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Lecture Notes on Schrodinger Equations
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Authors: Alexander Pankov (College of William and Mary, USA) 
Book Description:
...The selection of material, the presentation and the references given shall be most helpful to prepare the reader for further studies in spectral theory and mathematical physics...

Zentralblatt MATH 1128

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

   Series:
      Contemporary Mathematical Studies (Gaston M. N'Guerekata - Morgan State University - Series Editor)
   Binding: Hardcover
   Pub. Date: 2007
   ISBN: 1-60021-447-9
   Status: AV
  
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
AP At Press
AV Available
  
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Lecture Notes on Schrodinger Equations