Asymptotic localization of energy in non-disordered oscillator chains

We study two popular one-dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete non-linear Schrödinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter $ε>0$, is small. We rigorously establish that the thermal conductivity of the chains has a non-perturbative origin, with respect to the coupling constant $ε$, and we provide strong evidence that it decays faster than any power law in $ε$ as $ε\rightarrow 0$. The weak coupling regime also translates into a high temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature.

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