Yi Zhu, Department of Mathematical Sciences, Tsinghua University, China
Three-fold Weyl points for the periodic Schrödinger operator
Weyl points are degenerate points on the spectral bands at which energy bands intersect conically. They are the origins of many novel physical phenomena and have attracted much attention recently. In this talk, we investigate the existence of such points in the spectrum of the 3-dimensional Schrödinger operator H = −Δ+V (x) with V (x) being in a large class of periodic potentials. To the best of our knowledge, this is the first result on the existence of Weyl points for a broad family of 3d continuous Schrödinger equations. Indeed, we give very general conditions on the potentials which ensure the existence of 3-fold Weyl points on the associated energy bands. Different from 2-dimensional Dirac points where two adjacent band surfaces touch each other conically, the 3-fold Weyl points are conically intersection points of two energy bands with an extra band sandwiched in between. We give the required conditions and provide a comprehensive proof of such 3-fold Weyl points, which extends the Fefferman-Weinstein's strategy on the analysis of conical spectral points (JAMS 2012) to a higher dimension and to higher multiplicities. This talk is based on the joint work with H. Guo and M. Zhang at Tsinghua university.