The authors consider the existence of Weyl points in the \(L^2(\mathbb{R}^3)\)-spectrum of the three-dimensional Schrödinger operator \[ H=-\triangle+V(\mathbf{x}) \ \ (\mathbf{x}\in\mathbb{R}^3) \] where \(V(\mathbf{x})\) is a real-valued and periodic potential. The admissible potentials play an essential role in this consideration. In other words \(V(\mathbf{x})\in C^{\infty}(\mathbb{R}^3)\) has certain symmetries defined in the paper. Note that the Weyl points are degenerate, i.e. points on the spectral bands at which energy bands intersect conically. By Floquet-Bloch theory discussed by P. Kuchment [Floquet theory for partial differential equations. Basel: Birkhäuser Verlag (1993; Zbl 0789.35002); Bull. Am. Math. Soc., New Ser. 53, No. 3, 343–414 (2016; Zbl 1346.35170)] as well P. Kuchment and S. Levendorskiĭ [Trans. Am. Math. Soc. 354, No. 2, 537–569 (2002; Zbl 1058.35174)], the spectrum of \(H\) in \(L^2(\mathbb{R}^3)\) is the union of all energy bands \(E_b(\mathbf{k})\) (\(b\geq 1\)), for all \(\mathbf{k}\) in the Brillouin zone. For given \(V(\mathbf{x})\), two energy bands may intersect with each other conically at some \(\mathbf{k}\). This degenerate point \(\mathbf{k}\) in the 3-D energy bands is called a Weyl point. There exist different Weyl points depending on the multiplicity of degeneracy. It turns out that the threefold Weyl points are conically intersection points of two energy bands with an extra band sandwiched in between. Taking into account that the threefold and three-dimensional conical structures, must be ensured it is introduced some new symmetry on the considered model.
Finally it is shown numerical simulations on typical potentials which illustrate the main ideas in the paper.