ThreeFold Weyl Points for the Schrödinger Operator with Periodic Potentials

Weyl points are degenerate on the spectral bands at which energy intersect conically. They origins of many novel physical phenomena and have attracted much attention recently. In this paper, we investigate existence such in spectrum three-dimensional Schrödinger operator $H = - \Delta +V(x)$ with $V(x)$ being a large class periodic potentials. Specifically, give very general conditions potentials ensure threefold associated bands. Different from two-dimensional honeycomb structures possess Dirac where two adjacent band surfaces touch each other conically, conically intersection an extra sandwiched between. To conical structures, more delicate, new symmetries required. As consequence, techniques combining used to justify under proposed. This paper provides comprehensive proof points. particular, role symmetry endowed potential is carefully analyzed. Our extends analysis higher dimension multiplicities. We also provide some numerical simulations typical demonstrate our analysis.