## On the existence of eigenvalues of the Schrödinger operator H-$$\lambda$$ W in a gap of $$\sigma$$ (H).(English)Zbl 0594.34022

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.
Reviewer: B.Konopelchenko

### MSC:

 34L99 Ordinary differential operators

### Keywords:

Schrödinger operator; eigenvalues; colour of crystals
Full Text:

### References:

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