{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/54524"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/54524","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Spectral properties of relativistic and non-relativistic Krönig- Penney Hamiltonians with short-range impurities","abstract":"In this work, we investigate the spectrum of the non-relativistic Krönig-Penney Hamiltonian H<sub>α</sub>= -d²/dx² +αΣ<sub>m∈Z</sub>δ(-(2m+1)π) perturbed by a short-range potential λW and the spectrum of its relativistic counterpart obtained by replacing the Schrödinger Hamiltonian H<sub>α</sub> with its relativistic analogue H̅<sub>α</sub>. 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<sub>α</sub> + 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.","abstract_html":"In this work, we investigate the spectrum of the non-relativistic Krönig-Penney Hamiltonian H&lt;sub&gt;α&lt;/sub&gt;= -d²/dx² +αΣ&lt;sub&gt;m∈Z&lt;/sub&gt;δ(-(2m+1)π) perturbed by a short-range potential λW and the spectrum of its relativistic counterpart obtained by replacing the Schrödinger Hamiltonian H&lt;sub&gt;α&lt;/sub&gt; with its relativistic analogue H̅&lt;sub&gt;α&lt;/sub&gt;. 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 &quot;impurity states&quot; 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&lt;sub&gt;α&lt;/sub&gt; + 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 &gt; 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.","abstract_has_math":false,"creators":["Fassari, Silvestro"],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Klaus, Martin"],"committee_members":["Zweifel, Paul F.","Slawny, Joseph","Day, M.","Polewczak, J."],"year":1989,"date_issued":"1989","date_published":"1989","updated_at":"2026-07-22T22:19:33Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/54524","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Klaus, Martin"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Zweifel, Paul F.","Slawny, Joseph","Day, M.","Polewczak, J."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Fassari, Silvestro"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2015-07-10T20:00:21Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2015-07-10T20:00:21Z"]},{"key":"dc:date.issued","label":"Date","values":["1989"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/54524"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this work, we investigate the spectrum of the non-relativistic Krönig-Penney Hamiltonian H<sub>α</sub>= -d²/dx² +αΣ<sub>m∈Z</sub>δ(-(2m+1)π) perturbed by a short-range potential λW and the spectrum of its relativistic counterpart obtained by replacing the Schrödinger Hamiltonian H<sub>α</sub> with its relativistic analogue H̅<sub>α</sub>. 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<sub>α</sub> + 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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Spectral properties of relativistic and non-relativistic Krönig- Penney Hamiltonians with short-range impurities"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Klaus, Martin"],"dc:contributor.committeemember":["Zweifel, Paul F.","Slawny, Joseph","Day, M.","Polewczak, J."],"dc:contributor.department":["Mathematics"],"dc:creator":["Fassari, Silvestro"],"dc:date.accessioned":["2015-07-10T20:00:21Z"],"dc:date.available":["2015-07-10T20:00:21Z"],"dc:date.issued":["1989"],"dc:description.abstract":["In this work, we investigate the spectrum of the non-relativistic Krönig-Penney Hamiltonian H<sub>α</sub>= -d²/dx² +αΣ<sub>m∈Z</sub>δ(-(2m+1)π) perturbed by a short-range potential λW and the spectrum of its relativistic counterpart obtained by replacing the Schrödinger Hamiltonian H<sub>α</sub> with its relativistic analogue H̅<sub>α</sub>. 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<sub>α</sub> + 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."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/54524"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Spectral properties of relativistic and non-relativistic Krönig- Penney Hamiltonians with short-range impurities"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:33Z"}