Renormalization of Schrödinger equation for potentials with inverse-square singularities: Generalized Trigonometric Pöschl-Teller model
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912681977446400 |
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| author | da Silva, U. Camara |
| author_facet | da Silva, U. Camara |
| contents | We introduce a renormalization procedure necessary for the complete description of the energy spectra of a one-dimensional stationary Schrödinger equation with a potential that exhibits inverse-square singularities. We apply and extend the methods introduced in our recent paper on the hyperbolic Pöschl-Teller potential (with a single singularity) to its trigonometric version. This potential, defined between two singularities, is analyzed across the entire bidimensional coupling space. The fact that the trigonometric Pöschl-Teller potential is supersymmetric and shape-invariant simplifies the analysis and eliminates the need for self-adjoint extensions in certain coupling regions. However, if at least one coupling is strongly attractive, the renormalization is essential to construct a discrete energy spectrum family of one or two parameters. We also investigate the features of a singular symmetric double well obtained by extending the range of the trigonometric Pöschl-Teller potential. It has a non-degenerate energy spectrum and eigenstates with well-defined parity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Renormalization of Schrödinger equation for potentials with inverse-square singularities: Generalized Trigonometric Pöschl-Teller model da Silva, U. Camara Quantum Physics High Energy Physics - Theory Mathematical Physics We introduce a renormalization procedure necessary for the complete description of the energy spectra of a one-dimensional stationary Schrödinger equation with a potential that exhibits inverse-square singularities. We apply and extend the methods introduced in our recent paper on the hyperbolic Pöschl-Teller potential (with a single singularity) to its trigonometric version. This potential, defined between two singularities, is analyzed across the entire bidimensional coupling space. The fact that the trigonometric Pöschl-Teller potential is supersymmetric and shape-invariant simplifies the analysis and eliminates the need for self-adjoint extensions in certain coupling regions. However, if at least one coupling is strongly attractive, the renormalization is essential to construct a discrete energy spectrum family of one or two parameters. We also investigate the features of a singular symmetric double well obtained by extending the range of the trigonometric Pöschl-Teller potential. It has a non-degenerate energy spectrum and eigenstates with well-defined parity. |
| title | Renormalization of Schrödinger equation for potentials with inverse-square singularities: Generalized Trigonometric Pöschl-Teller model |
| topic | Quantum Physics High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2503.12715 |