Schrödinger equation for two quasi-exactly solvable potentials

We apply solutions of Heun’s general equation to the stationary Schrödinger equation with two quasi-exactly solvable elliptic potentials which depend on a real parameter $\mathrm{\ell }$. We get finite-series solutions from power series expansions for Heun’s equation if $\mathrm{\ell }$ is an integer, except if $\mathrm{\ell }=-1,-2,-3,-4$. If $\mathrm{\ell }\mathrm{\ne }-5\mathrm{/}2$ is half an odd integer, we obtain finite series in terms of hypergeometric functions. The quasi-exact solvability is expressed by the finite series solutions. However, for any value of $\mathrm{\ell }$, we find infinite-series eigenfunctions which are convergent and bounded for all values of the independent variable.