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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 Schroedinger 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 Schroedinger operators with periodic matrix potentials. It thoroughly examines eigenvalues, eigenfunctions, and spectral expansions for systems of one-dimensional Schroedinger 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 Schroedinger operators with complex-valued periodic potentials, and systematically advances to specialized cases such as the Mathieu-Schroedinger 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.