A Laplace transform approach to find the exact solution of the N-dimensional Schr\"odinger equation with Mie-type potentials and construction of Ladder operators

TL;DR

This paper presents a method using Laplace transforms to solve the N-dimensional Schrödinger equation with Mie-type potentials exactly, constructs ladder operators, and explores their Lie algebra structure.

Contribution

It introduces a Laplace transform approach for exact solutions of N-dimensional Schrödinger equations with Mie-type potentials and constructs associated ladder operators with Lie algebra analysis.

Findings

Exact bound state solutions obtained

Ladder operators constructed and analyzed

Lie algebra identified as SU(1,1)

Abstract

The second order N-dimensional Schrodinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation. Exact bound state solutions are obtained using convolution or Faltungs theorem. The Ladder operators are also constructed for the Mie-type potentials in N- dimensions. Lie algebra associated with these operators are studied and it is found that they satisfy the commutation relations for the SU(1,1) group.