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 \(1/\lambda \) as \(\lambda \rightarrow 0\) . 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 [14].
