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

Abstract: The Schr\"{o}dinger 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 $\psi(z)=\phi(z)u(z)$ with $z=z(x)$. The Schr\"{o}dinger 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 Schr\"{o}dinger 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 $\rho=z'(x)$ as before. We get a more general solution $z(x)$ through integrating $(z')^2=\alpha_{1}z^2+\beta_{1}z+\gamma_{1}$ directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail.