Sharp decay thresholds for eigenvalues of discrete Schrödinger operators on ℤ2
Abstract
We study eigenvalues of discrete Schrödinger operators H=Δ+V on ℤ2, where Δ is the uncentered Laplacian, i.e., the un-normalized adjacency operator of ℤ2 and V decays at infinity. By Weyl's theorem, the essential spectrum of H is [−4,4]. We determine the sharp decay thresholds for existence of eigenvalues in three distinct spectral regimes. While it is natural to expect different behavior at the spectral edge λ=±4, and the bulk, there is a further distinction between the regular energies 0<|λ|<4 and the interior critical energy λ=0 stemming from the reducibility of the corresponding Fermi surface in the latter case. For every 0<|λ|<4, we construct potentials V satisfying |V(n)|≤ C|n|−1 for which λ is an eigenvalue of Δ+V, and prove absence of eigenvalues when |V(n)|≤ C|n|−1−ε for some ε>0. At λ=0, the critical power changes and we construct potentials V satisfying |V(n)|≤ C|n| −2 for which $0$ is an eigenvalue of Δ+V, as well as prove absence when |V(n)|≤ C|n|−2−ε for any ε>0. Finally, at each spectral edge, we show that, for every K≥3, an eigenvalue can be created by potentials supported on exactly K sites, whereas a potential supported on at most two sites cannot create an edge eigenvalue. The proofs combine Green-function expansions and moment cancellation, Hilbert-space-valued iterations, discrete Carleman estimates, and a uniform Green-kernel estimate.
Related
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BibTeX
@misc{ganguly2026sharp,
title = {{Sharp decay thresholds for eigenvalues of discrete Schrödinger operators on $\mathbbZ^2$}},
author = {Shirshendu Ganguly and Wencai Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15908},
eprint = {2609.15908},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15908},
}
RIS
TY - UNPB
TI - Sharp decay thresholds for eigenvalues of discrete Schrödinger operators on ℤ^2
AU - Shirshendu Ganguly
AU - Wencai Liu
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2609.15908
UR - https://arxiv.org/abs/2609.15908
AB - We study eigenvalues of discrete Schrödinger operators H=Δ+V on ℤ^2, where Δ is the uncentered Laplacian, i.e., the un-normalized adjacency operator of ℤ^2 and V decays at infinity. By Weyl's theorem, the essential spectrum of H is [−4,4]. We determine the sharp decay thresholds for existence of eigenvalues in three distinct spectral regimes. While it is natural to expect different behavior at the spectral edge λ=±4, and the bulk, there is a further distinction between the regular energies 0<|λ|<4 and the interior critical energy λ=0 stemming from the reducibility of the corresponding Fermi surface in the latter case. For every 0<|λ|<4, we construct potentials V satisfying |V(n)|≤ C|n|^−1 for which λ is an eigenvalue of Δ+V, and prove absence of eigenvalues when |V(n)|≤ C|n|^−1−ε for some ε>0. At λ=0, the critical power changes and we construct potentials V satisfying |V(n)|≤ C|n| ^−2 for which $0$ is an eigenvalue of Δ+V, as well as prove absence when |V(n)|≤ C|n|^−2−ε for any ε>0. Finally, at each spectral edge, we show that, for every K≥3, an eigenvalue can be created by potentials supported on exactly K sites, whereas a potential supported on at most two sites cannot create an edge eigenvalue. The proofs combine Green-function expansions and moment cancellation, Hilbert-space-valued iterations, discrete Carleman estimates, and a uniform Green-kernel estimate.
ER -