Treatment of $N$-dimensional Schr\"{o}dinger equation for anharmonic potential via Laplace transform

TL;DR

This paper applies the Laplace transform method to solve the $N$-dimensional Schrödinger equation with an anharmonic potential, deriving eigenfunctions and energy levels efficiently and confirming results with existing methods.

Contribution

It introduces a novel application of Laplace transform to the $N$-dimensional anharmonic Schrödinger equation, providing a straightforward derivation of eigenvalues and eigenfunctions.

Findings

Eigenvalues and eigenfunctions derived using Laplace transform

Results consistent with previous methods

Simplified approach to solving anharmonic potentials

Abstract

First time anharmonic potential $V(r)=ar^2+br-\frac{c}{r} \,,(a>0) $ is examined for $N$-dimensional Schr\"{o}dinger equation via Laplace transformation method. In transformed space, the behavior of the Laplace transform at the singular point of the differential equation is used to study the eigenfunctions and the energy eigenvalues.The results are easy to derive and identical with those obtained by other methods.