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

TL;DR

This work develops a two-singularity renormalization framework for the one-dimensional Schrödinger equation with the trigonometric Pöschl-Teller potential, extending prior renormalization results from the hyperbolic case. By regularizing each inverse-square singularity, the authors introduce independent scales that induce dimensional transmutation and break asymptotic conformal symmetry, yielding one- or two-parameter families of discrete spectra across the full coupling space. Supersymmetry and shape invariance fix boundary data in the region $g_s,g_c\ge-1/4$, while strong-attractive regions require renormalization to obtain a well-defined spectrum, including Efimov-like bound states. The results have implications for AdS fluctuations and holography, and the methodology provides a generalizable alternative to self-adjoint-extension approaches for potentials with multiple inverse-square singularities.

Abstract

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.

Figures (12)

• Figure 1: Possible forms of the potential (\ref{['V_PT_trig']}): (a) $g_s,g_c>0$ (b) $g_s,g_c<0$; (c) $g_s<0$ and $g_c>0$.
• Figure 2: Coupling space diagram. The white line $g_s=g_c$ separates the two symmetric half-planes. Red region: strong-repulsive and strong-repulsive. Brown region: strong-repulsive and weak-medium. Blue region: strong-repulsive and strong-attractive. Pink region: weak-medium and weak-medium. Yellow region: weak-medium and strong-attractive. Green region: strong-repulsive and strong-repulsive.
• Figure 3: Updated coupling space diagram after the use of supersymmetry. Gray region: boundary conditions are well-defined. In the other sectors, renormalization is necessary.
• Figure 4: Curves of $\Theta_1(k)\in[-\pi,\pi)$, equation (\ref{['g_1']}), for $|\nu_s|=2\pi$ and $\nu_{c}=3.5$. (a) Positive energies ($k>0$); (b) Negative energies ($k\rightarrow i\kappa$). In both cases, the spectrum is the black points where the curves touch the line $\theta_{\nu_s}$, chosen here as $\pi/4$.
• Figure 5: