Spectral band degeneracies of $\frac{\pi}{2}-$rotationally invariant periodic Schr\'odinger operators
| Autor: | Keller, Rachael T., Marzuola, Jeremy L., Osting, Braxton, Weinstein, Michael I. |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | The dynamics of waves in periodic media is determined by the band structure of the underlying periodic Hamiltonian. Symmetries of the Hamiltonian can give rise to novel properties of the band structure. Here we consider a class of periodic Schr\"odinger operators, $H_V=-\Delta+V$, where $V$ is periodic with respect to the lattice of translates $\Lambda=\mathbb{Z}^2$. The potential is also assumed to be real-valued, sufficiently regular and such that, with respect to some origin of coordinates, inversion symmetric (even) and invariant under $\pi/2$ rotation. The present results are the $\mathbb{Z}^2-$ analogue of results obtained for conical degenerate points (Dirac points) in honeycomb structures. Our proofs make use of the framework developed by Fefferman-Weinstein and Fefferman-Lee-Thorp-Weinstein. Comment: 52 pages, 13 figures; This article was published in Multiscale Model. Simul., 16(4), 1684--1731 (2017). In this updated arXiv version we correct the statement and proof of Corollary 4.2. Clarifying edits were also made in the statements of Corollaries 5.4 and C.1 |
| Databáze: | arXiv |
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