Table of contents for Determining spectra in quantum theory / Michael Demuth, M. Krishna.


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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



Library of Congress subject headings for this publication: Potential theory (Mathematics)Scattering (Mathematics)Spectral theory (Mathematics)Operator theory