Shape-invariant Potentials and Singular Spaces

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 P\"oschl--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 P\"oschl--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 Schr\"odinger equation with the Eckart potential, which is also exactly solvable.