From One-Dimensional to Multidimensional

Abstract

In this chapter, we construct a perturbation theory for two and three-dimensional Schrodinger operators with a periodic potential. Moreover, we describe the complexity of the multidimensional case by comparing it with the one-dimensional case. We show how this complexity increases when one passes from one to two and three-dimensional. This way helps to understand the reason for the complexity of the perturbation theory of the multidimensional periodic Schrodinger operator. It also helps reading of this and the next chapters. Thus, there are two main aims of this chapter. The first main aim is the direct study of the physically interesting two and three-dimensional cases. The second main aim is to help to read the complicated perturbation theory for the d-dimensional case (\(d>3)\) which is given in the next chapter, since some main ideas of the multidimensional perturbation theory are explained in this chapter. This chapter consists of five sections. First section is the introduction, where we describe briefly the scheme of this chapter, introduce the formulas essentially used in this and the next chapters and demonstrate their applications on the one-dimensional case. This demonstration helps us to explain the necessity of the division of the Bloch eigenvalues into two groups, non-resonance ones and resonance ones, in the multidimensional case. Moreover, it gives an explanation guiding to find the asymptotic formulas for the non-resonance eigenvalues of the multidimensional case. More precisely, in the introduction, we give the brief proof of the asymptotic formulas for the Bloch eigenvalues and Bloch functions in the one-dimensional case. Then in Conclusions 2.1–2.3, we describe briefly some ideas of the proofs of these formulas and stress the complexity of the multidimensional case by comparing it with the one-dimensional case. In Sect. 2.2, we give definitions and obtain asymptotic formulas for the both groups (non-resonance and resonance) of the Bloch eigenvalues of the two and three-dimensional periodic Schrodinger operators. In Sect. 2.3, we obtain asymptotic formulas for the Bloch functions. In Sect. 2.4, we construct and investigate the large part of the isoenergetic surfaces in the high energy region which implies the validity of the Bethe–Sommerfeld conjecture. Finally, in Sect. 2.5, we consider the nonsmooth potentials.