Shape-invariant Potentials and Singular Spaces

TL;DR

This paper investigates two-dimensional dilaton-gravity brane-worlds with singular backgrounds within the Mann-Morsink-Sikkema-Steele (MMSS) framework. Using a first-order superpotential formalism, it constructs 2D analogues of Gremm’s thick brane and analyzes linear scalar perturbations, revealing that canonical matter yields a singular Poschl-Teller II stability potential with no bound states, whereas a non-canonical twinlike model produces a Poschl-Teller I potential with an infinite tower of bound states. A second analytic solution, inspired by holographic GPPZ flow, leads to an Eckart potential in the perturbation equation, which also has no bound states. The results demonstrate exact solvability of the perturbation spectra and highlight the role of background singularities and non-canonical dynamics in shaping the discrete bound-state structure, with potential holographic interpretations of the dual IR sector.

Abstract

In this work, we present two brane-world-type solutions in a two-dimensional (2D) dilaton gravity model with singular space-time backgrounds. By employing a first-order superpotential formalism, we first construct the 2D analogues of the thick brane solution previously given by Gremm and analyze the corresponding linear scalar perturbations. We show that for a model with canonical scalar matter fields, the effective potential of the linear perturbation equation is a singular Pöschl--Teller~II type, which does not admit bound states. However, for a model with non-canonical scalar fields, the effective potential becomes an exactly solvable Pöschl--Teller~I potential, which has an infinite tower of normalizable bound states. We also present a second analytic solution inspired by the work of Girardello \emph{et al.}, but with non-canonical scalar field. In this case, the linear perturbation equation is a Schrödinger equation with the Eckart potential, which is also exactly solvable.

Figures (3)

• Figure 1: Plots of the effective potentials $V(z)$ for scalar perturbations in the 2D canonical model with $n=1,2$.
• Figure 2: Plots of the warp factor $A(r;B,1)$, the scalar curvature $R(r;B,1)$, the effective potential $V_{+}(r;B,1)$, its partner potential $V_{-}(r;B,1)$, the wave functions of $V_{+}(r;1,1)$ and $V_{-}(r;1,1)$, respectively. The blue solid line at the bottom of the last panel is the infinite square well potential $V_{-}(r;1,1)$.
• Figure 3: Plots of the warp factor $A(r;{{c}})$, the scalar curvature $R(r;{{c}})$, the effective potential $V_{+}(r;{{c}})$ and its partner potential $V_{-}(r;{{c}})$.