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 $H=-Δ+V$ with a confining potential on $\mathbb{R}<sup>2$. The method combines domain truncation to a finite disk $D(R)$ on which the restricted eigenvalue problem is solved with a rigrous 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 $V_1(x)=(|x|<sup>2-1)<sup>2$ and the Cartesian double-well $V_2(x)=(x_1<sup>2-1)<sup>2+x_2<sup>2$. To author's knowledge, this paper reports the first rigorous eigenvalue bounds for Schrödinger operators on an unbounded domain.