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arXiv · 2212.09183

Schr\"odinger equation for two quasi-exactly solvable potentials

Bartolomeu D B Figueiredo
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Abstract

We apply solutions of Heun's general equation to the stationary Schr\"odinger equation with two quasi-exactly solvable elliptic potentials which depend on a real parameter $\ell$. We get finite-series solutions from power series expansions for Heun's equation if $\ell$ is an integer, except if $\ell=-1,-2,-3,-4$. If $\ell\neq-5/2$ 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 $\ell$, we find infinite-series eigenfunctions which are convergent and bounded for all values of the independent variable.

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Bartolomeu D B Figueiredo. 2022-12-18. Schr\"odinger equation for two quasi-exactly solvable potentials. https://arxiv.org/abs/2212.09183

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