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 BetheSommerfeld 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.

Details a unique method for deriving the asymptotic formulas for Bloch eigenvalues and Bloch functions in arbitrary dimensions Provides an introduction to the theoretical base for solving the inverse problems of the multidimensional Schrödinger operator Opens up new horizons for spectral analysis of quantum mechanical operators Represents a useful resource on mathematical problems in solid state quantum physics

Autorentext
Oktay Veliev received his B.S. degree in Mathematics in 1977 and Ph.D. degree in Mathematics in 1980 from Moscow State University, earning a Doctor of Sciences degree in 1989. From 1980 to 1983, he was a researcher and then senior researcher (19831988) at the Institute of Mathematics of the Academy of Sciences of Azerbaijan SSR. At Baku State University (Azerbaijan) he has been an Associate Professor (19881991), a Professor (19911992), and Head of the Department of Functional Analysis (19921997). Between 1993 and 1997, he was President of the Azerbaijan Mathematical Society. He was a visiting Professor at the University of Nantes (France), the Institute of Mathematics at the ETH (Switzerland), and Sussex University (England). From 1997 to 2002 he was a Professor at Dokuz Eylul University (Turkey), and since 2003 has been a Professor at Dogus University (Turkey). He has received grants from the American Mathematical Society and the International Science Foundation (Grant No. MVVOOO).

Inhalt
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.