Non-Self-Adjoint Schrödinger Operator with a Periodic Potential Spectral Theories for Scalar and Vectorial Cases and Their Generalizations

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.

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.

Oktay Veliev received his B.S. degree in Mathematics in 1977 and Ph.D. degree in Mathematics in 1980 from Moscow State University, earning a Doctor of Sciences degree in 1989. From 1980 to 1983, he was a researcher and then a senior researcher at the Institute of Mathematics of the Academy of Sciences of Azerbaijan SSR. At Baku State University (Azerbaijan), he has been Associate Professor, Professor, and Head of the Department of Functional Analysis. Between 1993 and 1997, he was President of the Azerbaijan Mathematical Society. He was Visiting Professor at the University of Nantes, the Institute of Mathematics at the ETH, and Sussex University. From 1997 to 2002, he was Professor at Dokuz Eylul University and since 2003 has been Professor at Dogus University.