{"id":{"repo_id":"baylor","oai_identifier":"oai:baylor-ir.tdl.org:2104/10658"},"canonical_url":"https://search.dev.ndltd.org/etd/baylor/oai:baylor-ir.tdl.org:2104/10658","repository":{"repo_id":"baylor","name":"Baylor University","base_url":"https://baylor-ir.tdl.org/server/oai/request"},"display":{"title":"The sixth-order Krall differential expression and self-adjoint operators.","abstract":"We first provide an overview of classical GNK Theory for symmetric, or symmatrizable, differential expressions in L2((a, b);w). Then we review how this theory was applied to find a self-adjoint operator in L2 \\mu(-1, 1) generated by the sixth-order Lagrangian symmetric Krall differential equation, as done by S. M. Loveland. We later construct the self-adjoint operator generated by the Krall differential equation in the extended Hilbert space L2(-1, 1) + C2 which has the Krall polynomials as (orthogonal) eigenfunctions. The theory we use to create this self-adjoint operator was developed recently by L. L. Littlejohn and R. Wellman as an application of the general Glazman-Krein-Naimark (GKN) Theorem discovered by W. N. Everitt and L. Markus using complex symplectic geometry. In order to explicitly construct this self-adjoint operator, we use properties of functions in the maximal domain in L2(-1, 1) of the Krall expression. As we will see, continuity, as a boundary condition, is forced by our construction of this self-adjoint operator. We also construct six additional examples of self-adjoint operators in an extended Hilbert space, three with a one-dimensional extension space and three with a two-dimensional extension space.","abstract_html":"We first provide an overview of classical GNK Theory for symmetric, or symmatrizable, differential expressions in L2((a, b);w). Then we review how this theory was applied to find a self-adjoint operator in L2 \\mu(-1, 1) generated by the sixth-order Lagrangian symmetric Krall differential equation, as done by S. M. Loveland. We later construct the self-adjoint operator generated by the Krall differential equation in the extended Hilbert space L2(-1, 1) + C2 which has the Krall polynomials as (orthogonal) eigenfunctions. The theory we use to create this self-adjoint operator was developed recently by L. L. Littlejohn and R. Wellman as an application of the general Glazman-Krein-Naimark (GKN) Theorem discovered by W. N. Everitt and L. Markus using complex symplectic geometry. In order to explicitly construct this self-adjoint operator, we use properties of functions in the maximal domain in L2(-1, 1) of the Krall expression. As we will see, continuity, as a boundary condition, is forced by our construction of this self-adjoint operator. We also construct six additional examples of self-adjoint operators in an extended Hilbert space, three with a one-dimensional extension space and three with a two-dimensional extension space.","abstract_has_math":false,"creators":["Elliott, Katie, 1991-"],"institution":"Baylor University.","degree_name":"Ph.D.","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Littlejohn, Lance, 1951-"],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-05","date_published":"2019-05","updated_at":"2026-07-24T01:08:10Z","subjects":["Self-adjoint operators.","Krall differential expression.","Extended Hilbert space."],"languages":["en"],"rights":["Baylor University works are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact libraryquestions@baylor.edu for inquiries about permission."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2104/10658","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Littlejohn, Lance, 1951-"]},{"key":"dc:creator","label":"Author","values":["Elliott, Katie, 1991-"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-07-29T13:29:12Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-07-29T13:29:12Z"]},{"key":"dc:date.issued","label":"Date","values":["2019-05"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"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":["Baylor University."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Self-adjoint operators.","Krall differential expression.","Extended Hilbert space."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Baylor University works are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact libraryquestions@baylor.edu for inquiries about permission."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/2104/10658"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We first provide an overview of classical GNK Theory for symmetric, or symmatrizable, differential expressions in L2((a, b);w). Then we review how this theory was applied to find a self-adjoint operator in L2 \\mu(-1, 1) generated by the sixth-order Lagrangian symmetric Krall differential equation, as done by S. M. Loveland. We later construct the self-adjoint operator generated by the Krall differential equation in the extended Hilbert space L2(-1, 1) + C2 which has the Krall polynomials as (orthogonal) eigenfunctions. The theory we use to create this self-adjoint operator was developed recently by L. L. Littlejohn and R. Wellman as an application of the general Glazman-Krein-Naimark (GKN) Theorem discovered by W. N. Everitt and L. Markus using complex symplectic geometry. In order to explicitly construct this self-adjoint operator, we use properties of functions in the maximal domain in L2(-1, 1) of the Krall expression. As we will see, continuity, as a boundary condition, is forced by our construction of this self-adjoint operator. We also construct six additional examples of self-adjoint operators in an extended Hilbert space, three with a one-dimensional extension space and three with a two-dimensional extension space."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["The sixth-order Krall differential expression and self-adjoint operators."]}]}],"canonical_facts":{"dc:contributor.advisor":["Littlejohn, Lance, 1951-"],"dc:creator":["Elliott, Katie, 1991-"],"dc:date.accessioned":["2019-07-29T13:29:12Z"],"dc:date.available":["2019-07-29T13:29:12Z"],"dc:date.issued":["2019-05"],"dc:description.abstract":["We first provide an overview of classical GNK Theory for symmetric, or symmatrizable, differential expressions in L2((a, b);w). Then we review how this theory was applied to find a self-adjoint operator in L2 \\mu(-1, 1) generated by the sixth-order Lagrangian symmetric Krall differential equation, as done by S. M. Loveland. We later construct the self-adjoint operator generated by the Krall differential equation in the extended Hilbert space L2(-1, 1) + C2 which has the Krall polynomials as (orthogonal) eigenfunctions. The theory we use to create this self-adjoint operator was developed recently by L. L. Littlejohn and R. Wellman as an application of the general Glazman-Krein-Naimark (GKN) Theorem discovered by W. N. Everitt and L. Markus using complex symplectic geometry. In order to explicitly construct this self-adjoint operator, we use properties of functions in the maximal domain in L2(-1, 1) of the Krall expression. As we will see, continuity, as a boundary condition, is forced by our construction of this self-adjoint operator. We also construct six additional examples of self-adjoint operators in an extended Hilbert space, three with a one-dimensional extension space and three with a two-dimensional extension space."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/2104/10658"],"dc:language.iso":["en"],"dc:rights":["Baylor University works are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact libraryquestions@baylor.edu for inquiries about permission."],"dc:subject":["Self-adjoint operators.","Krall differential expression.","Extended Hilbert space."],"dc:title":["The sixth-order Krall differential expression and self-adjoint operators."],"dc:type":["Thesis"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["Baylor University."]},"updated_at":"2026-07-24T01:08:10Z"}