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Sala de Conferințe a FacultățiiAbstract: This paper is concerned with the numerical approximation of the L² Dirichlet eigenpairs of the operator −Δ + V on a simply connected C² bounded domain Ω ⊂ ℝ² 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 C¹ radial potentials is challenging due to the lack of explicit basis functions analogous to Bessel functions. To overcome this difficulty, we consider the equation −Δu + Vu = λu on a ball containing Ω, 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 −Δu + Vu = λu on the ball and are independent of the domain Ω. Dirichlet eigenvalues of −Δ + V are then approximated by minimising the boundary values on ∂Ω among linear combinations of the basis functions and identifying those values of λ for which the computed minimum is sufficiently small. The proposed method is highly memory-efficient compared to the standard Finite Element approach.
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