# Exact normalized eigenfunctions for general deformed Hulth\'en potentials

## TL;DR

This paper derives exact eigenfunctions and normalization constants for the deformed Hulthén potential in quantum mechanics, and extends these solutions using Crum-Darboux transformations to more complex potentials.

## Contribution

It provides closed-form solutions and normalization formulas for the deformed Hulthén potential and its extensions, advancing analytical methods in quantum potential problems.

## Key findings

- Exact eigenfunctions for the deformed Hulthén potential derived.
- Closed-form normalization constants obtained for arbitrary q.
- Extended potentials solved using Crum-Darboux transformation.

## Abstract

The exact solutions of Schr\"odinger's equation with the deformed Hulth\'en potential $V_q(x)=-{\mu\, e^{-\delta\,x }}/({1-q\,e^{-\delta\,x}}),~ \delta,\mu, q>0$ are given, along with a closed--form formula for the normalization constants of the eigenfunctions for arbitrary $q>0$. The Crum-Darboux transformation is then used to derive the corresponding exact solutions for the extended Hulth\'en potentials $V(x)= -{\mu\, e^{-\delta\,x }}/({1-q\,e^{-\delta\,x}})+ {q\,j(j+1)\, e^{-\delta\,x }}/({1-q\,e^{-\delta\,x}})^2, j=0,1,2,\dots.$ A general formula for the new normalization condition is also provided.