On the quasi - exact solvability of a singular potential in D - dimensions; confined and unconfined

TL;DR

This paper derives quasi-exact solutions for a singular anharmonic potential in D dimensions, including confined cases, clarifying the nature of excited states and extending previous ground-state results.

Contribution

It provides new quasi-exact solutions for both ground and first excited states of a singular potential in D dimensions, including confined geometries, expanding the understanding of such systems.

Findings

Quasi-exact solutions for ground and first excited states are obtained.

Confined potential solutions are derived for cylindrical and spherical boundaries.

The first excited state solution by Dong and Ma is shown to be exotic.

Abstract

The D -dimensional quasi - exact solutions for the singular even - power anharmonic potential $V(q)=aq^2+bq^{-4}+cq^{-6}$ are reported. We show that whilst Dong and Ma's [5] quasi - exact ground - state solution (in D=2) is beyond doubt, their solution for the first excited state is exotic. Quasi - exact solutions for the ground and first excited states are also given for the above potential confined to an impenetrable cylindrical (D=2) or spherical (D=3) wall.