Supersymmetric partners of the one-dimensional infinite square well Hamiltonian
Entity
UAM. Departamento de Física Teórica de la Materia CondensadaPublisher
MDPIDate
2021-02-21Citation
10.3390/sym13020350
Symmetry-Basel 13.2 (2021): 350
ISSN
2073-8994 (online)DOI
10.3390/sym13020350Editor's Version
https://doi.org/10.3390/sym13020350Subjects
Point Interactions; Resolvent; Dirac Operator; FísicaRights
© 2021 by the authorsAbstract
We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator −d 2 / dx2 on L2 [−a, a], a > 0, that is, the one dimensional infinite square well. First of all, we classify these self-adjoint extensions in terms of several choices of the parameters determining each of the extensions. There are essentially two big groups of extensions. In one, the ground state has strictly positive energy. On the other, either the ground state has zero or negative energy. In the present paper, we show that each of the extensions belonging to the first group (energy of ground state strictly positive) has an infinite sequence of supersymmetric partners, such that the l-th order partner differs in one energy level from both the (l − 1)-th and the ( l + 1)-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of −d2 / dx2 come from a transcendental equation and are all infinite. For the case under our study, we determine the eigenvalues, which are also infinite, all the extensions have a purely discrete spectrum, and their respective eigenfunctions for all of its l-th supersymmetric partners of each extension
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Google Scholar:Gadella, Manuel
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Hernández Muñoz, José
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Nieto, Luis Miguel
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San Millán, Carlos
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