Long persistence of localization in a disordered anharmonic chain beyond the atomic limit

We establish rigorous bounds on the decorrelation time and thermal transport in the disordered Klein-Gordon chain with a quartic on-site potential, governed by a parameter $\lambda$. At $\lambda =0$, the chain is harmonic, and any form of transport is fully suppressed by Anderson localization. For the anharmonic system, at $\lambda >0$, our results show that decorrelation and transport can occur only on time scales that grow faster than any polynomial in $1\mathrm{/}\lambda$ as $\lambda \to 0$. From a technical perspective, the main novelty of our work is that we don’t restrict ourselves to the atomic limit. Instead, we develop perturbation theory around the harmonic system with a fixed harmonic interaction between nearby oscillators. This allows us to compare our mathematical results with previous numerical work and contribute to resolving an ongoing debate, as detailed in a companion paper arXiv:2308.10572.