Construction of Exactly Solvable Ring-Shaped Potentials

Abstract

<p style="text-align:justify;"> <span style="font-family:Verdana;">We propose a method for construction of exactly solvable ring-shaped potentials where</span><span style="font-family:Verdana;"> the</span><span style="font-family:Verdana;"> linear homogeneous second</span><span style="font-family:Verdana;">-</span><span style="font-family:Verdana;">order differential equation satisfied by special function is subjected to</span><span style="font-family:Verdana;"> the</span><span style="font-family:Verdana;"> extended transformation comprising a coordinate transformation and a functional transformation to retrieve</span><span style="font-family:Verdana;"> the</span><span style="font-family:Verdana;"> standard Schr?dinger polar angle equation form in non-relativistic quantum mechanics. By invoking plausible ansatze, exactly solvable ring-shaped potentials and corresponding angular wave functions are constructed. The method is illustrated using Jacobi and hypergeometric polynomials and the wave functions for the constructed ring-shaped potentials are normalized.</span> </p

Systematic search of exactly solvable ring-shaped potential using the transformation method

“…where ρ = − a + 1 2 − i b. The transformation function g(θ) from equation ( 13) and the Scarf potential from equation ( 24) are taken and η = 1 (only for mathematical simplifications) is used in identity (16); the regenerated ring-shaped potential [46] is found to be…”

“…Choosing η = 1 for mathematical simplification, and using equations ( 13) and ( 45) in ( 16) and comparing both sides, the regenerated ring-shaped potential is found to be [46]…”