Rigorous Eigenvalue Bounds for Schrödinger Operators with Confining Potentials on ${\mathbb{R}}^2$

Abstract

We propose a rigorous method for computing two-sided eigenvalue bounds of the Schrödinger operator $H=-\hat{I}”+V$ with a confining potential on ${\mathbb{R}}^2$. The method combines domain truncation to a finite disk $D(R)$ on which the restricted eigenvalue problem is solved with a rigorous eigenvalue bound, where Liu's eigenvalue bound along with the Composite Enriched Crouzeix--Raviart (CECR) finite element method proposed plays a central role. Two concrete potentials are studied: the radially symmetric ring potential $V_1(x)=(|x{|}^2-1{)}^2$ and the Cartesian double-well $V_2(x)=(x_1^2-1{)}^2+x_2^2$. To author's knowledge, this paper reports the first rigorous eigenvalue bounds for Schrödinger operators on an unbounded domain.