Non-Self-Adjoint Schrödinger Operator with a Periodic Potential
Spectral Theories for Scalar and Vectorial Cases and Their Generalizations
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Beschreibung
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.
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.
Solves the spectral expansion problem for periodic potentials in non-self-adjoint operators Offers complete spectral theory for the Mathieu–Schrödinger operator and PT-symmetric periodic optical potentials Introduces methods that open new avenues in mathematical physics, optics, and non-Hermitian quantum mechanics
Autor*in
Oktay Veliev
Themen in »Non-Self-Adjoint Schrödinger Operator with a Periodic Potential«
Non-self-adjoint Operators Schrödinger Operators Periodic Differential Operators PT-Symmetric Quantum Mechanics Periodic Optical Potential Spectral Analysis of the Schrödinger Operator Mathieu-Hill Operator Spectral Expansion of the Hill Operator Matheiu-Schrödinger Operator Non-Self-Adjoint Spectral Theory Schrödinger Operator with Periodic Potential Periodic Sturm-Liouville Operators Advanced Spectral Theory Applications Complex-Valued Optical Potentials Quantum Mechanics Spectral Classification
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“The exposition is self-contained and progresses from general spectral theory for arbitrary periodic complex potentials to specialized classes, matrix operators, and finally general periodic differential operators. The book is suitable for researchers working in spectral theory, non-self-adjoint operator theory, and mathematical physics, particularly those concerned with periodic structures and non-Hermitian phenomena.” (César R. de Oliveira, Mathematical Reviews, May, 2026)
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