The cost rate of nonlinear remote stabilization on the Aubry--André lattice: a reflected off-spectral exponent and the sharp identity for almost every phase

Abstract

We study the exponential rate $r(\alpha ,\lambda )$ of the energy ${\mathcal{E}}_N$ needed to steer a far site, at distance $N$, of an Aubry--André chain $H_{\lambda }$ via one boundary actuator with closed-loop margin $\alpha$. An exact eigenbasis reduction writes ${\mathcal{E}}_N$ as a Cauchy quadratic form ${\tilde{b}}^{\top }{C}^{-1}\tilde{b}$ in the boundary-amplitude ratios, whose rate is the off-spectral Lyapunov exponent of the transfer cocycle at the reflected band edge $z^{\star }=2E_{\min }-2\alpha -E_{\max }$, giving $r(\alpha ,\lambda )={\gamma }_{\lambda }(z^{\star })$. The rate lies in a bracket of width ${\log }_{+}(\lambda /2)$ whose ends coincide for $\lambda \le 2$, the spectrum having logarithmic capacity $\max (1,\lambda /2)$. We prove the identity unconditionally, for every phase, on the whole metallic--critical range $0<\lambda \le 2$: for $\lambda <2$ through subcritical almost reducibility as the sole external input, and at $\lambda =2$ because the Green's function there equals the Lyapunov exponent. For $\lambda >2$ the upper bound is unconditional, and the lower bound takes localization as its only external input: an inverse-free cocycle form makes ${\mathcal{E}}_N$ a cancellation-free positive sum, and a Christoffel--Darboux identity collapses its coefficients to $|c_k|=Q({\delta }_k)({\hat{\psi }}_N^{(k)}{)}^2$, where band-edge near-degeneracies cancel. With a three-distance lemma this yields $r={\gamma }_{\lambda }(z^{\star })$ at every Diophantine frequency and almost every phase, with gap $O(N^{-2/(2+\tau )})$ for type $\tau$ ($N^{-2/3}$ at bounded type), unconditionally for $\lambda \ge {\lambda }_1$ and under a polynomial-prefactor localization hypothesis for $2<\lambda <{\lambda }_1$. The relative gap $1-r/{\gamma }_{\lambda }(z^{\star })$ vanishes at both ends of the localized phase, with $g_{\mathbb{C}\setminus {\Sigma }_{\lambda }}(z^{\star })\to \mathrm{arccosh}3$ as $\lambda \to \infty$.