Spectral properties of relativistic and non-relativistic Krönig- Penney Hamiltonians with short-range impurities

TR Number
Date
1989
Journal Title
Journal ISSN
Volume Title
Publisher
Virginia Polytechnic Institute and State University
Abstract

In this work, we investigate the spectrum of the non-relativistic Krönig-Penney Hamiltonian Hα= -d²/dx² +αΣm∈Zδ(-(2m+1)π) perturbed by a short-range potential λW and the spectrum of its relativistic counterpart obtained by replacing the Schrödinger Hamiltonian Hα with its relativistic analogue H̅α.

The interesting feature of both spectra is that they have gaps and that bound states may occur in such gaps as a consequence of the presence of the short-range potential representing the impurity. Such bound states, often called "impurity states" in the solid state physics literature. are important with regard to the conductivity properties of solids

We show the existence of such bound states of Hα + λW in each sufficiently remote gap of its essential spectrum if the integral of W is different from zero and the 1 + 𝛅-moment of W is finite for some 𝛅 > 0. Furthermore, if the potential has a constant sign we prove that there is only one bound state in each sufficiently remote gap.

We shall see that in the relativistic case one may have more than one bound state in each remote gap under the same assumptions on W. Nevertheless, we shall see that such additional bound states cannot appear in the range of energies of solid state physics.

Description
Keywords
Citation