{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/37449"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/37449","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"The Born-Oppenheimer approximation in scattering theory","abstract":"We analyze the Schrödinger equation i𝜖 ¬<sup>2</sup>â /â tΨ = H(𝜖)Ψ, where H(â ¬) = - f24 Î x + h(X) is the hamiltonian of a molecular system consisting of nuclei with masses of order 𝜖¬<sup>-4</sup> and electrons with masses of order 1. The Born-Oppenheimer approximation consists of the adiabatic approximation to the motion of electrons and the semiclassical approximation to the time evolution of nuclei. The quantum propagator associated with this Schrödinger Equation is exp(-itH(â ¬)/â ¬<sup>2</sup>). We use the Born-Oppenheimer method to find the leading order asymptotic expansion in â ¬ to exp(_it~(t:»Ψ, i.e., we find Ψ(t) such that: (1) We show that if H(𝜖) describes a diatomic Molecule with smooth short range potentials, then the estimate (1) is uniform in time; hence the leading order approximation to the wave operators can be constructed. We also comment on the generalization of our method to polyatomic molecules and to Coulomb systems.","abstract_html":"We analyze the Schrödinger equation i𝜖 ¬&lt;sup&gt;2&lt;/sup&gt;â /â tΨ = H(𝜖)Ψ, where H(â ¬) = - f24 Î x + h(X) is the hamiltonian of a molecular system consisting of nuclei with masses of order 𝜖¬&lt;sup&gt;-4&lt;/sup&gt; and electrons with masses of order 1. The Born-Oppenheimer approximation consists of the adiabatic approximation to the motion of electrons and the semiclassical approximation to the time evolution of nuclei. The quantum propagator associated with this Schrödinger Equation is exp(-itH(â ¬)/â ¬&lt;sup&gt;2&lt;/sup&gt;). We use the Born-Oppenheimer method to find the leading order asymptotic expansion in â ¬ to exp(_it~(t:»Ψ, i.e., we find Ψ(t) such that: (1) We show that if H(𝜖) describes a diatomic Molecule with smooth short range potentials, then the estimate (1) is uniform in time; hence the leading order approximation to the wave operators can be constructed. We also comment on the generalization of our method to polyatomic molecules and to Coulomb systems.","abstract_has_math":false,"creators":["Kargol, Armin"],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematical Physics","degree_department":"Mathematical Physics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Hagedorn, George A."],"committee_members":["Bowden, Robert L.","Klaus, Martin","Slawny, Joseph","Zweifel, Paul F."],"year":1994,"date_issued":"1994-05-05","date_published":"1994-05-05","updated_at":"2026-07-22T22:19:16Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-03022006-093405"],"render_values":[{"text":"etd-03022006-093405","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/37449","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Hagedorn, George A."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Bowden, Robert L.","Klaus, Martin","Slawny, Joseph","Zweifel, Paul F."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematical Physics"]},{"key":"dc:creator","label":"Author","values":["Kargol, Armin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:09:54Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:09:54Z","2006-03-02"]},{"key":"dc:date.issued","label":"Date","values":["1994-05-05"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"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":["Mathematical Physics"]},{"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"]},{"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":["etd-03022006-093405"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/37449"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We analyze the Schrödinger equation i𝜖 ¬<sup>2</sup>â /â tΨ = H(𝜖)Ψ, where H(â ¬) = - f24 Î x + h(X) is the hamiltonian of a molecular system consisting of nuclei with masses of order 𝜖¬<sup>-4</sup> and electrons with masses of order 1. The Born-Oppenheimer approximation consists of the adiabatic approximation to the motion of electrons and the semiclassical approximation to the time evolution of nuclei. The quantum propagator associated with this Schrödinger Equation is exp(-itH(â ¬)/â ¬<sup>2</sup>). We use the Born-Oppenheimer method to find the leading order asymptotic expansion in â ¬ to exp(_it~(t:»Ψ, i.e., we find Ψ(t) such that: (1) We show that if H(𝜖) describes a diatomic Molecule with smooth short range potentials, then the estimate (1) is uniform in time; hence the leading order approximation to the wave operators can be constructed. We also comment on the generalization of our method to polyatomic molecules and to Coulomb systems."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["The Born-Oppenheimer approximation in scattering theory"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Hagedorn, George A."],"dc:contributor.committeemember":["Bowden, Robert L.","Klaus, Martin","Slawny, Joseph","Zweifel, Paul F."],"dc:contributor.department":["Mathematical Physics"],"dc:creator":["Kargol, Armin"],"dc:date.accessioned":["2014-03-14T21:09:54Z"],"dc:date.available":["2014-03-14T21:09:54Z","2006-03-02"],"dc:date.issued":["1994-05-05"],"dc:description.abstract":["We analyze the Schrödinger equation i𝜖 ¬<sup>2</sup>â /â tΨ = H(𝜖)Ψ, where H(â ¬) = - f24 Î x + h(X) is the hamiltonian of a molecular system consisting of nuclei with masses of order 𝜖¬<sup>-4</sup> and electrons with masses of order 1. The Born-Oppenheimer approximation consists of the adiabatic approximation to the motion of electrons and the semiclassical approximation to the time evolution of nuclei. The quantum propagator associated with this Schrödinger Equation is exp(-itH(â ¬)/â ¬<sup>2</sup>). We use the Born-Oppenheimer method to find the leading order asymptotic expansion in â ¬ to exp(_it~(t:»Ψ, i.e., we find Ψ(t) such that: (1) We show that if H(𝜖) describes a diatomic Molecule with smooth short range potentials, then the estimate (1) is uniform in time; hence the leading order approximation to the wave operators can be constructed. We also comment on the generalization of our method to polyatomic molecules and to Coulomb systems."],"dc:description.degree":["Ph. D."],"dc:format.medium":["BTD"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["etd-03022006-093405"],"dc:identifier.uri":["http://hdl.handle.net/10919/37449"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["The Born-Oppenheimer approximation in scattering theory"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematical Physics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:16Z"}