{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/115344"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/115344","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Spectral theory: minimizing Robin gaps, and singular limits for sign-changing weights","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-11 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-11 without embargo terms","abstract_has_math":false,"creators":["Kielty, Derek"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Laugesen, Richard S","Bronski, Jared","Kirr, Eduard","Zharnitsky, Vadim"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-05","date_published":"2022-05","updated_at":"2026-07-22T22:24:54Z","subjects":["spectral theory","partial differential equations","ordinary differential equations","spectral gaps","indefinite weight"],"languages":["en","eng"],"rights":["Copyright 2022 Derek Kielty"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/115344","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Laugesen, Richard S","Bronski, Jared","Kirr, Eduard","Zharnitsky, Vadim"]},{"key":"dc:creator","label":"Author","values":["Kielty, Derek"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-05","2022-03-18"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["spectral theory","partial differential equations","ordinary differential equations","spectral gaps","indefinite weight"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2022 Derek Kielty"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/115344"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-11 without embargo terms","The student, Derek Kielty, accepted the attached license on 2022-03-03 at 15:53.","The student, Derek Kielty, submitted this Dissertation for approval on 2022-03-03 at 16:06.","This Dissertation was approved for publication on 2022-03-18 at 10:14.","DSpace SAF Submission Ingestion Package generated from Vireo submission #17520 on 2022-11-11 at 13:04:29","The first part of this thesis considers the spectral gap of Schr\\\"odinger eigenvalue problems. Chapter 2 proves monotonicity results for the spectral gap with respect to the potential and the Robin parameter, for 1-dimensional Schr\\\"odinger operators. In particular, these results lead to sharp lower bounds on the spectral gap of operators with single-well potentials that generalize work of Cheng et al.\\ and Horv\\'ath. In higher dimensions, the spectral gap of the Neumann and Dirichlet Laplacians are each known to have a sharp positive lower bound among convex domains of a given diameter. Between these cases, for each positive value of the Robin parameter an analogous sharp lower bound on the spectral gap is conjectured. Chapter 3 shows the extension of this conjecture to each fixed negative Robin parameter fails, by constructing a family of double cone domains with an exponentially small spectral gap --- even though the first two eigenvalues themselves are polynomially large. The second part of this thesis considers the eigenvalue problem generated by a fixed differential operator with a sign-changing weight multiplying the eigenvalue term. Chapter 4 proves that as part of the weight is rescaled towards negative infinity on some subregion, the spectrum converges to that of the original problem restricted to the complementary region. On the interface between the regions the limiting problem acquires Dirichlet-type boundary conditions. The underlying theorem concerns eigenvalue problems for sign-changing bilinear forms on Hilbert spaces. The results are applied to a wide range of PDEs: second and fourth order equations with both Dirichlet and Neumann-type problems, and a problem where the eigenvalue appears in both the equation and the boundary condition."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Spectral theory: minimizing Robin gaps, and singular limits for sign-changing weights"]}]}],"canonical_facts":{"dc:contributor":["Laugesen, Richard S","Bronski, Jared","Kirr, Eduard","Zharnitsky, Vadim"],"dc:creator":["Kielty, Derek"],"dc:date":["2022-05","2022-03-18"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-11 without embargo terms","The student, Derek Kielty, accepted the attached license on 2022-03-03 at 15:53.","The student, Derek Kielty, submitted this Dissertation for approval on 2022-03-03 at 16:06.","This Dissertation was approved for publication on 2022-03-18 at 10:14.","DSpace SAF Submission Ingestion Package generated from Vireo submission #17520 on 2022-11-11 at 13:04:29","The first part of this thesis considers the spectral gap of Schr\\\"odinger eigenvalue problems. Chapter 2 proves monotonicity results for the spectral gap with respect to the potential and the Robin parameter, for 1-dimensional Schr\\\"odinger operators. In particular, these results lead to sharp lower bounds on the spectral gap of operators with single-well potentials that generalize work of Cheng et al.\\ and Horv\\'ath. In higher dimensions, the spectral gap of the Neumann and Dirichlet Laplacians are each known to have a sharp positive lower bound among convex domains of a given diameter. Between these cases, for each positive value of the Robin parameter an analogous sharp lower bound on the spectral gap is conjectured. Chapter 3 shows the extension of this conjecture to each fixed negative Robin parameter fails, by constructing a family of double cone domains with an exponentially small spectral gap --- even though the first two eigenvalues themselves are polynomially large. The second part of this thesis considers the eigenvalue problem generated by a fixed differential operator with a sign-changing weight multiplying the eigenvalue term. Chapter 4 proves that as part of the weight is rescaled towards negative infinity on some subregion, the spectrum converges to that of the original problem restricted to the complementary region. On the interface between the regions the limiting problem acquires Dirichlet-type boundary conditions. The underlying theorem concerns eigenvalue problems for sign-changing bilinear forms on Hilbert spaces. The results are applied to a wide range of PDEs: second and fourth order equations with both Dirichlet and Neumann-type problems, and a problem where the eigenvalue appears in both the equation and the boundary condition."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/115344"],"dc:language":["en","eng"],"dc:rights":["Copyright 2022 Derek Kielty"],"dc:subject":["spectral theory","partial differential equations","ordinary differential equations","spectral gaps","indefinite weight"],"dc:title":["Spectral theory: minimizing Robin gaps, and singular limits for sign-changing weights"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:54Z"}