https://doi.org/10.1140/epjc/s10052-025-15023-x
Regular Article - Theoretical Physics
Shape-invariant potentials and singular spaces
1
MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, School of Physics, Xi’an Jiaotong University, 710049, Xi’an, China
2
Department of Physics, Institute for Quantum Science and Technology (IQST), Shanghai University, 200444, Shanghai, China
a
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Received:
19
September
2025
Accepted:
31
October
2025
Published online:
11
November
2025
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 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.
© The Author(s) 2025
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