Non-self-adjoint Schrödinger Operator with a Periodic Potential

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.

1. Spectral Theory for the Schrödinger Operator with a Complex-Valued Periodic Potential.- 2. On the Special Potentials.- 3. On the Matheiu-Schrödinger Operator.- 4. PT-Symmetric Periodic Optical Potential.- Index.