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.
Highlights
We study the spectral properties of a half-periodic material, when this one is cut along any line
One can apply partial Bloch theory in this direction, and obtain that its spectrum has again the band-gap structure. This spectrum usually differs from σbulk due to the presence of edge modes. This is described by the edge spectrum σedge[θ] := σ H D [θ] \ σbulk
We focus on a dislocated version of the bulk operator
Summary
We study the spectral properties of a half-periodic material, when this one is cut along any line. Its spectrum σbulk is independent of θ, and has a band-gap structure by Bloch theory. One can apply partial Bloch theory in this direction, and obtain that its spectrum has again the band-gap structure. This spectrum usually differs from σbulk due to the presence of edge modes. In the incommensurable case, all gaps of σbulk are filled with edge spectrum This extends the previous work by Hempel and Kohlmann [7, 9], where the authors proved this filling gap phenomenon in the limit θ → 0. In the special case of Dirichlet boundary conditions, the Σ appearing in Theorem 1 is independent of θ, and equals the infimum of the bulk spectrum Σ = inf σbulk.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
