Spectral estimates of dynamically-defined and amenable operator families
We consider kernel operators defined by a dynamical system. The Hausdorff distance of spectra is estimated by the Hausdorff distance of subsystems. We prove that the spectrum map is $ \frac{1}{2} $-H\"older continuous provided the group action and kernel are Lipschitz continuous and the group has strict polynomial growth. Also, we prove that the continuity can be improved resulting in the spectrum map being Lipschitz continuous provided the kernel is instead locally-constant. This complements a 1990 result by J.~Avron; P.H.M.v.~Mouche; B.~Simon establishing that one-dimensional discrete quasiperiodic Schr\"odinger operators with Lipschitz continuous potentials, e.g., the Almost Mathieu Operator, exhibit spectral $ \frac{1}{2} $-H\"older continuity. Also, this complements a 2019 result by S.~Beckus; J.~Bellissard; H.~Cornean establishing that $ d $-dimensional discrete subshift Schr\"odinger operators with locally-constant potentials, e.g., the Fibonacci Hamiltonian, exhibit spectral Lipschitz continuity. Our work exposes the connection between the past two results, and the group, e.g., the Heisenberg group, needs not be the integer lattice nor abelian.