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Evgeny Korotyaev

Publications and source records attributed to Evgeny Korotyaev.

At least 19 recordsLinked to original sources
arXiv2026

Scattering for anisotropic potentials

We consider the scattering for the operator $H=H_o+V$, where the unperturbed operator $H_o$ is not assumed to be elliptic and the potential $V$ is anisotropic. Under some conditions on $H_o$ and $V$ we show that the wave operators for $H_o, H$ exist and are complete, $H$ has no singular continuous spectrum and the eigenvalues of $H$ can accumulate only to zero. For stronger conditions on $V$ the operator $H$ has finite number of eigenvalues only. Moreover, these results are applied to the invariance principle and for time-dependent potentials.

math-ph↗
arXiv2026

Inverse problem for the divisor of the good Boussinesq equation

A third-order operator with periodic coefficients is an L-operator in the Lax pair for the Boussinesq equation on a circle. The projection of the divisor of the Floquet solution poles for this operator coincides with the spectrum of the three-point Dirichlet problem. The sign of the norming constant of the three-point problem determines the sheet of the Riemann surface on which the pole lies. We solve the inverse problem for a third-order operator with three-point Dirichlet conditions when the spectrum and norming constant are known. We construct a mapping from the set of coefficients to the set of spectral data and prove that this mapping is an analytic bijection in the neighborhood of zero.

math-ph↗
arXiv2025

Estimates, asymptotics and trace formulas for periodic vector NLS equations, II

We consider a first order operator with a smooth periodic 3x3 matrix potential on the real line. It is the Lax operator for the periodic vector NLS equation. Its spectrum covers the real line and it is union of the spectral bands of multiplicity 3, separated by intervals (gaps) of multiplicity 1. We prove and describe the following: \\ $\cdot$ The geometry of the Riemann surface and its branch points. \\ $\cdot$ The asymptotics of branch points are determined and they are real at high energy. \\ $\cdot$ Trace formulas for integral of motions, including the Hamiltonian of the NLS equation. \\ $\cdot$ Estimates of the Hamiltonian in terms of gap lengths. The proof is based on the analysis of averaged quasi-momentum as a conformal mapping of the upper half plane on the domain on the upper half plane and on the asymptotics of the monodromy matrix and multipliers at high energy.

math-ph↗
arXiv2025

Inverse problems for ZS-operators and their isomorphisms

Consider two inverse problems for ZS-operators problems on the unit interval. It means that there are two corresponding mappings $F, f$ from a Hilbert space of potentials $H$ into their spectral data. They are called isomorphic if $F$ is a composition of $f$ and some isomorphism $U$ of $H$ onto itself. We consider isomorphic inverse problems for ZS-operators on the unit interval under basic boundary conditions and on the circle. The proof is based on the non-linear analysis and properties of the 4-spectra mapping constructed in our paper.

math.SP↗
arXiv2024

Spectral theory for periodic vector NLS equations

We consider a first order operator with a periodic 3x3 matrix potential on the real line. This operator appears in the problem of the periodic vector NLS equation. The spectrum of the operator covers the real line, it is union of the spectral bands of multiplicity 3, separated by spectral intervals of multiplicity 1. The main results of this work are the following: The Lyapunov function on the corresponding 2 or 3-sheeted Riemann surface is described. Necessary and sufficient conditions are given when the Riemann surface is 2-sheeted. The asymptotics of 2-periodic eigenvalues are determined. One constructs an entire function, which is positive on the spectrum of multiplicity 3 and is negative on its gaps. The Borg type results about inverse problems are solved.

math-ph↗
arXiv2024

Asymptotics of the divisor for the good Boussinesq equation

We consider a third order operator under the three-point Dirichlet condition. Its spectrum is the so-called auxiliary spectrum for the good Boussinesq equation, as well as the Dirichlet spectrum for the Schrödinger operator on the unit interval is the auxiliary spectrum for the periodic KdV equation. The auxiliary spectrum is formed by projections of the points of the divisor onto the spectral plane. We estimate the spectrum and the corresponding norming constants in terms of small operator coefficients. This work is the first in a series of papers devoted to solving the inverse problem for the Boussinesq equation.

math-ph↗
arXiv2024

Inverse problem for the L-operator in the Lax Pair of the Boussinesq equation on the circle

We consider a third-order non-self-adjoint operator, which is an $L$-operator in the Lax pair for the Boussinesq equation on the circle. We construct a mapping from the set of operator coefficients to the set of spectral data, similar to the corresponding mapping for the Hill operator constructed by E. Korotyaev. We prove that in a neighborhood of zero our mapping is analytic and one-to-one.

math-ph↗
arXiv2024

Inverse problem for 3-rd order operators under the 3-point Dirichlet conditions

We solve an inverse problem for a third order differential operator under the 3-point Dirichlet conditions. The third-order operator is an $L$-operator in the Lax pair for the good Boussinesq equation. We construct the mapping from the set of the coefficients to the set of spectral data. This mapping is an analytic bijection on a neighborhood of the zero.

math-ph↗
arXiv2024

Mc'Kean's transformation for 3-rd order operators

We consider a non-self-adjoint third order operator $(y''+py)'+py'+qy$ with 1-periodic coefficients $p,q$. This operator is the L-operator in the Lax pair for the good Boussinesq equation on the circle. In 1981, McKean introduced a transformation that reduces the spectral problem for this operator to a spectral problem for the Hill operator with a potential that depends analytically on the energy. In the present paper we are studying this transformation.

math-ph↗
arXiv2024

Resonances for vector-valued Jacobi operators on half-lattice

We study resonances for Jacobi operators on the half lattice with matrix valued coefficient and finitely supported perturbations. We describe a forbidden domain, the geometry of resonances and their asymptotics when the main coefficient of the perturbation (determining its length of support) goes to zero. Moreover, we show that\\ 1) Any sequence of points on the complex plane can be resonances for some Jacobi operators. In particular, the multiplicity of a resonance can be any number.\\ 2) The Jost determinant coincides with the Fredholm determinant up to the constant.\\ 3) The S-matrix on the a.c spectrum determines the perturbation uniquely.\\ 5) The value of the Jost matrix at any finite sequence of points on the a.c spectrum determine the Jacobi matrix uniquely. The length of this sequence is equals to the upper point of the support perturbation.

math.SP↗
arXiv2023

Trace formulas for magnetic Schrödinger operators on periodic graphs and their applications

We consider Schrödinger operators with periodic magnetic and electric potentials on periodic discrete graphs. The spectrum of such operators consists of a finite number of bands. We determine trace formulas for the magnetic Schrödinger operators. The traces of the fiber operators are expressed as finite Fourier series of the quasimomentum. The coefficients of the Fourier series are given in terms of the magnetic fluxes, electric potentials and cycles in the quotient graph from some specific cycle sets. Using the trace formulas we obtain new lower estimates of the total bandwidth for the magnetic Schrödinger operator in terms of geometric parameters of the graph, magnetic fluxes and electric potentials. We show that these estimates are sharp.

math.SP↗
arXiv2023

Resonances and inverse problems for energy-dependent potentials on the half-line

We consider Schrödinger equations with linearly energy-depending potentials which are compactly supported on the half-line. We first provide estimates of the number of eigenvalues and resonances for such complex-valued potentials under suitable regularity assumptions. Then, we consider a specific class of energy-dependent Schrödinger equations without eigenvalues, defined with Miura potentials and boundary conditions at the origin. We solve the inverse resonance problem in this case and describe sets of iso-resonance potentials and boundary condition parameters. Our strategy consists in exploiting a correspondance between Schrödinger and Dirac equations on the half-line. As a byproduct, we describe similar sets for Dirac operators and show that the scattering problem for Schrödinger equation or Dirac operator with an arbitrary boundary condition can be reduced to the scattering problem with the Dirichlet boundary condition.

math-ph↗
arXiv2023

Isomorphic inverse problems

Consider two inverse problems for Sturm-Liouville problems on the unit interval. It means that there are two corresponding mappings $F, f$ from a Hilbert space of potentials $H$ into their spectral data. They are called isomorphic if $F$ is a composition of $f$ and some isomorphism $U$ of $H$ onto itself. A isomorphic class is a collection of inverse problems isomorphic to each other. We consider basic Sturm-Liouville problems on the unit interval and on the circle and describe their isomorphic classes of inverse problems. For example, we prove that the inverse problems for the case of Dirichlet and Neumann boundary conditions are isomorphic. The proof is based on the non-linear analysis.

math-ph↗
arXiv2022

Two-sided estimates of total bandwidth for Schrödinger operators on periodic graphs

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We obtain two-sided estimates of the total bandwidth for the Schrödinger operators in terms of geometric parameters of the graph and the potentials. In particular, we show that these estimates are sharp. It means that these estimates become identities for specific graphs and potentials. The proof is based on the Floquet theory and trace formulas for fiber operators. The traces are expressed as finite Fourier series of the quasimomentum with coefficients depending on the potentials and cycles of the quotient graph from some specific cycle sets. In order to obtain our results we estimate these Fourier coefficients in terms of geometric parameters of the graph and the potentials.

math.SP↗
arXiv2022

Trace formulas for Schrödinger operators on periodic graphs

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We determine trace formulas for the Schrödinger operators. The proof is based on the decomposition of the Schrödinger operators into a direct integral and a specific representation of fiber operators. The traces of the fiber operators are expressed as finite Fourier series of the quasimomentum. The coefficients of the Fourier series are given in terms of the potentials and cycles in the quotient graph from some specific cycle sets. We also present the trace formulas for the heat kernel and the resolvent of the Schrödinger operators and the determinant formulas.

math.SP↗
arXiv2022

Inverse problems for Jacobi operators with finitely supported perturbations

We solve the inverse problem for Jacobi operators on the half lattice with finitely supported perturbations, in particular, in terms of resonances. Our proof is based on the results for the inverse eigenvalue problem for specific finite Jacobi matrices and theory of polynomials. We determine forbidden domains for resonances and maximal possible multiplicities of real and complex resonances.

math.SP↗
arXiv2021

Inverse scattering on the quantum graph for graphene

We consider the inverse scattering on the quantum graph associated with the hexagonal lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the S-matrix for all energies in any given open set in the continuous spectrum determines the potentials.

math-ph↗