On the Spectra of Sieved Schrödinger Operators
Jake Fillman, Alexandro Luna
TL;DR
The paper demonstrates that spectral dimension need not be invariant under sieving for Schrödinger operators, using the Fibonacci Hamiltonian as a test case. It develops a trace-map framework that links spectral properties to a hyperbolic dynamical system, showing that the sieved spectra $\Sigma_{\lambda}^{[\ell]}$ are zero-measure Cantor sets with $\dim_{\mathrm{H}}(\Sigma_{\lambda}^{[\ell]})=1$ for $\ell\ge2$ and $<1$ for $\ell=1$. For $\lambda>4$, there exists a fixed interval $K$ where the local dimension remains positive for all $\ell$ but vanishes in the limit $\ell\to\infty$, revealing a dramatic non-invariance under sieving. The analysis hinges on hyperbolicity of the trace map on invariant surfaces and on intersecting center-stable manifolds with a substitution-induced curve, yielding explicit dimension relations such as $\dim_{\mathrm{H}}^{\mathrm{loc}}(\Sigma_{\lambda}^{[\ell]},E)=\tfrac12\dim_{\mathrm{H}}\Omega_{V_0}$ for $V_0>0$ and $=1$ for $V_0=0$. Overall, the work exposes a genuinely new phenomenon for Schrödinger operators, distinct from CMV matrices, and highlights how spectral fractality can be altered by sieving in a controlled, dynamical-systems framework.
Abstract
We give a family of examples of discrete Schrödinger operators whose spectral dimension is not invariant under sieving. The examples are produced from the Fibonacci Hamiltonian, which is one of the main models of a one-dimensional quasicrystal. We also give a family of examples in which the local Hausdorff dimension tends to zero in some parts of the spectrum as the sieving parameter is sent to infinity.