Non-Self-Adjoint Schrödinger Operator with a Periodic Potential : Spectral Theories for Scalar and Vectorial Cases and Their Generalizations / by Oktay Veliev.

Format:

Book

Author/Creator:

Veliev, Oktay.

Series:

Physics and Astronomy Series

Language:

English

Subjects (All):

Mathematical physics.

Quantum theory.

Condensed matter.

Optics.

Theoretical, Mathematical and Computational Physics.

Quantum Physics.

Mathematical Methods in Physics.

Condensed Matter Physics.

Optics and Photonics.

Local Subjects:

Theoretical, Mathematical and Computational Physics.

Quantum Physics.

Mathematical Methods in Physics.

Condensed Matter Physics.

Optics and Photonics.

Physical Description:

1 online resource (777 pages)

Edition:

2nd ed. 2025.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2025.

Summary:

This book offers a comprehensive exploration of spectral theory for non-self-adjoint differential operators with complex-valued periodic coefficients, addressing one of the most challenging problems in mathematical physics and quantum mechanics: constructing spectral expansions in the absence of a general spectral theorem. It examines scalar and vector Schrödinger operators, including those with PT-symmetric periodic optical potentials, and extends these methodologies to higher-order operators with periodic matrix coefficients. The second edition significantly expands upon the first by introducing two new chapters that provide a complete description of the spectral theory of non-self-adjoint differential operators with periodic coefficients. The first of these new chapters focuses on the vector case, offering a detailed analysis of the spectral theory of non-self-adjoint Schrödinger operators with periodic matrix potentials. It thoroughly examines eigenvalues, eigenfunctions, and spectral expansions for systems of one-dimensional Schrödinger operators. The second chapter develops a comprehensive spectral theory for all ordinary differential operators, including higher-order and vector cases, with periodic coefficients. It also includes a complete classification of the spectrum for PT-symmetric periodic differential operators, making this edition the most comprehensive treatment of these topics to date. The book begins with foundational topics, including spectral theory for Schrödinger operators with complex-valued periodic potentials, and systematically advances to specialized cases such as the Mathieu–Schrödinger operator and PT-symmetric periodic systems. By progressively increasing the complexity, it provides a unified and accessible framework for students and researchers. The approaches developed here open new horizons for spectral analysis, particularly in the context of optics, quantum mechanics, and mathematical physics.

Contents:

1.Introduction and Overview

2.Spectral Theory for the Schr¨odinger Operator with a ComplexValued Periodic Potential

3.On the Special Potentials

4.On the Mathieu-Schr¨odinger Operator

5.PT-Symmetric Periodic Optical Potential

6.On the Schr¨odinger Operator with a Periodic Matrix Potential

7.Some Generalizations and Supplements.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Veliev, Oktay Non-Self-Adjoint Schrödinger Operator with a Periodic Potential

ISBN:

9783031902598

OCLC:

1527724696