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Table of Contents:
Preface
Notation
1. A Bit of Quantum Mechanics, pp. 1-10
1.1 Axioms
1.2 Quantization
1.3 Heisenberg Uncertainty Principle
1.4 Quantum Oscillator
2. Operators in Hilbert Spaces, pp. 11-27
2.1 Preliminaries
2.2 Symmetric and Self-adjoint Operators
2.3 Unitary Operators and Unitary Equivalence
2.4 Examples
2.5 Resolvent
2.6 Forms and Operators
3. Spectral Theorem for Self-adjoint Operators, pp. 29-39
3.1 Diagonalization for Self-adjoint Operators
3.2 Spectral Decomposition
3.3 Functional Calculus
3.4 Classification of Spectrum
4. Compact Operators and the Hilbert-Schmidt Theorem, pp. 41-58
4.1 Preliminaries
4.2 Compact Operators
4.3 Fredholm Operators
4.4 The Hilbert-Schmidt Theorem
4.5 Hilbert-Schmidt Operators
4.6 Trace Class Operators
4.7 A Step Apart: Polar Decomposition
4.8 Trace Class Operators, II
4.9 Trace and Kernel Function
5. Elements of Perturbation Theory, pp. 59-68
5.1 Introductory Examples
5.2 The Riesz Projector
5.3 The Kato Lemma
5.4 Perturbation of Eigenvalues
5.5 Relatively Compact Perturbations
6. Variational Principles, pp. 69-76
6.1 Glazman's Lemma
6.2 Minimax Principle
7. One-Dimensional Schrödinger Operator, pp. 77-97
7.1 Self-adjointness
7.2 Discreteness of Spectrum
7.3 Negative Eigenvalues
7.3.1. Dirichlet Eigenvalues
7.3.2. Neumann Boundary Condition
7.3.3. The Case of Whole Axis
7.4 Schrodinger Operator on a Finite Interval
7.5 Hamiltonians with Point Interaction
8. Multidimensional Schrödinger Operator, pp. 99-127
8.1 Self-adjointness
8.2 Discrete Spectrum
8.3 Essential Spectrum
8.4 Eigenfunctions and Spectrum
8.5 Decay of Eigenfunctions
8.6 Agmon's Metric
8.7 Steep Potential Well
8.7.1. Eigenvalues Below Essential Spectrum
8.7.2. Discreteness of Spectrum
9. Periodic Schrödinger Operator, pp. 129-140
9.1 Direct Integrals and Decomposable Operators
9.2 One Dimensional Case
9.3 Multidimensional Case
9.4 Further Results
9.5 Decaying Perturbations of Periodic Potentials
10.Quantum Graphs, pp. 141-151
10.1 Introducing Quantum Graphs
10.1.1. Metric Graphs
10.1.2. Hamiltonians
10.2 Finite Graphs
10.2.1. Discreteness of Spectrum
10.2.2. Reduction to Discrete Operators
10.3 Infinite Graphs
10.3.1. Schnol's Theorem
10.3.2. On Structure of Spectrum
11. Nonlinear Schrödinger Equation, pp. 153-169
11.1. Evolutionary Schrodinger Equation
11.1.1. Formal Discussion
11.1.2. Cauchy Problem
11.1.3. Standing Waves
11.2 Exponential Decay
11.3 Translation Invariant NLS
11.3.1. Dimension Greater than 1
11.3.2. One Dimensional Case
11.3.3. Discrete Translation Invariance
11.4 NLS with Potential Well
11.4.1. Infinite Well
11.4.2. Steep Well
A Sobolev Spaces and Elliptic Equations, pp.171-175
A1. Weak Derivative and Sobolev Spaces
A2. Embedding Theorems
A3. Elliptic Regularity and Maximum Principle
B Mountain Pass Theorem, pp. 177-178
References
Index |
