劉 雪峰

Spectral Galerkin methods are renowned for high-precision eigenvalue approximation, yet a rigorous lower bound obtained directly from a spectral discretisation has remained unavailable: the classical Kato and Weinstein--Temple enclosures do apply, but require a~priori information on a neighbouring eigenvalue. This paper resolves the issue by extending the author's projection-based framework for guaranteed lower eigenvalue bounds -- so far realised only through finite element methods -- to conforming spectral Galerkin methods. For trial spaces of exact eigenfunctions the required projection constant is the closed-form optimal value $C_N=λ_{M+1}^{-1/2}$, the inverse square root of the first omitted eigenvalue. For $-Δ+V$ with $0\le V\in L^\infty$, a \emph{projection-gap estimate} yields an explicit constant for the standard Galerkin matrix (exact at $V=0$), and a composite discretisation removes the $||V||_{L^\infty}$-dependence for large potentials. With Neumann domain truncation these give certified two-sided bounds on $R^d$; for two benchmark potentials on $R^2$ the spectral enclosures match or surpass certified finite element ones at two orders of magnitude fewer degrees of freedom. The same auxiliary-projector mechanism extends to singular potentials with an unbounded $L^\infty$ norm -- in particular to attractive Coulomb singularities in three dimensions, via a localised Hardy inequality -- which we develop in a companion paper.

We present, to the best of our knowledge, the first numerical algorithm for explicit, computable two-sided eigenvalue bounds for Schrödinger 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.

We propose a rigorous method for computing two-sided eigenvalue bounds of the Schrödinger operator $H=-Δ+V$ with a confining potential on $\mathbb{R}^2$. 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.