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Multiparameter perturbation theory of Fredholm operators applied to Bloch functions. (English. Russian original) Zbl 1197.47025
Math. Notes 86, No. 6, 767-774 (2009); translation from Mat. Zametki 86, No. 6, 819-828 (2009).
Let a linear Fredholm operator \(A(t)\) from a Banach space \(B_1\) to a Banach space \(B\), depending on parameters \(t=(t_1,\dots,t_n)\), be given and assume that \(A(t)\) is an analytic function of its parameters ranging in a connected domain \(\Omega\) of an \(n\)-dimensional complex space \({\mathbb C}^n\). It is sought to find the parameter values for which \(\text{dim}\;\text{Ker} A(t) \neq 0\), i.e., the equation \(A(t)u=0\) has a nonzero solution \(u\in B_1\). The set of such parameters \(t\) is denoted by \(\Lambda\).
In the paper under review, the author implements a general approach which allows locally to reduce the above problem to a finite-dimensional situation, namely, in a neighborhood of a given point \(t_0\), to find a finite-dimensional operator \(\widehat{g}(t): {\mathbb C}^m \to {\mathbb C}^m\) analytically depending on the parameters \(t\) and satisfying the condition that, for \(t\) sufficiently close to \(t_0\), \(\text{dim}\;\text{Ker}\;A(t) = \text{dim}\;\text{Ker}\;\widehat{g}(t)\). Thus the problem of solving the equation \(A(t)u = 0\) is reduced to the problem of investigating a finite-dimensional system of linear equations \(\widehat{g}(t) d = 0\), \(d \in {\mathbb C}^m\), which depend on the parameters \(t\).
This allows to obtain perturbation theory formulas for simple and conic points of the set \(\Lambda\) by using the ordinary implicit function theorem. 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.

47A55 Perturbation theory of linear operators
47A53 (Semi-) Fredholm operators; index theories
30D45 Normal functions of one complex variable, normal families
46E20 Hilbert spaces of continuous, differentiable or analytic functions
47F05 General theory of partial differential operators
35J10 Schrödinger operator, Schrödinger equation
Full Text: DOI
[1] I. Ts. Gokhberg and M. G. Krein, ”Fundamental aspects of defect numbers, root numbers and indexes of linear operators,” UspekhiMat. Nauk 12(2), 43–118 (1957) [in Russian]. · Zbl 0088.32101
[2] M. A. Shubin, Pseudodifferential Operators and Spectral Theory (Nauka, Moscow, 1978) [in Russian]. · Zbl 0451.47064
[3] V. V. Grushin, ”On a class of elliptic pseudodifferential operators degenerate on a submanifold,” Mat. Sb. 84(2), 163–195 (1971) [Math. USSR-Sb. 13, 155–185 (1971)]. · Zbl 0238.47038
[4] V. V. Grushin, ”Asymptotic behavior of the eigenvalues of the Schrödinger operator in thin closed tubes,” Mat. Zametki 83(4), 503–519 (2008) [Math. Notes 83 (3–4), 463–477 (2008)]. · Zbl 1152.35452
[5] V. I. Arnol’d, Supplementary Chaps. to the Theory of Ordinary Differential Equations (Nauka, Moscow, 1978) [in Russian].
[6] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4: Analysis of Operators (Academic Press, New York, 1979; Mir, Moscow, 1982). · Zbl 0405.47007
[7] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis and Self-Adjointness (Academic Press, New York, 1975;Mir, Moacow, 1978). · Zbl 0308.47002
[8] R. Saito, G. Dresslhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998).
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