On the existence of eigenvalues of the Schrödinger operator H-\(\lambda\) W in a gap of \(\sigma\) (H) (Q1076873)

The model problem for the one-electron theory of solids for the Schrödinger operator \(H=-\Delta +V(x)\) of the pure crystal and for W(x) the potential which describes the ''impurity'' is considered. The problem under study is the following: given an energy \(E\in R(\sigma (H))\) (where \(\sigma\) (H) is the spectrum of H), does there exist a real coupling constant \(\lambda\) so that \(E\in \sigma (H-\lambda W)\). A number of results asserting the existence of eigenvalues of the Schrödinger operator H-\(\lambda\) W in a gap \(\sigma\) (H) are proved. The existence of these eigenvalues is an important element in the theory of the colour of crystals. The basic theorems are proved in \(R^ n\); stronger results for \(n=1\) are presented.