Exact Solutions to the Schr\"{o}dinger Equation for the Inverse-Power Potential in Two Dimensions

TL;DR

This paper derives exact closed-form solutions for the two-dimensional Schrödinger equation with a complex inverse-power potential, expanding analytical understanding of quantum systems with such potentials.

Contribution

It provides the first exact solutions for the Schrödinger equation with a general inverse-power potential in two dimensions under specific parameter constraints.

Findings

Exact eigenfunctions and eigenvalues derived

Potential parameters constrained for solutions

Analytical framework applicable to similar potentials

Abstract

Utilizing an ${\it ansatz}$ for the eigenfunctions, we arrive at an exact closed form solution to the Schr\"{o}dinger equation with the inverse-power potential, $V(r)=ar^{-4}+br^{-3}+cr^{-2}+dr^{-1}$ in two dimensions, where the parameters of the potential $a, b, c, d$ satisfy a constraint.