From estimate to proof: certified ground-state energy bounds for singular Schrödinger operators on $R^3$
Year: 2026
Type: preprint
Abstract: We propose a rigorous computational method that certifies the separator required by Temple's inequality and its Lehmann--Goerisch refinements, and with it unconditional ground-state bounds for $H=-Δ+V$ on $R^3$ with attractive Coulomb singularities. For H$_2^+$ the method certifies $λ_2^{\rm eee}\ge-0.2035$ in place of the analytic $-\tfrac14$, and Temple's inequality with a global trial then gives an enclosure of width below $6.5\times10^{-51}$. For linear H$_3^{2+}$, whose central nucleus defeats the analytic separator, the certified widths are $1.3\times10^{-12}$ and $3.1\times10^{-13}$ at two nuclear separations. Such a separator, a lower bound on the second eigenvalue, is supplied by theory only for special configurations; elsewhere it is estimated and the bound is conditional. Here every hypothesis is verified in interval arithmetic at run time. Three constructions make this possible on an unbounded domain with a potential of unbounded $L^\infty$ norm: a two-sided Neumann/Dirichlet truncation with a computable remainder; a localized Hardy inequality and sharp form bound with explicit constants, restoring coercivity; and a projection stage that certifies lower bounds on the first two eigenvalues, the second serving as separator. On a box, a Lehmann--Goerisch step places $λ_1\in[-0.553291,-0.550740]$ for H$_2^+$. The coercivity constant, we show, must bound its own optimum from above. All drivers and certificates are released.
Source: arXiv (Cornell University)
Author: Liu, Xuefeng
Institution: Tokyo Woman's Christian University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204484474