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

We propose a rigorous method for computing two-sided eigenvalue bounds of the Schr\"odinger operator $H=-\Delta+V$ 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 Schr\"odinger operators on an unbounded domain.