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
o3 (1993), p. 301-313
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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, n° 3, 1993, Physique théorique
ABSTRACT. - We consider the limit h - 0 of the solution C
(t,
x,h)
ofSchrodinger equation:
We prove
that,
for anyinteger l >_ 2
and any initialcondition 03A6(0, x, h)
that
belongs
toSchwartz-class,
a solution C*(t,
x,h)
of the semiclassicalequation approximates C (t,
x,h)
such asRESUME. 2014 On considere la limite h - 0 de la
solution 03A6(t,
x,h)
de1’equation
deSchrodinger:
Nous prouvons que, pour tout nombre entier
1>2
et toute conditioninitiale O
(0,
x,h) qui appartient
aSchwartz-classe,
une solution O*(t,
x,h)
de
1’equation semi-classique approche 0 (t,
x,h)
tel queAnnales de l’lnstitut Henri Poincaré - Physique théorique - 0246-0211 1
302 T. ARAI
1. INTRODUCTION
It was shown in
[2]
that theapproximate
solutions to theSchrodinger equation
agree with the exact solution modulo errors on the order ofHowever,
the initial states of theequation
weremerely
dealt with certain Gaussian states or their finite linearcombinations.
In this paper we shall prove that G. A.Hagedorn’s
results also hold for moregeneral
stateswhich
belong
to Schwartz class if we somewhatmodify
several conditions.For
simplicity,
we will restrict attention to one space dimension. Ourproofs rely heavily
on the results of G. A.Hagedorn concerning
thesemiclassical behavior of certain Gaussian initial states.
We now introduce
enough
notations anddefinitions
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 existpositive
constantsM, C 1
andC2
such that -C2 _
V(x) _ C 1 eM x2 _ for
all x E IR.ASSUMPTION 1. 2. - We assume that and there exist
positive
constantsM, C 1
andC2
such that -C2
_ V(x) C 1 ( 1 + x I)M for
all XE R.
Also,
we assume that thequantum
Hamiltonianis
essentially self-adjoint
on theinfinitely
differentiable functions of com-pact support
inL2 (!R).
Under this Hamiltonian we shallstudy
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 whichsatisfy
Let a,
and 0 h _
1.Then , for j = 0, 1, 2,
..., wedefine
- .. , -
Here
Hj
denotes thej-th
order Hermitepolynomial
that isdefined by
And the branch
of
the square root will bespecified
in the context in whichthe
functions
are used. We notethat, for
anyfixed
valuesof A, B, h,
aand
r~, ~ (A, B, h,
a, r~,x) ~~° o
is acomplete
orthonormal basis inL2 ((1~).
The
following
theorem wasproved by
G. A.Hagedorn [2]
in 1981.Annales de l’lnstitut Henri Poincaré - Physique théorique
THEOREM 1. -
Suppose
V(x) satisfies Assumption
1. 1for
someinteger
l >_ 2. Let ao, ~0 E and let
Ao, Bo
E C whichsatisfy ( 1.1 ). Then, for
any T >0,
any J E N and any co, c l’ ..., cJ EC,
there existsC3
such that- J "’" i
whenever t E
[0, T]
and 0 ~ _ 1. Here[A (t),
B(t),
a{t),
11(t),
S(t)]
is theunique
bounded solution to the systemof 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 theunique
solution to the systemof
coupled ordinary differential equations
subject
to the initial conditionsCj(O, h) =Cj for 0 _ j J
andCj(O,
J + 1
_ j _
J + 3(l -1).
In thisequations,
denotesdk and ~ j, xk n ~
are
defined by
In Theorem
1, only
finite linear combinationsof ~ Pj }j 0
wastreated,
so that we shall
improve
this theorem to infinite linear combinations.For this purpose, we must
modify
the definition ofh),
because thesystem
ofordinary
differentialequation ( 1. 5)
isdepended
on J. Then wepropose to
replace {~(~ h) ) )1)
~Z- I~~C) ~~° o
that eachdj(t, h)
304 T.ARAI
is
independent
of J. We define each as follows: for each1,2,3,...},
where,
for each p=1, 2,
...,l -1, ap E { 3, 4,
..., l ~ 1},
andn p
isinteger except
no = 0 which satisfies -ap np a~.
We have the
following
main theorem.THEOREM 2. -
Suppose
V(x) satisfies Assumption 1 . 2 for
someinteger
l >_ 2. Let and letAo, Boec
whichsatisfy (1.1).
Leto
( c C)
be acomplex
sequence such that03A3|cj|.jp
~.for
all p > o.~=o
Then, for
T >0,
there existsC3
> 0 such thatwhenever t E
[0, T]
and 0 h 1. Here[A (t),
B(t),
a(t), ~ (t),
S(t)]
is theunique
solution to the systemof equation Justly
eachdj(t, h)
isdefined
in(1 6).
Remark 1. - It should be noted that the
resulting approximate dynamics
is not
unitary
under these~Cj
which are definededby (1. .6).
For thisfact,
see G. A.Hagedorn (2].
However we think that thisdisadvantage
can be
enough
to recoverby
the fact that J can be taken r.Remark 2. - It is
easily
showthat, by (1 6),
that we canput
Annales de /’Institut Henri Poincaré - Physique théorique
QUANTUM
where each
Fj(t.x)
is definedby
Remark 3. - For Schwartz class
function f~J(R),
letCj
be chosen so00 00
that
/M= ~ CjP (1, 1, 1, 0, 0, x), then,
we note forj=o j=o
all
~>0.
Therefore we notice that Theorem 1 is extended to the state because ofreplacing h)
withh).
2. SOME PRELIMINARY LEMMAS
Throughout
this section we mention threepreliminary
lemmas for theproof
of Theorem 2. The first lemmagives
the basic formula in semiclassi- calquantum
mechanics. Thisimportant
fact was obtainedby
G. A.Hage-
dorn in
[1].
The second one means thatoo
belongs
toL2-class.
The last lemma is the estimate in thepolynomial approximation
of thepotential
V(x).
LEMMA 2 . 1
(See
G. A.Hagedorn [1]).
- Let ao, 110 and letAo, B0 ~ C
whichsatisfy ( 1 .1 ). If V (x)
is apolynomial of degree
2 andV
(x) >__ - C, then, the following equality
holds:for
all t E!R and1, 2, ... ~ .
LEMMA 2 . 2. -
Suppose
thepotential V (x) satisfies Assumption
1. Ifor
some
integer
l >_ 2.Then,
there existsC (T, l ) > 0
such thatfor all j E ~ 0, 1, 2, ... ~
and all t E[0, T].
306 T. ARAI
Proof -
Thehypothesis imply
the existences of randR,
such thatand
for allt E [o, T]. Then,
for anyt E [o, T],
we seethat
c _
Now we can
easily
show thefollowing
estimate from the induction withrespect
to n E N: forallj=O, 1, 2,
... and allTherefore,
we see thatHence,
theinequality (2. 2)
wasproved.
0LEMMA 2. 3. -
Suppose
V(x) satisfies Assumption
1 . 2for
someinteger
~+ ~ 1
l >_ 2. Let
r>o, R>O,
and let~ Then,
k=0
and a ~
_ Rimplies
for 1, 2, ... }
whereAnnales de I’Institut Henri Poincaré - Physique théorique
QUANTUM
which each
D1 = D1 (I, R)
andD2
=D2 (C1, I, R, M) is defined by
Proof -
From theassumption
we can useTaylor’s
formula. Hence, we obtain
and jc 2014 ~ ~ 1 imply
And the
growth of C 1 ~ ( 1 +
means that there existp=max[(/+ 2)/2, M]
and suchthat |a|~R
andx2014a >__ 1 imply
Therefore,
we showthat, for and I a R,
Here,
from the estimate(2. 3),
we seethat,
Therefore,
we caneasily
obtain theinequality (2.4),
because of~//2 +1.
03. PROOF OF THEOREM 2
We shall divide the interval
[0, T]
intoN-pieces. Then,
we will be led to thefollowing
discrete timeanalogs
of theequation (i)
""(v)
andFj (t, x) :
Let
~(0)==~ ~N(0)=r~ A~(0)=Ao, BN (0) = Bo, SN (0) = 0
and
308 T.ARAI
And the discrete version of
Fj(t, x)
is thatl-1
By taking
Nsufficiently large
we can make these aN(n),
~N(n), AN (n), BN (n), SN (n)
andFN, j (n, x) approximate a (t), 11 (t),
A(t),
B(t),
S(t)
andFj(t, x)
ift = n T/N.
From Lemma 2 . 2 for wej=o
immediately
note thatTherefore,
we can combine the results of G. A.Hagedron [2]
and theprevious preliminary
lemmas.Then,
we can know thefollowing
results[I]
~fifl.
For all E >
0,
there existsN 1
> 0 such that N >N 1 implies that,
for all[I] AN (n)
isinvertible,
Annales de I’Institut Henri Poincaré - Physique theorique
QUANTUM
where
and
From
[II]
and[III],
we notice that theproof
will becomplete
if weshow the
following inequality: namely,
for all0 __ n _ N,
where
C3
is apositive
constant which isindependent
of E, N and h.310 T. ARAI 2
At first we
put Then,
fromt=0
Lemma 2 . 1 and
(2 .1 ),
we can calculatethat,
for0 n N -1,
x)
is thatThen,
we can estimate theL2-norm
of(n, x),
thatis,
there existsC2 (T, 1) > 0
such thatimplies
This fact can be shown as follows:
Annales de /’Institut Henri Poincaré - Physique theorique
Hence,
we have obtained theinequality (3 . 3).
Moreover,
from[IV], [V], (3.2)
and(3.3),
we shallinductively
prove thefollowing estimate,
thatis,
there existsC3 (T, /)>0
such that0 _ n N implies
The above
inequality
is trivial at n =0,
because ofSN (0)
=0, FN, j (0, x)
= 0and fFN,j(ü, x) = 0.
So we assume that thisinequality
holds until n = k.Then,
at n = k +1,
we see that312 T.ARAI
where
C3 (T, l)’
= max[C1 (T, I), C 2 (T, I), C 3 (T, I)].
Therefore,
we haveproved
that thisinequality
also holds for n = k + 1.Finally,
weput
the constantC3
= K.C3 (T, I)
> 0 which isindependent
of s,
N,
and ~.Then,
we caneasily
showthat,
for allOnN,
ACKNOWLEDGEMENTS
The author wishes to express sincere thanks to Professor S. Ishikawa of Keio
university
for his kindsuggestions.
Annales de l’Institut Henri Poincaré - Physique théorique
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.)