Includes bibliographical references (p. [203]-213) and index.

1 Measures and Transforms ................................. 1
1.1 Measures .............. ................................. 1
1.2 Fourier Transform ..................................... 5
1.3 The Wavelet Transform ................................. 7
1.4 Borel Transform .................. ...................... 16
1.5 Gesztesy-Krein-Simon Function ......................... 24
1.6 Notes ................. ............. ............ 25
2 Selfadjointness and Spectrum .............................. 29
2.1 Selfadjointness ........................... ............. 29
2.1.1 Linear Operators and Their Inverses ................. 29
2.1.2 Closed Operators ..................... .......... 30
2.1.3 Adjoint and Selfadjoint Operators ................... 32
2.1.4 Sums of Linear Operators .......................... 34
2.1.5 Sesquilinear Forms ................................ 35
2.2 Spectrum and Resolvent Sets ............................. 37
2.3 Spectral Theorem ..................................... 40
2.4 Spectral Measures and Spectrum .................. ....... 43
2.5 Spectral Theorem in the Hahn-Hellinger Form .............. 45
2.6 Components of the Spectrum ............................. 49
2.7 Characterization of the States in Spectral Subspaces ......... 53
2.8 Notes ................. ................... .......... 56
3 Criteria for Identifying the Spectrum ...................... 59
3.1 Borel Transform ...................................... 59
3.2 Fourier Transform ..................................... 68
3.3 Wavelet Transform ..................................... 69
3.4 Eigenfunctions ....................................... 70
3.5 Commutators ............................ ........... 72
3.6 Criteria Using Scattering Theory ....................... .. 80
3.6.1 Wave Operators .................................. 81
3.6.2 Stability of the Absolutely Continuous Spectra ........ 95
3.7 Notes .................................. ............104
4 Operators of Interest ...................................... 111
4.1 Unperturbed Operators ............... ............... . 111
4.1.1 Laplacians ..................... .................112
4.1.2 Unperturbed Semigroups and Their Kernels ..........119
4.1.3 Associated Processes . ........................... 120
4.1.4 Regular Dirichlet Forms, Capacities and Equilibrium
Potentials ......................... ..............121
4.2 Perturbed Operators ................ .................. 125
4.2.1 Deterministic Potentials ......................... . 125
4.2.2 Random Potentials ............................. . 133
4.2.3 Singular Perturbations .......................... . 135
4.3 Notes ................ .............................142
5 Applications .............................................153
5.1 Borel Transforms .......................................153
5.1.1 K otani Theory .................................... 153
5.1.2 Aizenman-Molchanov Method ...................... 160
5.1.3 Bethe Lattice ..................................... 172
5.1.4 Jaksid-Last Theorem ........................... . 181
5.2 Scattering ..........................................183
5.2.1 Decaying Random Potentials. ..................... .183
5.2.2 Obstacles and Potentials ........................ . 187
5.3 Notes ................................................196