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A NNALES DE L ’I. H. P., SECTION A

T AKAHIRO A RAI

On the large order asymptotics of general states in semiclassical quantum mechanics

Annales de l’I. H. P., section A, tome 59, n

o

3 (1993), p. 301-313

<http://www.numdam.org/item?id=AIHPA_1993__59_3_301_0>

© Gauthier-Villars, 1993, tous droits réservés.

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301

On the large order asymptotics of general states

in semiclassical quantum mechanics

Takahiro ARAI

Department of Mathematics, Faculty of Science and Technology, Keio Univ. 3-14-1 Hiyoshi Kohoku-ku Yokohama,

223 Japan

Vol. 59, 3, 1993, Physique théorique

ABSTRACT. - We consider the limit h - 0 of the solution C

(t,

x,

h)

of

Schrodinger equation:

We prove

that,

for any

integer l &#x3E;_ 2

and any initial

condition 03A6(0, x, h)

that

belongs

to

Schwartz-class,

a solution C*

(t,

x,

h)

of the semiclassical

equation approximates C (t,

x,

h)

such as

RESUME. 2014 On considere la limite h - 0 de la

solution 03A6(t,

x,

h)

de

1’equation

de

Schrodinger:

Nous prouvons que, pour tout nombre entier

1&#x3E;2

et toute condition

initiale O

(0,

x,

h) qui appartient

a

Schwartz-classe,

une solution O*

(t,

x,

h)

de

1’equation semi-classique approche 0 (t,

x,

h)

tel que

Annales de l’lnstitut Henri Poincaré - Physique théorique - 0246-0211 1

(3)

302 T. ARAI

1. INTRODUCTION

It was shown in

[2]

that the

approximate

solutions to the

Schrodinger equation

agree with the exact solution modulo errors on the order of

However,

the initial states of the

equation

were

merely

dealt with certain Gaussian states or their finite linear

combinations.

In this paper we shall prove that G. A.

Hagedorn’s

results also hold for more

general

states

which

belong

to Schwartz class if we somewhat

modify

several conditions.

For

simplicity,

we will restrict attention to one space dimension. Our

proofs rely heavily

on the results of G. A.

Hagedorn concerning

the

semiclassical behavior of certain Gaussian initial states.

We now introduce

enough

notations and

definitions

to allow us to state our main result.

ASSUMPTION 1 . 1. - We assume that

namely

V is 1 + 2-

th continuous

differentiable function,

and there exist

positive

constants

M, C 1

and

C2

such that -

C2 _

V

(x) _ C 1 eM x2 _ for

all x E IR.

ASSUMPTION 1. 2. - We assume that and there exist

positive

constants

M, C 1

and

C2

such that -

C2

_ V

(x) C 1 ( 1 + x I)M for

all XE R.

Also,

we assume that the

quantum

Hamiltonian

is

essentially self-adjoint

on the

infinitely

differentiable functions of com-

pact support

in

L2 (!R).

Under this Hamiltonian we shall

study

the evolu-

tion of states which are finite or infinite linear combinations of the Gaussian states

(A, B, 11,

a, 11,

x),

which are defined below.

DEFINITION 1. - Let A and B be non-zero

complex

numbers which

satisfy

Let a,

and 0 h _

1.

Then , for j = 0, 1, 2,

..., we

define

- .. , -

Here

Hj

denotes the

j-th

order Hermite

polynomial

that is

defined by

And the branch

of

the square root will be

specified

in the context in which

the

functions

are used. We note

that, for

any

fixed

values

of A, B, h,

a

and

r~, ~ (A, B, h,

a, r~,

x) ~~° o

is a

complete

orthonormal basis in

L2 ((1~).

The

following

theorem was

proved by

G. A.

Hagedorn [2]

in 1981.

Annales de l’lnstitut Henri Poincaré - Physique théorique

(4)

THEOREM 1. -

Suppose

V

(x) satisfies Assumption

1. 1

for

some

integer

l &#x3E;_ 2. Let ao, ~0 E and let

Ao, Bo

E C which

satisfy ( 1.1 ). Then, for

any T &#x3E;

0,

any J E N and any co, c l’ ..., cJ E

C,

there exists

C3

such that

- J "’" i

whenever t E

[0, T]

and 0 ~ _ 1. Here

[A (t),

B

(t),

a

{t),

11

(t),

S

(t)]

is the

unique

bounded solution to the system

of ordinary differential equations:

subject

to the initial conditions A

(0)

=

Ao,

B

(0)

=

Bo,

a

(0)

= ao, ~

(0)

= 110 and S

(0)

= 0.

The {cj (t, h)}J+3j=0

(l-1) is the

unique

solution to the system

of

coupled ordinary differential equations

subject

to the initial conditions

Cj(O, h) =Cj for 0 _ j J

and

Cj(O,

J + 1

_ j _

J + 3

(l -1).

In this

equations,

denotes

dk and ~ j, xk n ~

are

defined by

In Theorem

1, only

finite linear combinations

of ~ Pj }j 0

was

treated,

so that we shall

improve

this theorem to infinite linear combinations.

For this purpose, we must

modify

the definition of

h),

because the

system

of

ordinary

differential

equation ( 1. 5)

is

depended

on J. Then we

propose to

replace {~(~ h) ) )1)

~Z- I~

~C) ~~° o

that each

dj(t, h)

(5)

304 T.ARAI

is

independent

of J. We define each as follows: for each

1,2,3,...},

where,

for each p

=1, 2,

...,

l -1, ap E { 3, 4,

..., l ~ 1

},

and

n p

is

integer except

no = 0 which satisfies -

ap np a~.

We have the

following

main theorem.

THEOREM 2. -

Suppose

V

(x) satisfies Assumption 1 . 2 for

some

integer

l &#x3E;_ 2. Let and let

Ao, Boec

which

satisfy (1.1).

Let

o

( c C)

be a

complex

sequence such that

03A3|cj|.jp

~.

for

all p &#x3E; o.

~=o

Then, for

T &#x3E;

0,

there exists

C3

&#x3E; 0 such that

whenever t E

[0, T]

and 0 h 1. Here

[A (t),

B

(t),

a

(t), ~ (t),

S

(t)]

is the

unique

solution to the system

of equation Justly

each

dj(t, h)

is

defined

in

(1 6).

Remark 1. - It should be noted that the

resulting approximate dynamics

is not

unitary

under these

~Cj

which are defineded

by (1. .6).

For this

fact,

see G. A.

Hagedorn (2].

However we think that this

disadvantage

can be

enough

to recover

by

the fact that J can be taken r.

Remark 2. - It is

easily

show

that, by (1 6),

that we can

put

Annales de /’Institut Henri Poincaré - Physique théorique

(6)

QUANTUM

where each

Fj(t.x)

is defined

by

Remark 3. - For Schwartz class

function f~J(R),

let

Cj

be chosen so

00 00

that

/M= ~ CjP (1, 1, 1, 0, 0, x), then,

we note for

j=o j=o

all

~&#x3E;0.

Therefore we notice that Theorem 1 is extended to the state because of

replacing h)

with

h).

2. SOME PRELIMINARY LEMMAS

Throughout

this section we mention three

preliminary

lemmas for the

proof

of Theorem 2. The first lemma

gives

the basic formula in semiclassi- cal

quantum

mechanics. This

important

fact was obtained

by

G. A.

Hage-

dorn in

[1].

The second one means that

oo

belongs

to

L2-class.

The last lemma is the estimate in the

polynomial approximation

of the

potential

V

(x).

LEMMA 2 . 1

(See

G. A.

Hagedorn [1]).

- Let ao, 110 and let

Ao, B0 ~ C

which

satisfy ( 1 .1 ). If V (x)

is a

polynomial of degree

2 and

V

(x) &#x3E;__ - C, then, the following equality

holds:

for

all t E!R and

1, 2, ... ~ .

LEMMA 2 . 2. -

Suppose

the

potential V (x) satisfies Assumption

1. I

for

some

integer

l &#x3E;_ 2.

Then,

there exists

C (T, l ) &#x3E; 0

such that

for all j E ~ 0, 1, 2, ... ~

and all t E

[0, T].

(7)

306 T. ARAI

Proof -

The

hypothesis imply

the existences of rand

R,

such that

and

for all

t E [o, T]. Then,

for any

t E [o, T],

we see

that

c _

Now we can

easily

show the

following

estimate from the induction with

respect

to n E N: for

allj=O, 1, 2,

... and all

Therefore,

we see that

Hence,

the

inequality (2. 2)

was

proved.

0

LEMMA 2. 3. -

Suppose

V

(x) satisfies Assumption

1 . 2

for

some

integer

~+ ~ 1

l &#x3E;_ 2. Let

r&#x3E;o, R&#x3E;O,

and let

~ Then,

k=0

and a ~

_ R

implies

for 1, 2, ... }

where

Annales de I’Institut Henri Poincaré - Physique théorique

(8)

QUANTUM

which each

D1 = D1 (I, R)

and

D2

=

D2 (C1, I, R, M) is defined by

Proof -

From the

assumption

we can use

Taylor’s

formula. Hence, we obtain

and jc 2014 ~ ~ 1 imply

And the

growth of C 1 ~ ( 1 +

means that there exist

p=max[(/+ 2)/2, M]

and such

that |a|~R

and

x2014a &#x3E;__ 1 imply

Therefore,

we show

that, for and I a R,

Here,

from the estimate

(2. 3),

we see

that,

Therefore,

we can

easily

obtain the

inequality (2.4),

because of

~//2 +1.

0

3. PROOF OF THEOREM 2

We shall divide the interval

[0, T]

into

N-pieces. Then,

we will be led to the

following

discrete time

analogs

of the

equation (i)

""

(v)

and

Fj (t, x) :

Let

~(0)==~ ~N(0)=r~ A~(0)=Ao, BN (0) = Bo, SN (0) = 0

and

(9)

308 T.ARAI

And the discrete version of

Fj(t, x)

is that

l-1

By taking

N

sufficiently large

we can make these aN

(n),

~N

(n), AN (n), BN (n), SN (n)

and

FN, j (n, x) approximate a (t), 11 (t),

A

(t),

B

(t),

S

(t)

and

Fj(t, x)

if

t = n T/N.

From Lemma 2 . 2 for we

j=o

immediately

note that

Therefore,

we can combine the results of G. A.

Hagedron [2]

and the

previous preliminary

lemmas.

Then,

we can know the

following

results

[I]

~

fifl.

For all E &#x3E;

0,

there exists

N 1

&#x3E; 0 such that N &#x3E;

N 1 implies that,

for all

[I] AN (n)

is

invertible,

Annales de I’Institut Henri Poincaré - Physique theorique

(10)

QUANTUM

where

and

From

[II]

and

[III],

we notice that the

proof

will be

complete

if we

show the

following inequality: namely,

for all

0 __ n _ N,

where

C3

is a

positive

constant which is

independent

of E, N and h.

(11)

310 T. ARAI 2

At first we

put Then,

from

t=0

Lemma 2 . 1 and

(2 .1 ),

we can calculate

that,

for

0 n N -1,

x)

is that

Then,

we can estimate the

L2-norm

of

(n, x),

that

is,

there exists

C2 (T, 1) &#x3E; 0

such that

implies

This fact can be shown as follows:

Annales de /’Institut Henri Poincaré - Physique theorique

(12)

Hence,

we have obtained the

inequality (3 . 3).

Moreover,

from

[IV], [V], (3.2)

and

(3.3),

we shall

inductively

prove the

following estimate,

that

is,

there exists

C3 (T, /)&#x3E;0

such that

0 _ n N implies

The above

inequality

is trivial at n =

0,

because of

SN (0)

=

0, FN, j (0, x)

= 0

and fFN,j(ü, x) = 0.

So we assume that this

inequality

holds until n = k.

Then,

at n = k +

1,

we see that

(13)

312 T.ARAI

where

C3 (T, l)’

= max

[C1 (T, I), C 2 (T, I), C 3 (T, I)].

Therefore,

we have

proved

that this

inequality

also holds for n = k + 1.

Finally,

we

put

the constant

C3

= K.

C3 (T, I)

&#x3E; 0 which is

independent

of s,

N,

and ~.

Then,

we can

easily

show

that,

for all

OnN,

ACKNOWLEDGEMENTS

The author wishes to express sincere thanks to Professor S. Ishikawa of Keio

university

for his kind

suggestions.

Annales de l’Institut Henri Poincaré - Physique théorique

(14)

SEMICLASSICAL QUANTUM

REFERENCES

[1] G. A. HAGEDORN, Semiclassical Quantum Mechanics, I: The h ~ 0 Limit for Coherent States, Commun. Math. Phys., T. 71, 1980, pp. 77-93.

[2] G. A. HAGEDORN, Semiclassical Quantum Mechanics. III: The Large Order Asymptotics

and More General States, Ann. Phys., T. 135, 1981, pp. 58-70.

[3] G. A. HAGEDORN, Semiclassical Quantum Mechanics, IV: The Large Order Asymptotics

and more General States in more than One Dimension, Ann. Inst. H. Poincaré, T. 42, 1985, pp. 363-374.

[4] S. ROBINSON, The Semiclassical Limit of Quantum Dynamics, I: Time Evolution, J.

Math. Phys., T. 29, 1988, pp. 412-419.

[5] S. ROBINSON, The Semiclassical Limit of Quantum Dynamics, II: Scattering Theory,

Ann. Inst. H. Poincaré, T. 48, 1988, pp. 281-296.

( Manuscript received September 16, 1992.)

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