Energy Discontinuity of the Lyapunov Exponent for Quasiperiodic Schrödinger Operators with Gevrey Potentials
Year: 2026
Type: preprint
Abstract: We construct quasiperiodic Schrödinger operators with Gevrey potentials for which the Lyapunov exponent is discontinuous in the spectral energy. More precisely, let \(α\in\R\setminus\mathbb Q\) satisfy the Diophantine condition \(\|kα\|_{\T}\geqγ|k|^{-τ}\) for all \(k\in\Z\setminus\{0\}\), with \(γ>0\) and \(τ\geq1\). For every Gevrey index \(\mathfrak s>1+τ\) and every sufficiently large coupling \(λ\), we construct a real-valued potential \(v_λ\in G^{\mathfrak s}(\T,\R)\) and a sequence of energies \(E_j\downarrow0\) such that \[ L_{v_λ}(0)\geq(1-\varepsilon_λ)\logλ, \qquad \sup_{j\geq1} L_{v_λ}(E_j)\leq(1-c_*)\logλ, \] where \(L_{v_λ}(E)\) denotes the Lyapunov exponent of the Schrödinger operator with potential \(λv_λ\), \(c_*>0\) is an absolute constant, and \(\varepsilon_λ\to0\) as \(λ\to\infty\). For bounded-type frequencies (\(τ=1\)), the regularity range \(\mathfrak s>2\) is sharp up to the endpoint \(\mathfrak s=2\), since the Lyapunov exponent is known to be continuous in energy for \(1<\mathfrak s<2\).
Source: arXiv (Cornell University)
Authors: Xu, Jiahao, Zhang, Shuo
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223656450