Energy Discontinuity of the Lyapunov Exponent for Quasiperiodic Schrödinger Operators with Gevrey Potentials
Abstract
We construct quasiperiodic Schrödinger operators with Gevrey potentials for which the Lyapunov exponent is discontinuous in the spectral energy. More precisely, let α∈R∖ℚ satisfy the Diophantine condition |kα|T≥γ|k|−τ for all k∈Z∖{0}, with γ>0 and τ≥1. For every Gevrey index s>1+τ and every sufficiently large coupling λ, we construct a real-valued potential vλ∈ Gs(T,R) and a sequence of energies Ej↓0 such that Lvλ(0)≥(1−ελ)logλ, supj≥1 Lvλ(Ej)≤(1−c*)logλ, where Lvλ(E) denotes the Lyapunov exponent of the Schrödinger operator with potential λvλ, c*>0 is an absolute constant, and ελ→0 as λ→∞. For bounded-type frequencies (τ=1), the regularity range s>2 is sharp up to the endpoint s=2, since the Lyapunov exponent is known to be continuous in energy for 1<s<2.
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BibTeX
@misc{xu2026energy,
title = {{Energy Discontinuity of the Lyapunov Exponent for Quasiperiodic Schrödinger Operators with Gevrey Potentials}},
author = {Jiahao Xu and Shuo Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32623},
eprint = {2610.32623},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32623},
}
RIS
TY - UNPB
TI - Energy Discontinuity of the Lyapunov Exponent for Quasiperiodic Schrödinger Operators with Gevrey Potentials
AU - Jiahao Xu
AU - Shuo Zhang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32623
UR - https://arxiv.org/abs/2610.32623
AB - We construct quasiperiodic Schrödinger operators with Gevrey potentials for which the Lyapunov exponent is discontinuous in the spectral energy. More precisely, let α∈R∖ℚ satisfy the Diophantine condition |kα|_T≥γ|k|^−τ for all k∈Z∖{0}, with γ>0 and τ≥1. For every Gevrey index s>1+τ and every sufficiently large coupling λ, we construct a real-valued potential v_λ∈ G^s(T,R) and a sequence of energies E_j↓0 such that L_v_λ(0)≥(1−ε_λ)logλ, sup_j≥1 L_v_λ(E_j)≤(1−c_*)logλ, where L_v_λ(E) denotes the Lyapunov exponent of the Schrödinger operator with potential λv_λ, c_*>0 is an absolute constant, and ε_λ→0 as λ→∞. For bounded-type frequencies (τ=1), the regularity range s>2 is sharp up to the endpoint s=2, since the Lyapunov exponent is known to be continuous in energy for 1<s<2.
ER -