Multiparameter Perturbation Theory of Fredholm Operators Applied to Bloch Functions

In the present paper, a family of linear Fredholm operators depending on several parameters is considered. We implement a general approach, which allows us to reduce the problem of finding the set $\Lambda$ of parameters $t=(t_1,\dots,t_n)$ for which the equation $A(t)u=0$ has a nonzero solution to a finite-dimensional case. This allows us to obtain perturbation theory formulas for simple and conic points of the set $\Lambda$ by using the ordinary implicit function theorems. These formulas are applied to the existence problem for the conic points of the eigenvalue set $E(k)$ in the space of Bloch functions of the two-dimensional Schrödinger operator with a periodic potential with respect to a hexagonal lattice.