Multidimensional Periodic Schrödinger Operator : Perturbation Theory and Applications Chapter 1 Preliminary Facts Chapter 2 From One dimensional to Multidimensional Chapter 3 Asymptotic Formulas for the Bloch Eigenvalues and Bloch Functions. Chapter 4 Constructive Determination of the Spectral Invariants Chapter 5 Periodic Potential from the Spectral Invariants Chapter 6 Conclusions This book describes the direct and inverse problems of the multidimensional Schrödinger operator with a periodic potential a topic that is especially important in perturbation theory constructive determination of spectral invariants and finding the periodic potential from the given Bloch eigenvalues. It provides a detailed derivation of the asymptotic formulas for Bloch eigenvalues and Bloch functions in arbitrary dimensions while constructing and estimating the measure of the iso energetic surfaces in the high energy regime. Moreover it presents a unique method proving the validity of the Bethe–Sommerfeld conjecture for arbitrary dimensions and arbitrary lattices. Using the perturbation theory constructed it determines the spectral invariants of the multidimensional operator from the given Bloch eigenvalues. Some of these invariants are explicitly expressed by the Fourier coefficients of the potential making it possible to determine the potential constructively using Bloch eigenvalues as input data. Lastly the book presents an algorithm for the unique determination of the potential. This updated second edition includes an additional chapter that specifically focuses on lower dimensional cases providing the basis for the higher dimensional considerations of the chapters that follow