arXiv subjects
Publications and source records attributed to E. Korotyaev.
We solve inverse problems from the D-N map for the quantum graph on a finite domain in a square lattice and that on a hexagonal lattice, as well as inverse scattering problems from the S-matrix for a locally perturbed square lattice and a hexagonal lattice.
We consider normalized Laplacians and their perturbations by periodic potentials (Schrödinger operators) on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of non-degenerate bands) and a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graphs and show that they become identities for some class of graphs. We determine two-sided estimates on the length of the first spectral band and on the effective mass at the bottom of the spectrum of the Laplace and Schrödinger operators. In particular, these estimates yield that the first spectral band of Schrödinger operators is non-degenerate.
We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of the periodic graph, we introduce a subclass of minimal forms with a minimal number $\cI$ of edges in their supports on the period. We obtain a specific decomposition of the Laplacian into a direct integral in terms of minimal forms, where fiber Laplacians (matrices) have the minimal number $2\cI$ of coefficients depending on the quasimomentum and show that the number $\cI$ is an invariant of the periodic graph. Using this decomposition, we estimate the position of each band, the Lebesgue measure of the Laplacian spectrum and the effective masses at the bottom of the spectrum in terms of the invariant $\cI$ and the minimal forms. In addition, we consider an inverse problem: we determine necessary and sufficient conditions for matrices depending on the quasimomentum on a finite graph to be fiber Laplacians. Moreover, similar results for Schrödinger operators with periodic potentials are obtained.
We consider the third order operator with periodic coefficients on the real line. This operator is used in the integration of the non-linear evolution Boussinesq equation. For the minimal smoothness of the coefficients we prove that: 1) the operator is self-adjoint and it is decomposable into the direct integral, 2) the spectrum is absolutely continuous, fills the whole real axis, and has multiplicity one or three, 3) the Lyapunov function, analytic on a three-sheeted Riemann surface, is constructed and researched, 4) the spectrum of multiplicity three is bounded and it is described in terms of some entire function (the discriminant).
Consider the operator $ T=-{d^2dx^2}+x^2+q(x)$ in $L^2(\mathbb{R})$, where real functions $q$, $q'$ and $\int_0^xq(s)ds$ are bounded. In particular, $q$ is periodic or almost periodic. The spectrum of $T$ is purely discrete and consists of the simple eigenvalues $\{μ_n\}_{n=0}^\infty$, $μ_n<μ_{n+1}$. We determine their asymptotics $μ_n = (2n+1) + (2π)^{-1}\int_{-π}^πq(\sqrt{2n+1}\sinθ)dθ+ O(n^{-1/3})$.
We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on $\ell^2(\Z)$ of the form $(Hψ)_n= a_{n-1}ψ_{n-1}+b_nψ_n+a_nψ_{n+1}$, where $a_n=a_{n+q}$ and $b_n=b_{n+q}$ are periodic sequences of real numbers. The results are based on a study of the quasimomentum $k(z)$ corresponding to $H$. We consider $k(z)$ as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent over the gaps with the normalised traces of powers of $H$.