Table Of ContentExponentially Accurate Error Estimates of Quasiclassical
Eigenvalues
Julio H. Toloza
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Mathematical Physics
Dr. George Hagedorn, Chair
Dr. Lay Nam Chang
Dr. Martin Klaus
Dr. Beate Schmittmann
Dr. Werner Kohler
December 11, 2002
Blacksburg, Virginia
Keywords: Semiclassical Limit, Low-Lying Eigenvalues, Rayleigh-Schr¨odinger Perturbation
Theory, Quasimodes
Copyright 2002, Julio H. Toloza
Exponentially Accurate Error Estimates of Quasiclassical Eigenvalues
Julio H. Toloza
(ABSTRACT)
We study the behavior of truncated Rayleigh-Schr¨odinger series for the low-lying eigenvalues
ofthetime-independentSchr¨odingerequation,inthesemiclassicallimit(cid:126) (cid:38) 0. Undercertain
hypotheses on the potentialV(x), we prove that for any given small (cid:126) > 0 there is an optimal
truncation of the series for the approximate eigenvalues, such that the difference between an
approximate and actual eigenvalue is smaller than exp(−C/(cid:126)) for some positive constant C.
We also prove the analogous results concerning the eigenfunctions.
Dedication
To my mother.
iii
Acknowledgments
I am greatly indebted to my advisor Dr. George Hagedorn, who always has expressed confi-
dence in my work. His intellectual guidance and support have been crucial to complete this
dissertation.
I would like to thank Dr. Martin Klaus for his support to enrich and continue my academic
career.
I am also thankful to my former advisor Dr. Guido Raggio who introduced me to the Math-
ematical Physics.
A very special thanks to Chris Thomas. My life as a graduate student has been much easier
because of her ever timely help.
Finally, I want to express my gratitude to Natacha, for her patience, support, and under-
standing.
iv
Contents
1 Introduction 1
2 Preliminaries 8
2.1 R-S perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Some technical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Computation of the R-S coefficients 22
3.1 Degeneracy is preserved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Degeneracy is removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 The main estimate 41
5 Optimal truncation 52
6 Conclusion 64
v
6.1 Relaxing the hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 Asymptotics on the Born-Oppenheimer approximation . . . . . . . . . . . . 67
A Simplifying ξ (x) 69
N
Bibliography 71
Vita 75
vi
Chapter 1
Introduction
Perhaps one of the most elementary facts in Quantum Physics is that, for a sufficiently
deep potential well, the eigenvalue problem defined by the time-independent Schr¨odinger
equation admits normalizable solutions. That is, one expects that there are at least one
square-integrable function Ψ˜((cid:126);x) and a number E((cid:126)) that satisfy
(cid:126)2
H((cid:126))Ψ˜((cid:126);x) := (cid:20)− ∆ +V(x)(cid:21)Ψ˜(x) = E((cid:126))Ψ˜((cid:126);x), (1.1)
x
2
provided that the potential energy has a “deep enough” global minimum. Equivalently, if
one considers the Planck’s constant as a parameter, then the equation above is expected to
have solutions for small values of (cid:126) > 0. Since one looks for solutions near the bottom of the
potential well, this statement is often referred to as the existence of low-lying eigenvalues in
the semiclassical limit (cid:126) (cid:38) 0.
Along with the problem of existence of low-lying eigenvalues, one is also interested in the
1
Julio H. Toloza Chapter 1. Introduction 2
behavior of the corresponding perturbation series in powers of (cid:126), the so-called Rayleigh-
Schr¨odinger (R-S) series. It is well known that, in general, the R-S series are not convergent
but only asymptotic to the solutions of equation (1.1). However, one often wants to consider
truncations of these series as good approximations to the actual eigenvalues/eigenvectors.
This raises the natural question of whether or not one can find an optimal truncation that
minimizes the difference between the exact eigenvalues/eigenvectors and the corresponding
truncated R-S series.
In this dissertation we aim to find exponentially accurate asymptotics to the solutions of
(1.1). Weshallprovethat,undercertainconditionsofanalyticityandgrowthonthepotential
energy, one can truncate the R-S series so that the difference between the truncated series
andtheactualeigenvalue/eigenvectorcanbemadesmallerthanexp(−C/(cid:126))forsomepositive
constant C > 0. Our construction is based entirely on a straighforward application of the R-
S perturbation theory, as opposed to the technically awkward quantization of normal forms.
This latter technique is briefly described below. The results to be discussed here are already
published in two papers [27, 28]. The one-dimensional problem is considered in [27]. The
multidimensional problem, which involves degenerate perturbation theory, is discussed in
[28]. A review of results can be found in [29].
Rigorous results concerned with the discrete spectrum of (1.1), in the semiclassical limit
(cid:126) (cid:38) 0, were missing until not long ago. The first proof of existence of low-lying eigenvalues
andasymptoticR-SserieswaspresentedbyCombeset alin1983. Theirproof, whichinvolves
Dirichlet-Newmann bracketing and the Krein’s formula, only considers the one-dimensional
Julio H. Toloza Chapter 1. Introduction 3
problem. Shortly after, Simon gave another proof, based on geometric arguments, that is
valid in several dimensions [24]. In both cases, the potential energy V(x) is assumed to be
a sufficiently smooth function which has several global minima, each one admiting a non-
vanishing second order approximation. Those terms are separated from the potential energy
term, and a suitable scaling is made on the Schro¨dinger equation. The quadratic pieces along
with the kinetic energy term are treated as an unperturbed harmonic oscillator hamiltonian,
and the remainder is considered as a perturbation of it. Then, for small values of (cid:126) and
near the bottom of V(x), the whole hamiltonian is expected to be “close” to the harmonic
oscillator and therefore its low-lying eigenvalues should be also close to those of the harmonic
oscillator. Following the same underlying idea, but applying the so-called “twisting trick”
[22, Section IX.11], Howland presented another proof for the one-dimensional problem [11].
Along with the existence of low-lying eigenvalues, the aforementioned results state that the
low-lying eigenvalues of H((cid:126)) are given asymptotically by R-S series, in the sense that
N
(cid:12)E((cid:126))−(cid:126) E (cid:126)n(cid:12) ≤ C (cid:126)N+1+1 (1.2)
(cid:12) (cid:88) n 2(cid:12) N 2
(cid:12) (cid:12)
(cid:12) n=0 (cid:12)
(cid:12) (cid:12)
for any given N, and sufficiently small (cid:126) > 0. An analogous statement holds for the corre-
sponding eigenvectors. We also must mention that this problem has been studied by Helffer
and Sj¨ostrand in the framework of microlocal analysis (see below) [10].
The semiclassical limit of H((cid:126)) has also been studied from a rather different approach,
namely, by quantization of the canonical perturbation theory. In this context, the poten-
tial energy is split in the same way as described above. However, one now first considers
the perturbation problem of the classical hamiltonian and the properties of the associated
Julio H. Toloza Chapter 1. Introduction 4
perturbation series. Briefly speaking, one looks for approximate solutions to the canonical
equations for a hamiltonian of the form
h(A,φ) = h (A)+(cid:15)f(A,φ), (1.3)
0
where h (A) is the hamiltonian of a canonically integrable system (in our case, a harmonic
0
oscillator), already expressed as a function of the canonically conjugated action-angle vari-
ables A = (A ,...,A ) and φ = (φ ,...,φ ). The variables A ’s are essentially defined by
1 d 1 d i
the conserved quantities (for the harmonic oscillator, they are the energy contributions from
each coordinate x ). Each angle variable φ takes values in the unit circle T. It is clear that
i i
all the solutions to h (A), with the same initial datum A, wind around the d-dimensional
0
torus A × Td with constant frequencies ω = ∂h(A)/∂A . The perturbation f(A,φ) is as-
i i
sumed to be bounded below and sufficiently smooth. Then the Kolmogorov-Arnold-Moser
(KAM) theorem states that the solutions to the perturbed hamiltonian, for small (cid:15), also lie
on invariant tori that are close to the invariant tori of h (A), provided that h (A) satisfies
0 0
certain condition of non-degeneracy (see below). Moreover, for any given order N, one can
construct a canonical map (A,φ) → (A(cid:48),φ(cid:48)) such that (1.3) becomes
h(A(cid:48),φ(cid:48)) = h(N)(A(cid:48))+(cid:15)N+1f(N)(A(cid:48),φ(cid:48)),
(cid:15) (cid:15)
where h(N)(A(cid:48)) can be expressed in terms of the Birkhoff normal forms
(cid:15)
N
h(N)(A(cid:48)) = h (A(cid:48))+ (cid:15)nN(n)(A(cid:48)).
(cid:15) 0 (cid:88) 0
n=1
The corpus of mathematical results on canonical perturbation theory, which describes ge-
ometrical features in phase-space, is known as KAM theory. See e. g., [1, 5, 15]. The
