Discrete Spectrum in the Gaps of the Continuous one in the Large-Coupling-Constant Limit

Abstract

The present paper is related to the paper [1], however, it does not depend on it formally. When the gap in the spectrum of the unperturbed operator is semi-infinite (coincides with (−∞,0)) then the problem considered below corresponds to investigation of the function N(α,a,V) introduced in [1] for a >0 and a “regular” perturbation V(x) ≥ 0. As we shall show in the following, the problem can be reduced in general to the application of a simple abstract theorem. If the “control point” (see Section 2) is situated on the boundary of a gap and V is allowed to be “nonregular”, then substantially more special considerations are necessary. For the periodic Schroedinger operator this will be discussed in another paper. Here we restrict ourselves to some applications of the abstract theorem of Section 3.

Translated by the editors