Constructions of the soluble potentials for the non-relativistic quantum system by means of the Heun functions

Abstract

The Schrödinger equation $\psi "(x)+{\kappa }^2\psi (x)=0$ where ${\kappa }^2=k^2-V(x)$ is rewritten as a more popular form of a second order differential equation through taking a similarity transformation $\psi (z)=\varphi (z)u(z)$ with $z=z(x)$. The Schrödinger invariant $I_S(x)$ can be calculated directly by the Schwarzian derivative $\{z,x\}$ and the invariant $I(z)$ of the differential equation $u_{zz}+f(z)u_z+g(z)u=0$. We find an important relation for moving particle as ${\nabla }^2=-I_S(x)$ and thus explain the reason why the Schrödinger invariant $I_S(x)$ keeps constant. As an illustration, we take the typical Heun differential equation as an object to construct a class of soluble potentials and generalize the previous results through choosing different $\rho =z^{\prime }(x)$ as before. We get a more general solution $z(x)$ through integrating $(z^{\prime }{)}^2={\alpha }_1{z}^2+{\beta }_{1}z+{\gamma }_1$ directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail.

11 pages