Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian
We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo>&l...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/2/350 |
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| author | Manuel Gadella José Hernández-Muñoz Luis Miguel Nieto Carlos San Millán |
| author_facet | Manuel Gadella José Hernández-Muñoz Luis Miguel Nieto Carlos San Millán |
| author_sort | Manuel Gadella |
| collection | DOAJ |
| description | We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msup><mi>d</mi><mn>2</mn></msup><mo>/</mo><mi>d</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mrow><mo>[</mo><mo>−</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, 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 <i>ℓ</i>-th order partner differs in one energy level from both the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>ℓ</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-th and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>ℓ</mo><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msup><mi>d</mi><mn>2</mn></msup><mo>/</mo><mi>d</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> 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 <i>ℓ</i>-th supersymmetric partners of each extension. |
| first_indexed | 2024-03-09T00:39:15Z |
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| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T00:39:15Z |
| publishDate | 2021-02-01 |
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| series | Symmetry |
| spelling | doaj.art-442e5d1549984dabb10b07b8ac31c2b32023-12-11T17:54:52ZengMDPI AGSymmetry2073-89942021-02-0113235010.3390/sym13020350Supersymmetric Partners of the One-Dimensional Infinite Square Well HamiltonianManuel Gadella0José Hernández-Muñoz1Luis Miguel Nieto2Carlos San Millán3Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, 47011 Valladolid, SpainDepartamento de Física Teórica de la Materia Condensada, IFIMAC Condensed Matter Physics Center, Universidad Autónoma de Madrid, 28049 Madrid, SpainDepartamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, 47011 Valladolid, SpainDepartamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, 47011 Valladolid, SpainWe find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msup><mi>d</mi><mn>2</mn></msup><mo>/</mo><mi>d</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mrow><mo>[</mo><mo>−</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, 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 <i>ℓ</i>-th order partner differs in one energy level from both the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>ℓ</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-th and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>ℓ</mo><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msup><mi>d</mi><mn>2</mn></msup><mo>/</mo><mi>d</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> 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 <i>ℓ</i>-th supersymmetric partners of each extension.https://www.mdpi.com/2073-8994/13/2/350supersymmetric quantum mechanicsself-adjoint extensionsinfinite square wellcontact potentials |
| spellingShingle | Manuel Gadella José Hernández-Muñoz Luis Miguel Nieto Carlos San Millán Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian Symmetry supersymmetric quantum mechanics self-adjoint extensions infinite square well contact potentials |
| title | Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian |
| title_full | Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian |
| title_fullStr | Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian |
| title_full_unstemmed | Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian |
| title_short | Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian |
| title_sort | supersymmetric partners of the one dimensional infinite square well hamiltonian |
| topic | supersymmetric quantum mechanics self-adjoint extensions infinite square well contact potentials |
| url | https://www.mdpi.com/2073-8994/13/2/350 |
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