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

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1. Introduction

The exact solution of the Schrödinger equation with physical potentials has played an important role in quantum mechanics. This is due to the fact that an exact knowledge of the energy spectrum and the wave functions of the one-dimensional Schrödinger equation turns out to be very useful in many applied problems. Generally speaking, for a given external field, one of our main tasks is to show how to solve the differential equation through choosing suitable variables and then find its exact solutions expressed by some special functions. Here we focus on how to construct a class of the solvable potentials within the framework of the nonrelativistic Schrödinger equation. Up to now, a lot of similar works have been carried out with this stimulation [1–9]. For example, Lemieux and Bose presented an advanced analysis of the variable transformation and have listed 8 potentials [3] (written in French). Batic, Williams, and Nowakowski have discussed the general potential allowing reduction of the Schrödinger equation to the general Heun’s differential equation by an energy-independent transformation. In particular, Theorem 4.1 in [8] gives the most general form of this potential including 15 parameters instead of 14 parameters, which contain 10 parameters ( $g_{\mathrm{0,1},\mathrm{2,3},\mathrm{4}}$ and $f_{\mathrm{0,1},\mathrm{2,3},\mathrm{4}}$ ) and original parameters $\alpha ,\beta ,\gamma ,\delta ,ϵ,q$ involved in the general Heun equation but with a constraint also named as Fuchsian relation (11). In that work [8], they find that their results cover all results obtained by Bose, Lemieux, Natanzon, and Iwata [1–4, 10]. Iwata studied the soluble potentials in terms of the hypergeometric functions [10]. It is not surprising to see this since the special functions considered by them are only the special cases of the general Heun function. Just recently Ishkhanyan and his coauthors worked out a series of significant works by taking $\rho \propto (z-a_{\mathrm{1}}{)}^{m_{\mathrm{1}}}(z-a_{\mathrm{2}}{)}^{m_{\mathrm{2}}}($

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