Multidimensional Periodic Schr¨odinger Operator

Abstract

The book is devoted to the spectral theory of the multidimensional Schr¨odinger operator L(q) with a periodic potential q. This operator is a fundamental operator of the solid state physics and describes the motion of a particle in the bulk matter. The book consists of five chapters. The first chapter presents preliminary defnitions and statements to be used in the next chapters. Besides the author gives a brief discussion of what is known from the literature and what is presented in the book about the perturbation theory of L(q). In the second chapter the asymptotic formulas of arbitrary order for the Bloch eigenvalue and Bloch function of the periodic Schr¨odinger operator L(q) of arbitrary dimension is obtained. Moreover, the author constructed and estimated the measures of the isoenergetic surfaces in the high energy region which implies the validity of the Bethe-Sommerfeld conjecture for arbitrary dimension and arbitrary lattice. This conjecture was formulated in 1928 and claims that there exist only a finite number of gaps in the spectrum of L(q)

Keywords

• Multidimensional
• Periodic
• Schr¨odinger

Oktay Veliev. Multidimensional Periodic Schr¨odinger Operator Perturbation Theory and Applications . ISBN 3319166425. P.242, 2015, Publisher: Springer, Series: Springer Tracts in Modern Physics.

Abstract