2026 Volume 18 Pages 97-100
We propose a rigorous method for computing two-sided eigenvalue bounds of the Schrödinger operator H = −Δ+V with a confining potential on ℝ2. The method applies domain truncation and uses a Neumann eigenvalue problem on a truncated bounded domain Ω. The confinement of the potential provides the theoretical justification for the truncation error. On domain Ω, Liu’s eigenvalue bound together with the Composite Enriched Crouzeix–Raviart (CECR) finite element method plays a central role. Two concrete potentials are studied: the radially symmetric ring potential V1(x) = (|x|2 − 1)2 and the Cartesian double-well V2(x) = (x12 − 1)2 + x22. To the author’s knowledge, this paper gives the first rigorous eigenvalue bounds for Schrödinger operators with these two unbounded model potentials on an unbounded domain.
