Spectral band degeneracies of $\frac{\pi}{2}-$rotationally invariant periodic Schr\"odinger operators

TL;DR

This paper investigates how $rac{ ext{pi}}{2}$-rotational symmetry in periodic Schrödinger operators influences spectral degeneracies, extending known results from honeycomb structures to $ ext{Z}^2$-periodic potentials.

Contribution

It establishes the spectral degeneracy phenomena for $ ext{Z}^2$-periodic Schrödinger operators with $rac{ ext{pi}}{2}$-rotational symmetry, generalizing previous honeycomb lattice results.

Findings

Spectral degeneracies are characterized under $rac{ ext{pi}}{2}$-rotation symmetry.

Results extend Dirac point analysis from honeycomb to $ ext{Z}^2$-periodic potentials.

The framework of Fefferman-Weinstein is employed for proofs.

Abstract

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.