Schr\"odinger spectrum generated by the Cornell potential

TL;DR

This paper analyzes the eigenvalues of the d-dimensional Schrödinger equation with the Cornell potential using the envelope method and asymptotic iteration method, providing analytic bounds and accurate numerical results.

Contribution

It introduces the combined use of the envelope method and AIM to analyze the Cornell potential eigenvalues, offering new bounds and efficient computations.

Findings

Envelope method provides analytic bounds for eigenvalues.

AIM yields highly accurate numerical eigenvalues.

Scaling reduces the problem to a single parameter.

Abstract

The eigenvalues $E_{n\ell}^d(a,c)$ of the $d$-dimensional Schr\"odinger equation with the Cornell potential $V(r)=-a/r+c\,r$, $a,c>0$ are analyzed by means of the envelope method and the asymptotic iteration method (AIM). Scaling arguments show that it is sufficient to know $E(1,\lambda)$, and the envelope method provides analytic bounds for the equivalent complete set of coupling functions $\lambda(E)$. Meanwhile the easily-implemented AIM procedure yields highly accurate numerical eigenvalues with little computational effort.