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.