Mesh-free numerical method for Dirichlet eigenpairs of the Laplacian with potential

This paper is concerned with the numerical approximation of the $L^2$ Dirichlet eigenpairs of the operator $-\mathrm{\Delta }+V$ on a simply connected $C^2$ bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ containing the origin, where $V$ is a radial potential. We propose a mesh-free method inspired by the Method of Particular Solutions for the Laplacian (i.e. $V=0$). Extending this approach to general $C^1$ radial potentials is challenging due to the lack of explicit basis functions analogous to Bessel functions. To overcome this difficulty, we consider the equation $-\mathrm{\Delta }u+Vu=\lambda u$ on a ball containing $\mathrm{\Omega }$, without imposing boundary conditions, for a collection of values $\lambda$ forming a fine discretisation of the interval in which eigenvalues are sought. By rewriting the problem in polar coordinates and applying a Fourier expansion with respect to the angular variable, we obtain a decoupled system of ordinary differential equations. These equations are solved numerically using a one-dimensional Finite Element Method, yielding a family of basis functions that are solutions of the equation $-\mathrm{\Delta }u+Vu=\lambda u$ on the ball and are independent of the domain $\mathrm{\Omega }$. Dirichlet eigenvalues of $-\mathrm{\Delta }+V$ are then approximated by minimising the boundary values on $\mathrm{\partial }\mathrm{\Omega }$ among linear combinations of the basis functions and identifying those values of $\lambda$ for which the computed minimum is sufficiently small. The proposed method is highly memory-efficient compared to the standard Finite Element approach.