{"id":{"repo_id":"utc","oai_identifier":"oai:scholar.utc.edu:theses-1649"},"canonical_url":"https://search.dev.ndltd.org/etd/utc/oai:scholar.utc.edu:theses-1649","repository":{"repo_id":"utc","name":"University of Tennessee - Chattanooga","base_url":"https://scholar.utc.edu/do/oai/"},"display":{"title":"Krein's identity and trace formulas for half-line Schrodinger operators","abstract":"We consider self-adjoint extensions of the minimal operator generated by the differential expression $\\cL = - d^2/dx^2+V$ on the half-line $[0,\\infty)$, where $V$ is a real-valued function integrable with respect to the weight $1+x$. The self-adjoint extensions are of Schr\\\"odinger-type and form a one-parameter family formally given by $H_{\\alpha}=-d^2/dx^2+V$, $\\alpha\\in[0,\\pi)$, with the boundary condition $\\sin(\\alpha)f'(0) = \\cos(\\alpha)f(0)$ at $x=0$. We derive a formula that relates the resolvent operator of $H_{\\alpha}$ to the resolvent operator of $H_0$ in terms of the Jost solution corresponding to the underlying differential equation $\\mathcal{L}u=zu$. Combining this resolvent formula and properties of the Jost solution, we compute the trace of the difference of the resolvents of $H_{\\alpha}$ and the free operator $H_{\\alpha}^{(0)}$ with $V\\equiv 0$ in terms of the parameter $\\alpha$ and the Jost function for $\\cL u=zu$.","abstract_html":"We consider self-adjoint extensions of the minimal operator generated by the differential expression <span class=\"etd-inline-math\">\\cL = - d<sup>2</sup>/dx<sup>2</sup>+V</span> on the half-line $[0,\\infty)$, where $V$ is a real-valued function integrable with respect to the weight $1+x$. The self-adjoint extensions are of Schr\\&quot;odinger-type and form a one-parameter family formally given by <span class=\"etd-inline-math\">H<sub>&alpha;</sub>=-d<sup>2</sup>/dx<sup>2</sup>+V</span>, <span class=\"etd-inline-math\">&alpha;\\in[0,&pi;)</span>, with the boundary condition <span class=\"etd-inline-math\">\\sin(&alpha;)f&#x27;(0) = \\cos(&alpha;)f(0)</span> at $x=0$. We derive a formula that relates the resolvent operator of <span class=\"etd-inline-math\">H<sub>&alpha;</sub></span> to the resolvent operator of <span class=\"etd-inline-math\">H<sub>0</sub></span> in terms of the Jost solution corresponding to the underlying differential equation $\\mathcal{L}u=zu$. Combining this resolvent formula and properties of the Jost solution, we compute the trace of the difference of the resolvents of <span class=\"etd-inline-math\">H<sub>&alpha;</sub></span> and the free operator <span class=\"etd-inline-math\">H<sub>&alpha;</sub><sup>(0)</sup></span> with $V\\equiv 0$ in terms of the parameter <span class=\"etd-inline-math\">&alpha;</span> and the Jost function for $\\cL u=zu$.","abstract_has_math":true,"creators":["Sofo, Philip C."],"institution":"University of Tennessee at Chattanooga","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Nichols, Roger A.","Barioli, Francesco; Belinskiy, Boris P.; van der Merwe, Lucas C.","College of Arts and Sciences"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":null,"date_issued":"","date_published":null,"updated_at":"2026-07-24T05:46:51Z","subjects":["Differential operators","Schrödinger operator","Operator theory","Kreĭn spaces","Differential equations"],"languages":["English","eng"],"rights":[],"rights_urls":["https://rightsstatements.org/page/InC/1.0/?language=en"],"identifier_entries":[]},"links":{"outbound_url":"https://scholar.utc.edu/theses/495","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Nichols, Roger A.","Barioli, Francesco; Belinskiy, Boris P.; van der Merwe, Lucas C.","College of Arts and Sciences"]},{"key":"dc:creator","label":"Author","values":["Sofo, Philip C."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-05-01T07:00:00Z"]},{"key":"dc:publisher","label":"Institution","values":["University of Tennessee at Chattanooga","Chattanooga (Tenn.)"]},{"key":"dc:relation","label":"Dc Relation","values":["Masters Theses and Doctoral Dissertations"]},{"key":"dc:type","label":"Dc Type","values":["Masters theses","Text"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Differential operators","Schrödinger operator","Operator theory","Kreĭn spaces","Differential equations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://rightsstatements.org/page/InC/1.0/?language=en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholar.utc.edu/theses/495"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Dept. of Mathematics","M. S.; A thesis submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Master of Science."]},{"key":"dc:description.abstract","label":"Abstract","values":["We consider self-adjoint extensions of the minimal operator generated by the differential expression $\\cL = - d^2/dx^2+V$ on the half-line $[0,\\infty)$, where $V$ is a real-valued function integrable with respect to the weight $1+x$. The self-adjoint extensions are of Schr\\\"odinger-type and form a one-parameter family formally given by $H_{\\alpha}=-d^2/dx^2+V$, $\\alpha\\in[0,\\pi)$, with the boundary condition $\\sin(\\alpha)f'(0) = \\cos(\\alpha)f(0)$ at $x=0$. We derive a formula that relates the resolvent operator of $H_{\\alpha}$ to the resolvent operator of $H_0$ in terms of the Jost solution corresponding to the underlying differential equation $\\mathcal{L}u=zu$. Combining this resolvent formula and properties of the Jost solution, we compute the trace of the difference of the resolvents of $H_{\\alpha}$ and the free operator $H_{\\alpha}^{(0)}$ with $V\\equiv 0$ in terms of the parameter $\\alpha$ and the Jost function for $\\cL u=zu$."]},{"key":"dc:title","label":"Title","values":["Krein's identity and trace formulas for half-line Schrodinger operators"]}]}],"canonical_facts":{"dc:contributor":["Nichols, Roger A.","Barioli, Francesco; Belinskiy, Boris P.; van der Merwe, Lucas C.","College of Arts and Sciences"],"dc:creator":["Sofo, Philip C."],"dc:date":["2017-05-01T07:00:00Z"],"dc:description":["Dept. of Mathematics","M. S.; A thesis submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Master of Science."],"dc:description.abstract":["We consider self-adjoint extensions of the minimal operator generated by the differential expression $\\cL = - d^2/dx^2+V$ on the half-line $[0,\\infty)$, where $V$ is a real-valued function integrable with respect to the weight $1+x$. The self-adjoint extensions are of Schr\\\"odinger-type and form a one-parameter family formally given by $H_{\\alpha}=-d^2/dx^2+V$, $\\alpha\\in[0,\\pi)$, with the boundary condition $\\sin(\\alpha)f'(0) = \\cos(\\alpha)f(0)$ at $x=0$. We derive a formula that relates the resolvent operator of $H_{\\alpha}$ to the resolvent operator of $H_0$ in terms of the Jost solution corresponding to the underlying differential equation $\\mathcal{L}u=zu$. Combining this resolvent formula and properties of the Jost solution, we compute the trace of the difference of the resolvents of $H_{\\alpha}$ and the free operator $H_{\\alpha}^{(0)}$ with $V\\equiv 0$ in terms of the parameter $\\alpha$ and the Jost function for $\\cL u=zu$."],"dc:identifier":["https://scholar.utc.edu/theses/495"],"dc:language":["English","eng"],"dc:publisher":["University of Tennessee at Chattanooga","Chattanooga (Tenn.)"],"dc:relation":["Masters Theses and Doctoral Dissertations"],"dc:rights":["https://rightsstatements.org/page/InC/1.0/?language=en"],"dc:subject":["Differential operators","Schrödinger operator","Operator theory","Kreĭn spaces","Differential equations"],"dc:title":["Krein's identity and trace formulas for half-line Schrodinger operators"],"dc:type":["Masters theses","Text"]},"updated_at":"2026-07-24T05:46:51Z"}