Exact spectral and scattering solutions of the 1D Schrödinger equation in the Jacobi–Dunkl framework

Abstract

We investigate the one-dimensional nonrelativistic Schrödinger equation within the Jacobi–Dunkl framework. Although the Jacobi–Dunkl operator is well established in harmonic analysis and special-function theory, its systematic use in quantum-mechanical spectral and scattering problems remains comparatively limited. In this work, we develop an explicit parity-resolved formulation in which the reflection-induced deformation is incorporated directly at the operator level. The Jacobi–Dunkl momentum operator, Hamiltonian, weighted scalar product, and parity decomposition are constructed consistently in the associated weighted Hilbert space. For reflection-invariant potentials, the stationary Schrödinger equation is naturally separated into even and odd sectors. The odd sector acquires an additional effective hyperbolic contribution generated by the reflection part of the operator, whereas the even sector is governed by a different reduced equation. The formalism is first applied to the one-dimensional infinite quantum well, for which exact eigenfunctions and parity-dependent quantization conditions are obtained in terms of Jacobi-type hypergeometric functions. We then treat the hyperbolic Pöschl–Teller potential. In the bound-state sector, finite parity-dependent discrete spectra and normalized eigenfunctions expressed through Jacobi polynomials are derived. The corresponding scattering sector is also analyzed by reconstructing the full-line reflection and transmission amplitudes from the even- and odd parity channels. The results show that the Jacobi–Dunkl parameters restructure the spectrum, split the parity branches, modify the bound-state localization, and affect the scattering behavior. The ordinary Schrödinger results are recovered as a formal limiting benchmark, confirming the consistency of the construction.