Exact spectral solutions of Pöschl–Teller potentials via trigonometric and hyperbolic Dunkl operators

Abstract

Exact analytical solutions of the Schrödinger equation were derived for the symmetric trigonometric and hyperbolic Pöschl–Teller (PT) potentials using geometry-compatible trigonometric and hyperbolic Dunkl operators. In rational Dunkl operator formulations, inverse-square singularities arise from the geometric mismatch between the deformation kernel and the potential structure, destroying exact solvability and forcing Pekeris-type approximation schemes. By employing geometry-compatible operators, these singularities were eliminated at the operator level and exact solvability was restored without any approximation. The trigonometric and hyperbolic Dunkl operators, although rigorously defined in the mathematical literature, have not been systematically applied to quantum mechanical potential problems; the present work addressed this gap and identified operator–geometry compatibility as a key condition for exact solvability in Dunkl-deformed systems. Within the supersymmetric quantum mechanics framework, closed-form parity-resolved energy spectra and eigenfunctions were obtained in terms of Jacobi polynomials, and shape invariance was established to be preserved exactly under the deformation. The geometry-compatible Hilbert space weights \(|\sin (\alpha x)|^{2\mu }\) and \(|\sinh (\alpha x)|^{2\mu }\) were derived, replacing the rational Dunkl weight \(|x|^{2\mu }\). The corrected SUSY hierarchy exhibits alternating parity at successive steps, and the Jacobi-polynomial eigenfunctions depend explicitly on \(\mu \). The exact Robertson uncertainty lower bound is eigenstate-dependent; its simple near-origin form is only an approximation.