A New Approach for Higher Order Difference Equations and Eigenvalue problems via Physical Potentials

TL;DR

This paper introduces a novel variation of parameters method using delta exponential functions to solve higher order difference equations and eigenvalue problems, providing closed-form solutions for complex physical models.

Contribution

It presents a new approach to solving difference equations in closed form, applied to eigenvalue problems like Sturm-Liouville and Schrödinger equations with Coulomb potential.

Findings

Closed-form solutions for second-order Sturm-Liouville problems

Sum representation of solutions for Coulomb potential and hydrogen atom

Analytical solutions for fourth-order relaxation difference equations

Abstract

In this study, we give the variation of parameters method from a different viewpoint for the Nth order inhomogeneous linear ordinary difference equations with constant coefficient by means of delta exponential function . Advantage of this new approachment is to enable us to investigate the solution of difference equations in the closed form. Also, the method is supported with three difference eigenvalue problems, the second-order Sturm-Liouville problem, which is called also one dimensional Schr\"odinger equation, having Coulomb potential, hydrogen atom equation, and the fourth-order relaxation difference equations. We find sum representation of solution for the second order discrete Sturm-Liouville problem having Coulomb potential, hydrogen atom equation, and analytical solution of the fourth order discrete relaxation problem by the variation of parameters method via delta exponential and delta trigonometric functions .