'x.

(3.10)

y definition (§1,theorem 2.1), if x E Supp(HRUR) = Rx Supp U, then

(nR UR) (v, x) = E UR(V, Y ' Z),

YE1,IV]

that is, by (3.2)

(nRUR)(v, x) = E e('0-Vk' YUR(v, Z)'

YE1 11 VI

We substitute this expression for 11R UR and the analogous expression for

11RUR into (3.10). Using definition 3.4 of .N' and (3.7), we obtain the

formula

(U I U') =

UR(v, Z) - UR(v, Z )d'x

under the assumptions (i) and (ii).

mod Al

(3.11)

This formula (3.11) proves lemma 3.2 and is of use in proving the

following theorem, part 1 of which is clear.

THEOREM 3.2.

(Scalar product)

1°) Let V and V' be two lagrangian

manifolds in Z. The scalar product (- I -) is a function of pairs U and U'

of lagrangian functions on V and V' that is sesquilinear over F°.

(U U') is defined when Supp U n Supp U' is compact.

(U I U') e tl°, a module over .F' depending on V and V'.

(U U') depends only on the behavior of U and U' in a neighborhood of

Supp U n Supp U'.

(U I U') = 0 when Supp U n Supp U' = 0.

(U I U') and (U' l U) are conjugate.

(aU I U') = (U I a* U') when a and a* are adjoint lagrangian operators.

2°) Suppose V = V. Then (U I U') E .F° (algebra of formal numbers with

vanishing phase):

0 <, (U I U), equality implying U = 0. (3.12)

II,§2,3

121

The norm of U,

IUII = (UIU)112 E.°,

(3.13)

satisfies the triangle and Schwarz inequalities:

11U + U'II , IIUII

+ 11 U'11,

I(UIU')I,

IIUII'IIU'Ilin.Fo;

if the equality holds, then U and U' are proportional: either U = 0 or there

exists a c .F ° such that U' = a.U. Let

Ue(v, z) = a,,0(2)e°-O-() mod

(3.14)

11z

mod 1

(3.15)

where z denotes any one of the points of V projecting onto z in V; (3.15)

assumes V is given a 2-orientation that defines the Maslov index MR(z) and

arg d'x (1,§3,corollary 3); the lagrangian amplitude fo is independent of

the choice of R, but depends on the choice of >l, a regular measure on V such

that arg rl = 0. Then, using analogous notation for U' and denoting Rxz by

' (z)'1

(3.16)

xR,o(z)xe,°(z)d'x = flo(z)#0

is a regular differential form on V, which is independent of R, and

0(Z)flo(z), mod-.

(3.17)

(U I U') =

J aR,o(z)xR,o(z)d`x

= J

3°) Suppose V and V are transverse. Then

v'12 (UIU')E.F

I)112(UIU)

E [Hess(cp

T')]

"'OCR, 0(z)xeo(z)e"°[wR(=)

wK(=)] mod-

(3.18)

zE Vn V'

^ 12

nodziZ

mod-; (3.19)

zCVnV'

in (3.18),

122

II,§2,3

p and cp' are defined near Rz by

(P(x) = (PR(1 x'X n P),

(P'(x) _ 9R(RXlx n P')>

Hess(cp - cp) _ {Hesss[gp(x) - (P'(x)]}x

= R1Z>

in (3.19),

>G is the lagrangian phase of V,

ti4 e A4(Z) is the 2-oriented lagrangian plane tangent to P at 2,

m is the Maslov index defined mod 4 (I,§3,theorem 1),

'lo A '1o and d 21z are the measures on Z defined by (5.11) and (5.15) (§1).

Proof of (3.16). Use §1,(5.9).

Proof of (3.17). Using a sufficiently fine partition of unity Ej b* o b, = 1,

it suffices to prove (3.17) under the assumptions (i) and (ii) used in the

proof of lemma 3.2, which imply (3.11). Since V = V', formula (3.11) can

be written

(U I U') = j UR(v, z)URU

(v, z)d'x,

(3.20)

and hence proves (3.17).

Proof of (3.12). Using the same partition of unity, it suffices to prove

(3.12) when (3.20) holds. Then (3.12) is evident, as well as the following

more precise statement: If U # 0, then

co OE'(UI U) = Y

,whereseN,

0[2, > 0-

=2s V

Proof of (3.13).

Use the preceding formula and part 2 of §1,theorem 2.3.

Proof of the Schwarz inequality. Suppose U and U' are not proportional.

Then there exists a e .°" and s a N such that

U=aU'+

U",

where the lagrangian amplitudes $' and /3 of U' and U" are not propor-

tional. A classical calculation gives

U,, III

- I(U' I U")12].

U12.II U'l12

- I(UI

U')12

= IVI2s[I

U'II2.

I1,§2,3-II,§2,5

123

It then suffices to prove that the right-hand side is > 0 in 3F°. In other

words, it suffices to prove that U and U' satisfy the strict Schwarz in-

equality when their lagrangian amplitudes fl° and j30 ' are not proportional.

Thus, from (3.17), we have

U112.1

U' I

a(Z)12n

- I(U 101 2 =

I!l°(Z)12q. fy I

f°(Z)fia(z)n

Hence, from the classical Schwarz inequality,

11U112.1

U'112 - I(UIu')12 = po mod1,where 0 < paER.

Thus, by II,§1,theorem 2.3,

1(U I U')1 < 11 U

. I U111.

Proof of the triangle inequality. Use the Schwarz inequality.

Proof of 3°).

Using a partition of unity, we may assume that (U I U') is

expressed by (3.11). Then (3.18) follows from part 3 of theorem 5.1 (§1)

and (3.19) of theorem 5.3 (§1).

Remark 3.

Without assumption (i) (made in the proof of lemma 3.2),

the integral in the right-hand side of (3.20) diverges (see theorem 2.2 on

the structure of lagrangian functions). Thus definition 3.5 of (1) gives a

meaning to this divergent integral.

4. The Group Sp2(Z)

The group Sp2(Z) of automorphisms of the 2-symplectic geometry of Z

(see I,§3,4) clearly transforms lagrangian operators into lagrangian opera-

tors. The images of lagrangian functions under this group are lagrangian

functions. The group leaves the scalar product invariant.

5. Lagrangian Distributions

Lagrangian distributions are defined by replacing functions by distribu-

tions (§1,7) in definitions 2.1, 3.1, and 3.2. Theorems 2.1, 2.2 (the

!'R,

being

distributions), 2.3, and 3.1 apply to distributions just as to functions.

Definition 2.3 and theorem 2.4,2°) apply to the scalar product (U 1 U')