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The Darboux transformation and algebraic deformations of shape-invariant potentials

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dc.contributor.authorGómez-Ullate Otaiza, David
dc.contributor.authorKamran, Niky
dc.contributor.authorMilson, Robert
dc.date.accessioned2023-06-20T10:55:38Z
dc.date.available2023-06-20T10:55:38Z
dc.date.issued2004-02-06
dc.description©Iop science. The research of DGU is supported in part by a CRM-ISM Postdoctoral Fellowship and the Spanish Ministry of Education under grant EX2002-0176. The research of NK and RM is supported by the National Science and Engineering Research Council of Canada. The authors would like to thank Prof. González-López and Prof. Gesztesy for interesting discussions, as well as the referees, who made very interesting remarks on the first version of the paper
dc.description.abstractWe investigate the backward Darboux transformations (addition of the lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simple elementary functions. A countable family, m = 0, 1,.2,..., of deformations exists for each family of shape-invariant potentials. We prove that the m_th deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules P_(m)^(m) Ϲ P_(-m+1)^(m) Ϲ (...) , where P_n^(m) is a codimension m subspace of <1, z,..., z_(n)>. In particular, we prove that the first (m = 1) algebraic deformation of the shape-invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules P_n^(1) = <1, z_(2),..., z_(n)>. By construction, these algebraically deformed Hamiltonians do not have an sl(2) hidden symmetry algebra structure.
dc.description.departmentDepto. de Física Teórica
dc.description.facultyFac. de Ciencias Físicas
dc.description.refereedTRUE
dc.description.sponsorshipSpanish Ministry of Education
dc.description.sponsorshipNational Science and Engineering Research Council of Canada
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/30915
dc.identifier.doi10.1088/0305-4470/37/5/022
dc.identifier.issn0305-4470
dc.identifier.officialurlhttp://dx.doi.org/10.1088/0305-4470/37/5/022
dc.identifier.relatedurlhttp://iopscience.iop.org
dc.identifier.relatedurlhttp://arxiv.org/abs/quant-ph/0308062
dc.identifier.urihttps://hdl.handle.net/20.500.14352/51463
dc.issue.number5
dc.journal.titleJournal of physics A: Mathematical and general
dc.language.isoeng
dc.page.final1804
dc.page.initial1789
dc.publisherIop science
dc.relation.projectIDCRM-ISM Postdoctoral Fellowship
dc.relation.projectIDEX2002-0176
dc.rights.accessRightsopen access
dc.subject.cdu51-73
dc.subject.keywordDifferential-operators
dc.subject.keywordSchrodinger-operators
dc.subject.keywordFactorization method
dc.subject.keywordQuantum-mechanics
dc.subject.keywordSupersymmetry
dc.subject.keywordEquation
dc.subject.ucmFísica-Modelos matemáticos
dc.subject.ucmFísica matemática
dc.titleThe Darboux transformation and algebraic deformations of shape-invariant potentials
dc.typejournal article
dc.volume.number37
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