Explicit Two-Sided Eigenvalue Bounds for Schr\"odinger Operators with Singular Potentials via Finite Element Method

TL;DR

This paper introduces a novel numerical algorithm providing explicit, verified two-sided eigenvalue bounds for Schrödinger operators with singular potentials, applicable to unbounded domains and potentials like Coulomb's, verified through finite element methods.

Contribution

The authors develop the first explicit, verified finite element-based method for two-sided eigenvalue bounds of Schrödinger operators with singular potentials on unbounded domains.

Findings

Algorithm successfully computes bounds for Coulomb potentials.

Numerical experiments confirm convergence and accuracy.

Method applicable to hydrogen atom and molecular ion models.

Abstract

We present, to the best of our knowledge, the first numerical algorithm for explicit, computable two-sided eigenvalue bounds for Schr\"odinger operators H = -Delta + V on R^N, N = 2,3, in the presence of both an unbounded potential and an unbounded domain. "Explicit" here means that all constants and ingredients are derived in closed form from the mesh, the potential, and a small set of explicit inequalities (Payne-Weinberger, Hardy, and explicit bounded-domain Sobolev embeddings); the conversion to fully verified(IEEE-754-safe, interval-arithmetic) enclosures is a separate verification step and is left for future work. In particular, singular attractive potentials of Coulomb type, V(x) = -Z/|x|, which model the hydrogen atom and the H_2^+ molecular ion, are covered by the theory. The method combines domain truncation to a bounded domain D(R) containing {|x| <= R} with an extension of Liu's Composite Enriched Crouzeix-Raviart (CECR) finite element method to sign-indefinite potentials. Upper bounds come from the standard conforming Galerkin method; lower bounds come from the CECR construction, whose gap to the exact eigenvalue closes as the mesh is refined. Numerical experiments on the 2D single- and two-centred Coulomb potentials and on the 3D hydrogen atom and H_2^+ molecular ion illustrate the algorithm and confirm the predicted convergence.