From One-Dimensional to Multidimensional

In this chapter, we construct a perturbation theory for two and three dimensional Schrödinger operatorsSchrödinger operator 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 of the complexity of the perturbation theory of the multidimensional periodic Schrödinger operatorSchrödinger 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.