Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space

Abstract

<p style='text-indent:20px;'>In this article, we propose and analyze some novel spectral methods for the Schödinger equation (including the associated eigenvalue problem) with an inverse square potential on an arbitrary whole space <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^d$\end{document}</tex-math></inline-formula> for any dimension <inline-formula><tex-math id="M2">\begin{document}$d$\end{document}</tex-math></inline-formula>. We start from the investigation that the radial component of the eigenfunctions, corresponding to spherical harmonics of degree <inline-formula><tex-math id="M3">\begin{document}$n$\end{document}</tex-math></inline-formula>, of the Schrödinger operator <inline-formula><tex-math id="M4">\begin{document}$\displaystyle -Δ u + \frac{c^2}{r^2}u$\end{document}</tex-math></inline-formula> can be expressed by Bessel functions of fractional orders <inline-formula><tex-math id="M5">\begin{document}$α_n = \sqrt{c^2+(n+d/2-1)^2}$\end{document}</tex-math></inline-formula> together with the multiplier <inline-formula><tex-math id="M6">\begin{document}$r^{1-\frac{d}{2}}$\end{document}</tex-math></inline-formula>. This knowledge helps us to construct the Müntz-Hermite functions as the basis functions to fit the singularities of the eigenfunctions. In return, a novel spectral method is then proposed for solving the Schrödinger eigenvalue problem efficiently. Further, a Galerkin spectral approximation using genuine Hermite functions with a distinct Müntz sequence <inline-formula><tex-math id="M7">\begin{document}$\{α_n = α+n+d/2-1\}$\end{document}</tex-math></inline-formula> is also proposed for the Schrödinger source problem with a singular solution of type <inline-formula><tex-math id="M8">\begin{document}$r^{α}$\end{document}</tex-math></inline-formula>. Optimal error estimates are then established rigorously for both the source and eigenvalue problems. In contrast to classic Hermite spectral methods using tensorial basis functions, our new methods possess an exponential order of convergence for such singular problems while offer a banded structure of the stiffness and mass matrices. Finally, numerical experiments illustrate the efficiency and spectral accuracy of our new methods.