PreprintarXiv licencearXiv
On the Emergence of Discrete Spectrum for Weakly Disordered Schrödinger Operators
Abstract
We investigate the spectral properties of the Anderson operator perturbed by a localized negative potential, −V. Specifically, we analyze the random Schrödinger operator defined by H = −Δ+ve ∑n ωn χn − V, where the unperturbed operator exhibits a disordered energy landscape. Our primary focus is to establish precise estimates on the number of negative eigenvalues (bound states) induced by the attractive perturbation. By analyzing the competition between Anderson localization and the binding capacity of the potential, we provide quantitative bounds on the discrete spectrum. These results offer new insights into how randomness enhances the eigenvalue bounds.
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BibTeX
@misc{safronov2026emergence,
title = {{On the Emergence of Discrete Spectrum for Weakly Disordered Schrödinger Operators}},
author = {Oleg Safronov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.20505},
eprint = {2606.20505},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.20505},
}
RIS
TY - UNPB
TI - On the Emergence of Discrete Spectrum for Weakly Disordered Schrödinger Operators
AU - Oleg Safronov
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2606.20505
UR - https://arxiv.org/abs/2606.20505
AB - We investigate the spectral properties of the Anderson operator perturbed by a localized negative potential, −V. Specifically, we analyze the random Schrödinger operator defined by H = −Δ+ve ∑_n ω_n χ_n − V, where the unperturbed operator exhibits a disordered energy landscape. Our primary focus is to establish precise estimates on the number of negative eigenvalues (bound states) induced by the attractive perturbation. By analyzing the competition between Anderson localization and the binding capacity of the potential, we provide quantitative bounds on the discrete spectrum. These results offer new insights into how randomness enhances the eigenvalue bounds.
ER -