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

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

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Polytechnic Institute and State University

Year dc:date.issued

1989

Author dc:creator

• Fassari, Silvestro

Chair dc:contributor.committeechair

• Klaus, Martin

Committee members dc:contributor.committeemember

• Zweifel, Paul F.
• Slawny, Joseph
• Day, M.
• Polewczak, J.

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/10919/54524

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/54524