Renormalization of Schrödinger equation for potentials with inverse-square singularities: Generalized Trigonometric Pöschl-Teller model

• URL: http://arxiv.org/abs/2503.12715v1
• Date: Mon, 17 Mar 2025 00:56:28 GMT
• Title: Renormalization of Schrödinger equation for potentials with inverse-square singularities: Generalized Trigonometric Pöschl-Teller model
• Authors: U. Camara da Silva,
• Abstract summary: We introduce a renormalization procedure necessary for the complete description of the energy spectra of a one-dimensional stationary Schr"odinger equation with a potential that exhibits inverse-square singularities.<n>We also investigate the features of a singular symmetric double well obtained by extending the range of the trigonometric P"oschl-Teller potential.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a renormalization procedure necessary for the complete description of the energy spectra of a one-dimensional stationary Schr\"odinger equation with a potential that exhibits inverse-square singularities. We apply and extend the methods introduced in our recent paper on the hyperbolic P\"oschl-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 P\"oschl-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 P\"oschl-Teller potential. It has a non-degenerate energy spectrum and eigenstates with well-defined parity.

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