Renormalization of Schr\"odinger equation for potentials with inverse-square singularities: Generalized Trigonometric P\"oschl-Teller model

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.