Exact and approximate bound state solutions of the Schrödinger equation with a class of Kratzer-type potentials in the global monopole spacetime

TL;DR

The authors address bound-state spectra of a non-relativistic charged particle in the global monopole spacetime under Kratzer-type potentials, incorporating a topology-induced self-interaction. They solve the radial Schrödinger equation in two cases: the standard Kratzer potential and a screened modified Kratzer potential, employing the Frobenius method to reduce the problem to confluent and Gauss hypergeometric equations and to obtain explicit eigenfunctions. The resulting energies depend on the monopole parameter $\alpha$ and, in the screened case, on the screening parameter $\delta$, with bound states existing under well-defined quantization conditions. The work highlights how spacetime topology and short-range screening modify molecular-potential-like bound states, offering insights for analogue gravity and topological defect physics in quantum systems.

Abstract

This work investigates the motion of a non-relativistic charged particle within the spacetime of a global monopole. We introduce the Schrödinger equation to describe the particle's motion with two interactions by considering the Kratzer and the screened modified Kratzer potential. The problem's eigenfunctions and eigenvalues are obtained by deriving and solving the radial equation. The effective potential encompasses both the Kratzer and electrostatic self-interaction potential and leads to bound states solutions. The energy spectrum is investigated, particularly emphasizing its dependence on the system's physical parameters. The screened modified Kratzer potential and the screened self-interaction potential reveal an important role in influencing both the effective potential and the energy spectrum. Additionally, it also accommodates the existence of bound states. All these behaviors are illustrated with graphs and discussed in detail.

Figures (4)

• Figure 1: Energy levels as a function of the monopole parameter $\alpha$ for different values of $l$. In (a), for $n=0$, (b) for $n=1$, and (c) $n=2$. In both cases, we using $A=0.5$, $D=1$$q=1$, $\hbar=1$ and $M=1$
• Figure 2: Effective potential (\ref{['pt']}) as a function of $r$. The solid line is for $\delta=0$ (which leads to the effective potential (\ref{['veffND']})), the dashed line is for $\delta=0.1$, and the dotted line is for $\delta=0.2$. In (a), we set $l=2$ and use different values of $\alpha$. In (b), we set $\alpha=0.5$ and use different values of $l$. In both cases, we are using $A=2$ and $D=4$.
• Figure 3: Energy levels as a function of the monopole parameter $\alpha$ for different values of $l$. In (a), for $n=0$, (b) for $n=1$, and (c) $n=2$. In both cases, we using $A=2$, $D=4$, $q=1$, $\hbar=1$, $M=1$ and $\delta=0.001$.
• Figure 4: Sketch of $\mathcal{E}_{01}$ as a function of the monopole parameter $\alpha$ for different values of $\delta$. We use $A=2$, $D=4$, $q=1$, $\hbar=1$ and $M=1$.