Sharp Computational Bounds for Spectral Types of Schrödinger Operators
Abstract
We prove sharp bounds for determining the spectral-type decomposition of Schrödinger operators in the spirit of Smale's program on the foundations of computation. For explicit one-dimensional self-adjoint Schrödinger operators H=−d2/dx2+V on L2(ℝ), where V∈ C∞(ℝ;ℝ) is given by a finite description of all derivatives and derivative bounds, the pure point and absolutely continuous spectral sets cannot, in general, be recovered by any single limiting procedure. The singular continuous spectral set is strictly harder: it cannot, in general, be recovered by two nested limiting procedures. Analytic constructions of dichotomies realize the lower bounds: Gordon-type repetitions for pure point spectrum, high barriers for absolutely continuous spectrum, and an inverse spectral construction for singular continuous spectrum based on Riesz products, moment-killing perturbations, and a computational Gelfand--Levitan scheme. The finite-description framework also implies corresponding limitations on what can be certified in fixed formal systems (e.g., when used in computer-assisted proofs). Conversely, using wavelet-based certified computation, we prove matching upper bounds for broad classes of self-adjoint differential operators on ℝd with coefficients of locally bounded variation and quantitative local variation control: two limits suffice for the pure point and absolutely continuous parts, and three for the singular continuous part. This provides a sharp hierarchy for spectral types.
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BibTeX
@misc{colbrook2026sharp,
title = {{Sharp Computational Bounds for Spectral Types of Schrödinger Operators}},
author = {Matthew J. Colbrook and George Coote},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08945},
eprint = {2609.08945},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08945},
}
RIS
TY - UNPB
TI - Sharp Computational Bounds for Spectral Types of Schrödinger Operators
AU - Matthew J. Colbrook
AU - George Coote
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08945
UR - https://arxiv.org/abs/2609.08945
AB - We prove sharp bounds for determining the spectral-type decomposition of Schrödinger operators in the spirit of Smale's program on the foundations of computation. For explicit one-dimensional self-adjoint Schrödinger operators H=−d^2/dx^2+V on L^2(ℝ), where V∈ C^∞(ℝ;ℝ) is given by a finite description of all derivatives and derivative bounds, the pure point and absolutely continuous spectral sets cannot, in general, be recovered by any single limiting procedure. The singular continuous spectral set is strictly harder: it cannot, in general, be recovered by two nested limiting procedures. Analytic constructions of dichotomies realize the lower bounds: Gordon-type repetitions for pure point spectrum, high barriers for absolutely continuous spectrum, and an inverse spectral construction for singular continuous spectrum based on Riesz products, moment-killing perturbations, and a computational Gelfand--Levitan scheme. The finite-description framework also implies corresponding limitations on what can be certified in fixed formal systems (e.g., when used in computer-assisted proofs). Conversely, using wavelet-based certified computation, we prove matching upper bounds for broad classes of self-adjoint differential operators on ℝ^d with coefficients of locally bounded variation and quantitative local variation control: two limits suffice for the pure point and absolutely continuous parts, and three for the singular continuous part. This provides a sharp hierarchy for spectral types.
ER -