Closed-form solutions for the Schrödinger wave equation with non-solvable potentials: A perturbation approach

Haifa Ibrahim Alrebdi,Thabit Barakat

doi:10.1515/phys-2021-0028

Abstract

Abstract To obtain closed-form solutions for the radial Schrödinger wave equation with non-solvable potential models, we use a simple, easy, and fast perturbation technique within the framework of the asymptotic iteration method (PAIM). We will show how the PAIM can be applied directly to find the analytical coefficients in the perturbation series, without using the base eigenfunctions of the unperturbed problem. As an example, the vector Coulomb ( ∼ 1 / r ) \left( \sim 1\hspace{0.1em}\text{/}\hspace{0.1em}r) and the harmonic oscillator ( ∼ r 2 ) \left( \sim {r}^{2}) plus linear ( ∼ r ) \left( \sim r) scalar potential parts implemented with their counterpart spin-dependent terms are chosen to investigate the meson sectors including charm and beauty quarks. This approach is applicable in the same form to both the ground state and the excited bound states and can be easily applied to other strongly non-solvable potential problems. The procedure of this method and its results will provide a valuable hint for investigating tetraquark configuration.

Highlights

The lack of closed-form solutions to the radial Schrödinger wave equation with many potential models makes the study of such potentials one of the most popular theoretical laboratories for examining the validity of the various approximation techniques based on perturbative and non-perturbative approaches [6,7,8,9,10,11]
PIAM is unlike the other perturbation methods, with no constraints on the coupling constants or the quantum numbers involved in the phenomenological potential, and even it avoids sticking to numerical computations at
We presented the static potential, which is composed of scalar and vector parts, implemented with their counterpart spin-dependent correction terms

Summary

Introduction

The experimental observations of the exotic states in the charm sector with cccc content at Belle [2] and BESIII [3], as well as the structures of the form QqqQ in ref. [4,5], have opened new interest to restudy the hadronic structures and their spectroscopy with different phenomenological potential models. Many of the existing theoretical work in the literature can obtain the hadron masses by solving numerically the two-body radial Schrödinger wave equation with Cornellinspired potentials, and the respective spin-dependent interaction terms are added and treated by the standard perturbation technique [12]. The computational schedule in those works is very complicated and can be hardly extended to study other potential models On this basis, the present work is aimed to solve the radial Schrödinger wave equation for heavy quark–antiquark structures with the vector Coulomb potential plus linear and harmonic scalar potentials parts, implemented in the same form with their counterpart spin-dependent terms. PIAM can yields very accurate and rapidly converging eigenvalues, and it is applicable in the same form to both the ground and excited energy states. We give a summary and concluding remarks for possible future works on tetraquark masses

Theoretical framework

Numerical results for heavy mass spectrum of quark–antiquark structures

Findings

Conclusion