From estimate to proof: certified ground-state energy bounds for singular Schrödinger operators on R3
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 R3 with attractive Coulomb singularities. For H2+ the method certifies λ2eee≥−0.2035 in place of the analytic −1/4, and Temple's inequality with a global trial then gives an enclosure of width below 6.5×10−51. For linear H32+, whose central nucleus defeats the analytic separator, the certified widths are 1.3×10−12 and 3.1×10−13 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∞ 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 λ1∈[−0.553291,−0.550740] for H2+. The coercivity constant, we show, must bound its own optimum from above. All drivers and certificates are released.
Related
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BibTeX
@misc{liu2026from,
title = {{From estimate to proof: certified ground-state energy bounds for singular Schrödinger operators on $R^3$}},
author = {Xuefeng Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.25760},
eprint = {2608.25760},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25760},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - From estimate to proof: certified ground-state energy bounds for singular Schrödinger operators on R^3
AU - Xuefeng Liu
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2608.25760
UR - https://arxiv.org/abs/2608.25760
AB - 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^eee≥−0.2035 in place of the analytic −1/4, and Temple's inequality with a global trial then gives an enclosure of width below 6.5×10^−51. For linear H_3^2+, whose central nucleus defeats the analytic separator, the certified widths are 1.3×10^−12 and 3.1×10^−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^∞ 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∈[−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.
ER -