Constructions of the Soluble Potentials for the Nonrelativistic Quantum System by Means of the Heun Functions

Abstract

The Schrödinger equation ψ′′(x)+κ2ψ(x)=0 where κ2=k2-V(x) is rewritten as a more popular form of a second order differential equation by taking a similarity transformation ψ(z)=ϕ(z)u(z) with z=z(x). The Schrödinger invariant IS(x) can be calculated directly by the Schwarzian derivative z,x and the invariant I(z) of the differential equation uzz+f(z)uz+g(z)u=0. We find an important relation for a moving particle as ∇2=-IS(x) and thus explain the reason why the Schrödinger invariant IS(x) keeps constant. As an illustration, we take the typical Heun’s differential equation as an object to construct a class of soluble potentials and generalize the previous results by taking different transformation ρ=z′(x) as before. We get a more general solution z(x) through integrating (z′)2=α1z2+β1z+γ1 directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail. The results are also compared with those obtained by Bose, Lemieux, Batic, Ishkhanyan, and their coworkers. It should be recognized that a subtle and different choice of the transformation z(x) also related to ρ will lead to difficult connections to the results obtained from other different approaches.