Discrete spectra for confined and unconfined -a/r + b r^2 potentials in d dimensions

TL;DR

This paper derives exact and approximate solutions for the Schrödinger equation with Coulomb plus harmonic oscillator potentials in multiple dimensions, considering both unconfined and confined cases, using the asymptotic iteration method.

Contribution

It provides new exact analytic solutions and parametric energy dependencies for a class of d-dimensional potentials, including confinement effects, using the asymptotic iteration method.

Findings

Exact solutions for unconfined potentials in all space.

Approximate solutions for confined potentials inside spherical boxes.

Explicit dependence of eigenenergies on potential parameters and confinement radius.

Abstract

Exact solutions to the d-dimensional Schroedinger equation, d\geq 2, for Coulomb plus harmonic oscillator potentials V(r)=-a/r+br^2, b>0 and a\ne 0 are obtained. The potential V(r) is considered both in all space, and under the condition of spherical confinement inside an impenetrable spherical box of radius R. With the aid of the asymptotic iteration method, the exact analytic solutions under certain constraints, and general approximate solutions, are obtained. These exhibit the parametric dependence of the eigenenergies on a, b, and R. The wave functions have the simple form of a product of a power function, an exponential function, and a polynomial. In order to achieve our results the question of determining the polynomial solutions of the second-order differential equation (\sum_{i=0}^{k+2}a_{k+2,i}r^{k+2-i})y"+(\sum_{i=0}^{k+1}a_{k+1,i}r^{k+1-i})y'-(\sum_{i=0}^{k}\tau_{k,i}r^{k-i})y=0 for k=0,1,2 is solved.