Non-self-adjoint Schrödinger Operator with a Periodic Potential

✍ Scribed by Oktay Veliev

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 301
• Category: Library

✦ Synopsis

This book gives a complete spectral analysis of the non-self-adjoint Schrödinger operator with a periodic complex-valued potential. Building from the investigation of the spectrum and spectral singularities and construction of the spectral expansion for the non-self-adjoint Schrödinger operator, the book features a complete spectral analysis of the Mathieu-Schrödinger operator and the Schrödinger operator with a parity-time (PT)-symmetric periodic optical potential. There currently exists no general spectral theorem for non-self-adjoint operators; the approaches in this book thus open up new possibilities for spectral analysis of some of the most important operators used in non-Hermitian quantum mechanics and optics. Featuring detailed proofs and a comprehensive treatment of the subject matter, the book is ideally suited for graduate students at the intersection of physics and mathematics.

✦ Table of Contents

Preface
Contents
1 Introduction and Overview
2 Spectral Theory for the Schrödinger Operator with a Complex-Valued Periodic Potential
2.1 Introduction
2.2 On the Floquet Theory and Spectrum of L(q)
2.3 Asymptotic Formulas for the Bloch Eigenvalues and Bloch Functions
2.3.1 Uniform Asymptotic Formulas for the Isolated Eigenvalues
2.3.2 Asymptotic Formulas for the Isolated Pairs
2.3.3 On the Numerations of the Bloch Eigenvalues and Bands
2.4 Spectral Singularities and ESS of the Operator L(q)
2.5 Spectral Expansion for the Non-self-adjoint Operator L(q)
2.6 The Necessity of the Brackets and P.V. Integrals and Criteria for the Elegant Expansion
2.7 On the Asymptotic Spectrality of L(q)
2.7.1 Spectral Singularities and Spectrality
2.7.2 On the Asymptotically Spectral Potentials
2.8 Appendices
3 On the Special Potentials
3.1 On the Even Potentials
3.2 On the PT-Symmetric Potentials
3.2.1 General Properties of the Spectrum
3.2.2 On the Bands Γn for Large n
3.2.3 Reality and Non-Reality of the Bands and Spectrality of L(q)
3.2.4 Finite Zone PT-Symmetric Periodic Potentials
3.2.5 Conclusions, Notes and References
3.3 Pure Complex-Valued Potentials with Pure Real Spectrum
3.3.1 On the Bloch Eigenvalues and Bloch Function
3.3.2 On the Inverse Problem
3.3.3 On the Spectral Singularities and ESS
4 On the Mathieu-Schrödinger Operator
4.1 Introduction
4.2 Asymptotic Formulas for the Isolated Eigenvalues and Isospectrality
4.3 Asymptotic Formulas for the Pairs of the Eigenvalues and Spectrality
4.4 On the ESS and Spectral Expansion of H(a,b)
4.5 On Simplicity of the Periodic and Antiperiodic Eigenvalues and Elegant Spectral Expansion
5 PT-Symmetric Periodic Optical Potential
5.1 Introduction
5.2 On the Bloch Eigenvalues
5.3 On the Bands and Components of the Spectrum
5.4 Spectral Singularities, ESS and Spectral Expansion
5.5 Finding the Second Critical Point
5.6 On the Small Perturbations
5.7 Conclusions and Notes
Index