This book convenes and deepens generic results about spectral measures, many of them available so far in scattered literature. It starts with classic topics such as Wiener lemma, Strichartz inequality, and the basics of fractal dimensions of measures, progressing to more advanced material, some of them developed by the own authors.
A fundamental concept to the mathematical theory of quantum mechanics, the spectral measure relates to the components of the quantum state concerning the energy levels of the Hamiltonian operator and, on the other hand, to the dynamics of such state. However, these correspondences are not immediate, with many nuances and subtleties discovered in recent years.
A valuable example of such subtleties is found in the so-called “Wonderland theorem” first published by B. Simon in 1995. It shows that, for some metric space of self-adjoint operators, the set of operators whose spectral measures are singular continuous is a generic set (which, for some, is exotic). Recent works have revealed that, on top of singular continuity, there are other generic properties of spectral measures. These properties are usually associated with a number of different notions of generalized dimensions, upper and lower dimensions, with dynamical implications in quantum mechanics, ergodicity of dynamical systems, and evolution semigroups. All this opens ways to new and instigating avenues of research.
Author(s): Moacir Aloisio , Silas L. Carvalho , César R. de Oliveira
Series: Latin American Mathematics Series
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 246
City: Cham
Tags: quantum theory, spectral analysis, elliptic equations, classical measure theory, fractals, quantum logic, singular continuous spectra, Hausdorff dimension, packing dimension, Wonderland theorem, quasi ballistic dynamics, Schrödinger semigroups
Preface
Contents
0 Book's Outline
0.1 Motivation
0.2 Part I
0.3 Part II
1 Spectrum and Dynamics: Some Basic Concepts
1.1 First Basic Concepts
1.1.1 Standard Spectral Classification
1.1.2 Fractal Measures
1.1.3 Dimensions of Measures
1.2 Semigroups
1.2.1 General Definitions
1.2.2 Asymptotics of C0-Semigroups: A Short Account
1.2.3 Joint Resolution of Identity and Normal Semigroups
1.3 Wonderland Theorem
1.3.1 Proof of Wonderland Theorem
Part I Quantum Models
2 Correlation Dimension
2.1 Correlation Dimension and Return Projections
2.2 Gδ Sets For Correlation Dimension
2.3 Generic Correlation Dimension
2.4 Applications
2.4.1 Bounded Operators: Norm Convergence
2.4.2 Bounded Operators: Strong Convergence
2.4.3 Discrete Schrödinger Operators
3 Fractal Measures and Dynamics
3.1 Local Dimensions of Measures and Dynamics
3.2 Fractal Decomposition of Measures
3.3 Dynamics of UαHC and UαHS Measures
3.4 Lower Bounds on Moments
3.4.1 b-adic Intervals
3.4.2 Proof of Theorem 3.26(i)
3.4.3 Proof of Theorem 3.26(iii)
4 Escaping Probabilities and Quasiballistic Dynamics
4.1 Laplace Average Moments
4.2 Escaping Probabilities
4.3 The SULE Condition and Quasilocalization
4.4 Gδ Sets for Dynamical Exponents
4.5 Applications to Schrödinger Operators
4.5.1 Bounded Potentials
4.5.2 Analytic Quasiperiodic Potentials
4.5.3 Unbounded Discrete Schrödinger Operators
5 Generalized Dimensions and Dynamics
5.1 Dynamics of Pure Point Operators
5.2 Box-Counting Dimension
5.3 Thick Point Spectrum and Upper Fractal Dimensions
5.3.1 Proof of Proposition 5.8
5.4 Generic Dimensions for the Hydrogen Atom
5.4.1 Proof of Theorem 5.15
5.5 A Dimensional Heritage
5.6 Dimensions and Moments
5.6.1 Proof of Theorem 5.23
Proof of Theorem 5.34
Proof of Theorem 5.32
5.7 Generic Dimensions for Some Schrödinger Operators
Part II Ergodic Theory and Semigroups
6 Generic Scales of Weak Mixing
6.1 Ergodic Dynamical Systems
6.2 Typical Automorphisms in the Weak Topology
6.2.1 Proof of Rohlin's Lemma
6.2.2 Proof of Halmos' Conjugacy Lemma
6.3 Refined Scales of Weak Mixing
6.3.1 Relation to Correlation Dimensions
6.3.2 Generic Behavior
7 Asymptotics of C0-Semigroups
7.1 Normal Semigroups
7.1.1 Polynomial Decay Rates Spectral Properties
7.1.2 Generic Decay Rates
7.1.3 Stability and Spectrum
7.2 C0-Semigroups in Hilbert Spaces
7.2.1 Generic Decay Rates
7.2.2 Applications
Wave Equation with Localized Viscoelasticity
Thermoelastic Systems of Bresse Type
Damped Wave Equation on the Torus
8 Generic Stability for Self-Adjoint Semigroups
8.1 Generic Stability
8.2 Proof of Theorem 8.1
8.3 A Typical Spectral Property
8.4 Schrödinger Equation Spectrum
Reference
Index
