Spectral theory and differential operators

Cover
Preface
1 Tbe fundamental ideas
1.1 Unbounded linear operators
1.2 Self-adjointness
1.3 Multiplication operators
1.4 Relatively bounded perturbations
Exercises
2 Tbe spectral tbeorem
2.1 Introduction
2.2 The Helffer-Sjöstrand formula
2.3 The first spectral theorem
2.4 Invariant and cyc1ic subspaces
2.5 The L² spectral representation
2.6 Norm resolvent convergence
Exercises
3 Translation invariant operators
3.1 Introduction
3.2 Schwartz space
3.3 The Fourier transform
3.4 Distributions
3.5 Differentiai operators
3.6 Some L^p estimates
3.7 The Sobolev spaces W^{n,2}(R^N)
Exercises
4 Tbe variational metbod
4.1 Classification of the spectrum
4.2 Compact operators
4.3 Positivity and fractional powers
4.4 Closed quadratic forms
4.5 The variational formulae
4.6 Lower bounds on eigenvalues
Exercises
5 Furtber spectral results
5.1 The Poisson problem
5.2 The heat equation
5.3 The Hardy inequality
5.4 Singular elliptic operators
5.5 The biharmonic operator
Exercises
6 Diricblet boundary conditions
6.1 Dirichlet boundary conditions
6.2 The Dirichlet Laplacian
6.3 The general case
Exercises
7 Neumann boundary conditions
7.1 Properties of the W^{1,2} spaces
7.2 Neumann boundary conditions
7.3 Computation of eigenvalues
Exercises
8 Schrödinger operators
8.1 Introduction
8.2 Definition of the operators
8.3 The positive spectrum
8.4 Compact perturbations
8.5 The negative spectrum
8.6 Double wells
Exercises
Bibliography
Notation index
Index