{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/51851"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/51851","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Eigenvalue Statistics for Random Block Operators","abstract":"The Schrodinger Hamiltonian for a single electron in a crystalline solid with independent, identically distributed (i.i.d.) single-site potentials has been well studied. It has the form of a diagonal potential energy operator, which contains the random variables, plus a kinetic energy operator, which is deterministic. In the less-understood cases of multiple interacting charge carriers, or of correlated random variables, the Hamiltonian can take the form of a random block-diagonal operator, plus the usual kinetic energy term. Thus, it is of interest to understand the eigenvalue statistics for such operators. In this work, we establish a criterion under which certain random block operators will be guaranteed to satisfy Wegner, Minami, and higher-order estimates. This criterion is phrased in terms of properties of individual blocks of the Hamiltonian. We will then verify the input conditions of this criterion for a certain quasiparticle model with i.i.d. single-site potentials. Next, we will present a progress report on a project to verify the same input conditions for a class of one-dimensional, single-particle alloy-type models. These two results should be sufficient to demonstrate the utility of the criterion as a method of proving Wegner and Minami estimates for random block operators.","abstract_html":"The Schrodinger Hamiltonian for a single electron in a crystalline solid with independent, identically distributed (i.i.d.) single-site potentials has been well studied. It has the form of a diagonal potential energy operator, which contains the random variables, plus a kinetic energy operator, which is deterministic. In the less-understood cases of multiple interacting charge carriers, or of correlated random variables, the Hamiltonian can take the form of a random block-diagonal operator, plus the usual kinetic energy term. Thus, it is of interest to understand the eigenvalue statistics for such operators. In this work, we establish a criterion under which certain random block operators will be guaranteed to satisfy Wegner, Minami, and higher-order estimates. This criterion is phrased in terms of properties of individual blocks of the Hamiltonian. We will then verify the input conditions of this criterion for a certain quasiparticle model with i.i.d. single-site potentials. Next, we will present a progress report on a project to verify the same input conditions for a class of one-dimensional, single-particle alloy-type models. These two results should be sufficient to demonstrate the utility of the criterion as a method of proving Wegner and Minami estimates for random block operators.","abstract_has_math":false,"creators":["Schmidt, Daniel F."],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Elgart, Alexander"],"committee_members":["Hagedorn, George A.","Klaus, Martin","Ball, Joseph A."],"year":2015,"date_issued":"2015-04-28","date_published":"2015-04-28","updated_at":"2026-07-22T22:18:57Z","subjects":["Eigenvalue statistics","Wegner estimate","Minami estimate","Schrodinger operator"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:4806"],"render_values":[{"text":"vt_gsexam:4806","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/51851","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Elgart, Alexander"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Hagedorn, George A.","Klaus, Martin","Ball, Joseph A."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Schmidt, Daniel F."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2015-04-29T08:02:11Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2015-04-29T08:02:11Z"]},{"key":"dc:date.issued","label":"Date","values":["2015-04-28"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"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":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Eigenvalue statistics","Wegner estimate","Minami estimate","Schrodinger operator"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"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.other","label":"Dc Identifier Other","values":["vt_gsexam:4806"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/51851"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The Schrodinger Hamiltonian for a single electron in a crystalline solid with independent, identically distributed (i.i.d.) single-site potentials has been well studied. It has the form of a diagonal potential energy operator, which contains the random variables, plus a kinetic energy operator, which is deterministic. In the less-understood cases of multiple interacting charge carriers, or of correlated random variables, the Hamiltonian can take the form of a random block-diagonal operator, plus the usual kinetic energy term. Thus, it is of interest to understand the eigenvalue statistics for such operators. In this work, we establish a criterion under which certain random block operators will be guaranteed to satisfy Wegner, Minami, and higher-order estimates. This criterion is phrased in terms of properties of individual blocks of the Hamiltonian. We will then verify the input conditions of this criterion for a certain quasiparticle model with i.i.d. single-site potentials. Next, we will present a progress report on a project to verify the same input conditions for a class of one-dimensional, single-particle alloy-type models. These two results should be sufficient to demonstrate the utility of the criterion as a method of proving Wegner and Minami estimates for random block operators."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Eigenvalue Statistics for Random Block Operators"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Elgart, Alexander"],"dc:contributor.committeemember":["Hagedorn, George A.","Klaus, Martin","Ball, Joseph A."],"dc:contributor.department":["Mathematics"],"dc:creator":["Schmidt, Daniel F."],"dc:date.accessioned":["2015-04-29T08:02:11Z"],"dc:date.available":["2015-04-29T08:02:11Z"],"dc:date.issued":["2015-04-28"],"dc:description.abstract":["The Schrodinger Hamiltonian for a single electron in a crystalline solid with independent, identically distributed (i.i.d.) single-site potentials has been well studied. It has the form of a diagonal potential energy operator, which contains the random variables, plus a kinetic energy operator, which is deterministic. In the less-understood cases of multiple interacting charge carriers, or of correlated random variables, the Hamiltonian can take the form of a random block-diagonal operator, plus the usual kinetic energy term. Thus, it is of interest to understand the eigenvalue statistics for such operators. In this work, we establish a criterion under which certain random block operators will be guaranteed to satisfy Wegner, Minami, and higher-order estimates. This criterion is phrased in terms of properties of individual blocks of the Hamiltonian. We will then verify the input conditions of this criterion for a certain quasiparticle model with i.i.d. single-site potentials. Next, we will present a progress report on a project to verify the same input conditions for a class of one-dimensional, single-particle alloy-type models. These two results should be sufficient to demonstrate the utility of the criterion as a method of proving Wegner and Minami estimates for random block operators."],"dc:description.degree":["Ph. D."],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:4806"],"dc:identifier.uri":["http://hdl.handle.net/10919/51851"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Eigenvalue statistics","Wegner estimate","Minami estimate","Schrodinger operator"],"dc:title":["Eigenvalue Statistics for Random Block Operators"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:18:57Z"}