Geodesic
Multiparameter Perturbation Theory of Fredholm Operators Applied to Bloch Functions
Matematičeskie zametki, Tome 86 (2009) no. 6, pp. 819-828

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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, Bloch function, two-dimensional Schrödinger operator, Hilbert space, analytic function.
Mots-clés : hexagonal lattice 
@article{MZM_2009_86_6_a1,
     author = {V. V. Grushin},
     title = {Multiparameter {Perturbation} {Theory} of {Fredholm} {Operators} {Applied} to {Bloch} {Functions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {819--828},
     publisher = {mathdoc},
     volume = {86},
     number = {6},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_86_6_a1/}
}
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V. V. Grushin. Multiparameter Perturbation Theory of Fredholm Operators Applied to Bloch Functions. Matematičeskie zametki, Tome 86 (2009) no. 6, pp. 819-828. http://geodesic.mathdoc.fr/item/MZM_2009_86_6_a1/