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 Λ of parameters t = (t 1, ..., 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 Λ 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.
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Original Russian Text © V. V. Grushin, 2009, published in Matematicheskie Zametki, 2009, Vol. 86, No. 6, pp. 819–828.
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Grushin, V.V. Multiparameter perturbation theory of Fredholm operators applied to bloch functions. Math Notes 86, 767–774 (2009). https://doi.org/10.1134/S0001434609110194
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DOI: https://doi.org/10.1134/S0001434609110194