Determining Spectra in Quantum Theory

by Demuth, Michael; Krishna, M.

• ISBN13: 9780817643669
• ISBN10: 0817643664
• Format: Hardcover
• Copyright: 2005-07-01
• Publisher: Birkhauser

Summary

The spectral theory of Schrödinger operators, in particular those with random potentials, continues to be a very active field of research. This work focuses on various known criteria in the spectral theory of selfadjoint operators in order to identify the spectrum and its components á la Lebesgue decomposition. Key features and topics:* Well-developed exposition of criteria that are especially useful in determining the spectra of deterministic and random Schrödinger operators occurring in quantum theory* Systematically uses measures and their transforms (Fourier, Borel, wavelet) to present a unifying theme* Establishes criteria for identifying the spectrum* Examines a series of applications to show point spectrum and continuous spectrum in some models of random operators* Presents a series of spectral-theoretic results for the perturbed operators introduced in the earlier chapters with examples of localization and delocalization in the theory of disordered systems* Presents modern criteria (using wavelet transform, eigenfunction decay) that could be used to do spectral theory* Unique work in book form combining the presentation of the deterministic and random cases, which will serve as a platform for further research activities This concise unified presentation is aimed at graduate students and researchers working in the spectral theory of Schrödinger operators with either fixed or random potentials in particular. However, given the large gap that this book fills in the literature, it will serve a wider audience of mathematical physicists in its contribution to works in spectral theory.

Table of Contents

Preface	v
Measures and Transforms	1	(28)
Measures	1	(4)
Fourier Transform	5	(2)
The Wavelet Transform	7	(9)
Borel Transform	16	(8)
Gesztesy--Krein--Simon ξ Function	24	(1)
Notes	25	(4)
Selfadjointness and Spectrum	29	(30)
Selfadjointness	29	(8)
Linear Operators and Their Inverses	29	(1)
Closed Operators	30	(2)
Adjoint and Selfadjoint Operators	32	(2)
Sums of Linear Operators	34	(1)
Sesquilinear Forms	35	(2)
Spectrum and Resolvent Sets	37	(3)
Spectral Theorem	40	(3)
Spectral Measures and Spectrum	43	(2)
Spectral Theorem in the Hahn--Hellinger Form	45	(4)
Components of the Spectrum	49	(4)
Characterization of the States in Spectral Subspaces	53	(3)
Notes	56	(3)
Criteria for Identifying the Spectrum	59	(52)
Borel Transform	59	(9)
Fourier Transform	68	(1)
Wavelet Transform	69	(1)
Eigenfunctions	70	(2)
Commutators	72	(8)
Criteria Using Scattering Theory	80	(24)
Wave Operators	81	(14)
Stability of the Absolutely Continuous Spectra	95	(9)
Notes	104	(7)
Operators of Interest	111	(42)
Unperturbed Operators	111	(14)
Laplacians	112	(7)
Unperturbed Semigroups and Their Kernels	119	(1)
Associated Processes	120	(1)
Regular Dirichlet Forms, Capacities and Equilibrium Potentials	121	(4)
Perturbed Operators	125	(17)
Deterministic Potentials	125	(8)
Random Potentials	133	(2)
Singular Perturbations	135	(7)
Notes	142	(11)
Applications	153	(50)
Borel Transforms	153	(30)
Kotani Theory	153	(7)
Aizenman--Molchanov Method	160	(12)
Bethe Lattice	172	(9)
Jaksic--Last Theorem	181	(2)
Scattering	183	(13)
Decaying Random Potentials	183	(4)
Obstacles and Potentials	187	(9)
Notes	196	(7)
References	203	(12)
Index	215

Excerpts

"The spectral theory of Schrodinger operators, in particular those with random potentials, continues to be a very active field of research. This work focuses on various known criteria in the spectral theory of selfadjoint operators in order to identify the spectrum and its components a la Lebesgue decomposition." "This concise unified presentation is aimed at graduate students and researchers working in the spectral theory of Schrodinger operators with either fixed or random potentials in particular. However, given the large gap that this book fills in the literature, it will serve a wider audience of mathematical physicists in its contribution to works in spectral theory."--BOOK JACKET.