The book describes the direct problems and the inverse problem of the multidimensional Schrödinger operator with a periodic potential. This concerns perturbation theory and constructive determination of the spectral invariants and finding the periodic potential from the given Bloch eigenvalues. The unique method of this book derives the asymptotic formulas for Bloch eigenvalues and Bloch functions for arbitrary dimension. Moreover, the measure of the iso-energetic surfaces in the high energy region is construct and estimated. It implies the validity of the Bethe-Sommerfeld conjecture for arbitrary dimensions and arbitrary lattices. Using the perturbation theory constructed in this book, the spectral invariants of the multidimensional operator from the given Bloch eigenvalues are determined. Some of these invariants are explicitly expressed by the Fourier coefficients of the potential. This way the possibility to determine the potential constructively by using Bloch eigenvalues as input data is given. In the end an algorithm for the unique determination of the potential is given.

Presents an introduction to the theoretical base for solving the inverse problems of the multidimensional Schrödinger operator Opens new horizons for spectral analysis of quantum mechanical operators Covers the constructive determination of spectral invariants A useful resource for mathematical problems of solid state quantum physics Includes supplementary material: sn.pub/extras

Inhalt
Preface.- Asymptotic Formulas for the Bloch Eigenvalues and Bloch Functions.- Constructive Determination of the Spectral Invariants.- Periodic Potential from the Spectral Invariants.- Conclusions.