Beyond energy bounds: comparing Schrödinger eigenstates
Abstract
We derive eigenstate comparisons from ordered Schrödinger operators, extending energy bounds to overlaps, spatial distributions and nodes. For three form-ordered operators with discrete comparison ground states, suppose the upper ground-state energy lies in the lower operator's ground-state gap. Three comparison spectral values then bound all three squared ground-state overlaps and guarantee a simple discrete intermediate ground state. Related gap estimates control low-energy subspaces and individual excited-state overlaps under additional gap conditions. In one dimension and radially, a single crossing of the potential difference relative to the energy shift orders ground-state likelihood ratios and monotone observables; a nondecreasing difference suffices. The results cover asymmetric wells and shortened right Dirichlet walls, with coordinatewise comparisons for submodular reference potentials and coordinatewise nondecreasing perturbations. A two-sided Sturm argument orders corresponding internal nodes for nondecreasing potential differences, even between orthogonal one-dimensional excited states. For Yukawa ground states, Hulthén comparisons bound overlaps, radial moments and the density at the nucleus; the latter two-sided bounds hold for screening 0<δ<2/√6. Optimizing scaled Hulthén majorants gives a lower bound of approximately $1.13611$ for n2δc(n,0), where δc(n,0) is the critical screening for the nth s-wave bound state. This bound is optimal within that two-parameter comparison family.
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BibTeX
@misc{wang2026beyond,
title = {{Beyond energy bounds: comparing Schrödinger eigenstates}},
author = {Xiao-Ke Wang and Han Jia and Shengjun Wu and Zeng-Bing Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39136},
eprint = {2610.39136},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39136},
}
RIS
TY - UNPB
TI - Beyond energy bounds: comparing Schrödinger eigenstates
AU - Xiao-Ke Wang
AU - Han Jia
AU - Shengjun Wu
AU - Zeng-Bing Chen
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39136
UR - https://arxiv.org/abs/2610.39136
AB - We derive eigenstate comparisons from ordered Schrödinger operators, extending energy bounds to overlaps, spatial distributions and nodes. For three form-ordered operators with discrete comparison ground states, suppose the upper ground-state energy lies in the lower operator's ground-state gap. Three comparison spectral values then bound all three squared ground-state overlaps and guarantee a simple discrete intermediate ground state. Related gap estimates control low-energy subspaces and individual excited-state overlaps under additional gap conditions. In one dimension and radially, a single crossing of the potential difference relative to the energy shift orders ground-state likelihood ratios and monotone observables; a nondecreasing difference suffices. The results cover asymmetric wells and shortened right Dirichlet walls, with coordinatewise comparisons for submodular reference potentials and coordinatewise nondecreasing perturbations. A two-sided Sturm argument orders corresponding internal nodes for nondecreasing potential differences, even between orthogonal one-dimensional excited states. For Yukawa ground states, Hulthén comparisons bound overlaps, radial moments and the density at the nucleus; the latter two-sided bounds hold for screening 0<δ<2/√6. Optimizing scaled Hulthén majorants gives a lower bound of approximately $1.13611$ for n^2δ_c(n,0), where δ_c(n,0) is the critical screening for the nth s-wave bound state. This bound is optimal within that two-parameter comparison family.
ER -