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 \(α\in\R\setminus\mathbb Q\) 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 \(1<\mathfrak s<2\).