This article is cited in 22 papers

Multiparameter Perturbation Theory of Fredholm Operators Applied to Bloch Functions

V. V. Grushin

Moscow State Institute of Electronics and Mathematics

Abstract: 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.

Keywords: multiparameter perturbation theory, Fredholm operator, hexagonal lattice, Bloch function, two-dimensional Schrödinger operator, Hilbert space, analytic function.

UDC: 517.958

Received: 11.11.2008

DOI: 10.4213/mzm6575

 English version: Mathematical Notes, 2009, 86:6, 767–774

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