On the Spectra of Sieved Schr\"odinger Operators

TL;DR

This paper demonstrates that the spectral dimension of certain Schr"odinger operators, derived from Fibonacci models, changes under sieving, revealing new spectral properties in quasicrystal models.

Contribution

It introduces examples of Schr"odinger operators where spectral dimension varies with sieving, highlighting non-invariance in spectral properties of quasicrystal models.

Findings

Spectral dimension is not invariant under sieving for certain operators.

Local Hausdorff dimension can tend to zero in parts of the spectrum.

Examples are based on Fibonacci Hamiltonian models.

Abstract

We give a family of examples of discrete Schr\"odinger 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.