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

Abstract

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 R d for any dimension d. We start from the investigation that the radial component of the eigenfunctions, corresponding to spherical harmonics of degree n, of the Schrödinger operator −∆u + c 2 r 2 u can be expressed by Bessel functions of fractional orders αn = c 2 + (n + d/2 − 1) 2 together with the multiplier r 1− d 2. 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 {αn = α + n + d/2 − 1} is also proposed for the Schrödinger source problem with a singular solution of type r α. 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