Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian
| Autor: | Luis Miguel Nieto, Manuel Gadella, C. San Millán, Jose Hernández-Muñoz |
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| Přispěvatelé: | UAM. Departamento de Física Teórica de la Materia Condensada |
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Physics and Astronomy (miscellaneous) Transcendental equation General Mathematics infinite square well FOS: Physical sciences Dirac Operator 01 natural sciences symbols.namesake Point Interactions Condensed Matter::Superconductivity self-adjoint extensions 0103 physical sciences Computer Science (miscellaneous) 010306 general physics Resolvent Eigenvalues and eigenvectors Mathematical Physics Mathematics supersymmetric quantum mechanics Quantum Physics 010308 nuclear & particles physics Group (mathematics) lcsh:Mathematics Order (ring theory) Física Mathematical Physics (math-ph) Eigenfunction Mathematics::Spectral Theory Differential operator lcsh:QA1-939 Chemistry (miscellaneous) symbols contact potentials Ground state Hamiltonian (quantum mechanics) Quantum Physics (quant-ph) |
| Zdroj: | Symmetry Volume 13 Issue 2 Symmetry, Vol 13, Iss 350, p 350 (2021) |
| DOI: | 10.48550/arxiv.2104.08617 |
| Popis: | We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator $-d^2/dx^2$ on $L^2[-a,a]$, $a>0$, 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 $\ell$-th order partner differs in one energy level from both the $(\ell-1)$-th and the $(\ell+1)$-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of $-d^2/dx^2$ 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 $\ell$-th supersymmetric partners of each extension. 17 pages, 1 table and 5 figures |
| Databáze: | OpenAIRE |
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