Spectral properties of periodic systems cut at an angle

TL;DR

This paper studies the spectral properties of a 2D Schrödinger operator with a periodic structure cut at an angle, revealing that incommensurate cuts fill spectral gaps with edge states, unlike the commensurate case.

Contribution

It proves that incommensurate cuts eliminate spectral gaps by filling them with edge spectrum, contrasting with the band-gap structure in the commensurate case.

Findings

In commensurate cases, the spectrum exhibits a band-gap Bloch structure.

In incommensurate cases, spectral gaps are filled with edge spectrum.

Edge spectrum presence depends on the angle of the cut.

Abstract

We consider a semi-periodic two-dimensional Schr\"odinger operator which is cut at an angle. When the cut is commensurate with the periodic lattice, the spectrum of the operator has the band-gap Bloch structure. We prove that in the incommensurable case, there are no gaps: the gaps of the bulk operator are filled with edge spectrum.