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Ⅻ：PERTURBATION OF POINT SPECTRA 1

1．Finite-dimensional perturbation theory 1

Appendix Algebraic and geometric multiplicity of eigenvalues of finite matrices 9

2．Regular perturbation theory 10

3．Asymptotic perturbation theory 25

4．Summability methods in perturbation theory 38

5．Spectral concentration 45

6．Resonances and the Fermi golden rule 51

Notes 60

Problems 69

ⅩⅢ：SPECTRAL ANALYSIS 75

1．The min-max principle 75

2．Bound states of Schr？dinger operators Ⅰ：Quantitative methods 79

3．Bound states of Schr？dinger operators Ⅱ：Qualitative theory 86

4．Isσdisc（H）finite or infinite？ 86

B．Bounds on N（V）in the central case 90

C．Bounds on N（V）in the general two-body case 98

4．Locating the essential spectrum Ⅰ：Weyl's theorem 106

5．Locating the essential spectrum Ⅲ：The HVZ theorem 120

6．The absence of singular continuous spectrum Ⅰ：General theory 136

7．The absence of singular continuous spectrum Ⅱ：Smooth perturbations 141

A．Weakly coupled quantum systems 151

B．Positive commutators and repulsive potentials 157

C．Local smoothness and wave operators for repulsive potentials 163

8．The absence of singular continuous spectrum Ⅲ：Weighted L2 spaces 168

9．The spectrum of tensor products 177

10．The absence of singular continuous spectrum Ⅳ：Dilation analytic potentials 183

11．Properties of eigenfunctions 191

12．Nondegeneracy of the ground state 201

Appendix 1 The Beurling-Deny criteria 209

Appendix 2 The Levy-Khintchine formula 212

13．Absence of positive eigenvalues 222

Appendix Unique continuation theorems for Schr？dinger operators 239

14．Compactness criteria and operators with compact resolvent 244

15．The asymptotic distribution of eigenvalues 260

16．Schr？dinger operators with periodic potentials 279

17．An introduction to the spectral theory of non-self-adjoint operators 316

Notes 338

Problems 364

List of Symbols 387

Index 389