Energy spectra and scattering state solutions of a particle confined in a Langevin-type function

In this paper, we investigated the bound and scattering state solutions of the Schrödinger equation under the modified Langevin function. Using the functional analysis method, analytical expressions for the energy spectra, scattering phase shift and total cross section were obtained. Other approximations, including the WKB and perturbation methods with the Greene-Aldrich approximation scheme, are employed to validate the solutions. The resulting energy spectra are in good agreement with the numerical results of the Matrix-Numerov method. The scattering state solution reveals that the phase shift increases with angular momentum quantum number. The results show that the centrifugal barrier and the attractive nature of the potential function significantly influence both phase shift and the energy spectra. Notably, the total cross-section displays resonant features and the Ramsauer-Townsend effect. The variations in the total cross sections give rise to densely packed and short-lived sparse resonances, as well as a third region where both resonance and the Ramsauer-Townsend effects are suppressed, resulting in a saturated pulsating scattering pattern.