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
\(\frac{1}{2}\)-Hölder 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 Schrödinger operators with Lipschitz continuous
potentials, e.g., the Almost Mathieu Operator, exhibit spectral
\(\frac{1}{2}\)-Hölder continuity. Also, this complements a 2019
result by S. Beckus; J. Bellissard; H. Cornean establishing that
\(d\)-dimensional discrete subshift Schrödinger 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. This is joint work with
Siegfried Beckus.
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