Asymptotic spectral analysis and tunnelling for a class of difference operators [Elektronische Ressource] / von Elke Rosenberger

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• Mathematics

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• Publié par: universitat_potsdam
• Publié le: 01 janvier 2006
• Nombre de lectures: 17
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Institut fur Mathematik
Mathematische Physik: Semiklassik und Asymptotik
Asymptotic Spectral Analysis and Tunnelling
for a class of Di erence Operators
Dissertation
zur Erlangung des akademischen Grades
“doctor rerum naturalium”
(Dr.rer.nat.)
in der Wissenschaftsdisziplin “Mathematische Physik”
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultat
der Universitat Potsdam
von
Elke Rosenberger
Potsdam, den 8. April 2006Contents
Chapter 1. Introduction 1
1.1. Denition of the Operator Class 1
1.2. General Strategy and Main Results 2
1.3. Classi cation and Motivation 4
1.4. Structure of this work 6
1.5. Open Questions related to this work 9
Chapter 2. Stability of the spectrum 11
2.1. Notations and Preliminaries 11
2.2. Harmonic Approximation of the Spectrum of H 15ε
2.3. Probabilistic Operator 32
Chapter 3. Construction of asymptotic expansions 37
3.1. Hypothesis and motivation 37
3.2. Solution of the Eikonal Equation 39
3.3. Transformation of the variable and formal symbol spaces 41
3.4. Construction of asymptotic expansions 49
3.5. Constr of Asymptotic Expansions in x and ε 54
Chapter 4. Finsler Distance associated to H 61ε
4.1. Denition and Properties of Finsler Manifold and Finsler Metric 61
4.2. Finsler Function adapted to a hyperregular Hamiltonian 64
4.3. Geodesics as base integral curves of the associated vector eld 72
4.4. Application to H and the Eikonal (in)-equality 76ε
Chapter 5. Weighted estimates for Dirichlet eigenfunctions 81
5.1. Preliminary Results 81
5.2. Weighted Estimates 83
Chapter 6. Interaction between multiple wells 89
6.1. Setting 89
6.2. Distance of the Eigenspaces 94
6.3. The Interaction Matrix 100
6.4. The “Spectrum” of one well 111
6.5. Comparison of exact and asymptotic Dirichlet eigenfunctions 118
6.6. Asymptotic eigenfunctions and the interaction matrix 122
Appendix A. Technical details and supplementary computations 127
A.1. The discrete Fourier transform 127
A.2. Simultaneous diagonalization of two quadratic forms 131
A.3. Kinetic Energy as translation operator 132
A.4. Unitary Transformation 132
A.5. Direct computation of w w 133
A.6. Direct proof of Lemma 2.12 134
A.7. Valuation onK1 135
2
Appendix B. Symbolic Calculus in the discrete setting 137
dB.1. Pseudo-di erential operators on the lattice ( εZ) 137
B.2. Stationary phase and applications 141
iii CONTENTS
dB.3. Norm estimates for operators on (εZ) in microlocal approximation 148
2 dB.4. De nition of Pseudo-dierential Operators on L (R ) 153
B.5. Analogue of the Persson Theorem in the discrete setting 154
Bibliography 161CHAPTER 1
Introduction
The central topic of this thesis is the investigation of a rather general class of families of
2 ddierence operators H , parameterized by a small parameter ε, ε>0. They act on ‘ ((εZ) ), theε
dsquare summable functions on the lattice (εZ) .
Wearegoingtoanalyzetheasymptoticbehaviorasε→0ofthespectraandtheeigenfunctions
of these operators.
Inspired by the paper of Hel er and Sj ostrand [ 33], we give sharp estimates for interactions
between di erent “wells” (minima) of the potential energy, in particular for the discrete tunnelling
e ect.
While the continuous case has been exhaustively explored (see for example Hel er-Sostranj d
[33], [34], [35], [36]), there exist very few results in the discrete setting (see Hel er-Sj d
[37], [38], [39] for the one dimensional Harper equation) and none, known to the author, in the
generality presented here.
For a multiple well potential energy, the interaction between di erent wells is analyzed by
comparing the eigenvalues of local operators at the wells with the eigenvalues of the original
operator. Eigenvalues of the direct sum of the local operators, which are degenerate, correspond
to eigenvalues of the original operator H , which are exponentially close to each other. Thus weε
can say that the coupling of the wells induces a splitting of degenerate eigenvalues.
Furthermore,takingthematrix-representationofH withrespecttothebasisofeigenfunctionsε
of the decoupled operators located at the wells, the non-diagonal terms describe the interaction
and thus the tunnelling between these wells.
1.1. De nition of the Operator Class

2 dWe are going to analyze a discrete Hamilton operator H , acting on ‘ (εZ) , the space ofε
dsquaresummablefunctionsonthed-dimensionalε-scaledlattice(εZ) . Thelatticeparameterε>0
takes the role of a small parameter, analogously to the Planck constant in Schrodin ger operators
in the semi-classical setting. Thus we always assume thatε is small and construct expansions with
respect to ε in the limit ε→0.
The operator H is given byε
H =(T +V ) where (1.1)ε ε ε
X
T = a (x)ε
d∈(εZ)
dand V is a multiplication operator. The operator denotes a translation, i.e. for x,∈(εZ)ε
u(x)=u(x+).
As a function ofthe lattice pointx,a is assumed to be slowly varying, i.e.,a together withall its
derivatives should be bounded uniformly with respect to . The summand a , which is in fact a0 0
multiplication operator, is chosen such that T can be interpreted as generalized kinetic energy inε
the sense of De