Preliminary Facts

In this chapter, we present some definitions and statements from the points of view of both physicists and mathematicians to be used in the next chapters. We mean especially the definitions of the lattices, periodic functions, Brillouin zones, Schrödinger operatorSchrödinger operator, Bloch eigenvalues, Bloch functions, diffraction planes, band structures, and Fermi surfaces. Moreover, we try to explain the transition between these notions due to the understanding of the physicists and mathematicians. Besides, we give a brief discussion of what is known from the literature and what is presented in the book about the perturbation theory of the multidimensional Schrödinger operatorSchrödinger operator with a periodic potential. For this aim, we consider the large Bloch eigenvalues and the corresponding Bloch functions of the one-dimensional periodic Schrödinger operatorSchrödinger operator by the approach of Chap.  2 , since it helps to compare the well-known one-dimensional case with the multidimensional case and to see the complexity of the results obtained in this book.