Necessary and sufficient conditions for the inverse problem of one class ordinary differential operators with complex periodic coefficients

On the Differential Operators with Periodic Matrix Coefficients

In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and quasiperiodic boundary conditions. Then using these asymptotic formulas, we find conditions on the coefficients for which the number of gaps in the spectrum of the self-adjoint differential operator with the periodic matrix coefficients is finite.

Spectral Expansion Series with Parenthesis for the Nonself-adjoint Periodic Differential Operators

In this paper we construct the spectral expansion for the differential operator generated in all real line by ordinary differential expression of arbitrary order with periodic complex-valued coefficients by introducing new concepts as essential spectral singularities and singular quasimomenta and using the series with parenthesis. Moreover, we find a criteria for which the spectral expansion coincides with the Gelfand expansion for the self-adjoint case.

Uniform Convergence of the Spectral Expansion for a Differential Operator with Periodic Matrix Coefficients

In this paper, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and the quasiperiodic boundary conditions. Using these asymptotic formulas, we find conditions on the coefficients for which the root functions of this operator form a Riesz basis. Then we obtain the uniformly convergent spectral expansion of the differential operators with the periodic matrix coefficients

线性周期常微分方程和延迟微分方程的切比雪夫配置法：后验估计

Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates

提出一种针对线性ODE和DDE问题的切比雪夫配置法，通过后验估计评估解精度，并推广到DDE系统以可靠估计稳定性，主要贡献是证明了单值算子特征值的后验估计定理。

$2$-modified characteristic Fredholm determinants, Hill's method, and the periodic Evans function of Gardne

Using the relation established by Johnson--Zumbrun between Hill's method of aproximating spectra of periodic-coefficient ordinary differential operators and a generalized periodic Evans function given by the $2$-modified characteristic Fredholm determinant of an associated Birman--Schwinger system, together with a Volterra integral computation introduced by Gesztesy--Makarov, we give an explicit connection between the generalized Birman--Schwinger-type periodic Evans function and the standard Jost function-type periodic Evans function defined by Gardner in terms of the fundamental solution of …

Characterization of the periodic and antiperiodic spectra of non-self-adjoint Dirac operators

The necessary and sufficient conditions are given for a sequence of complex numbers to be the periodic (or antiperiodic) spectrum of non-self-adjoint Dirac operator.

The discrete spectrum for complex perturbations of periodic Jacobi matrices

We study spectrum inclusion regions for complex Jacobi matrices which are compact perturbations of real periodic Jacobi matrices. The condition sufficient for the lack of discrete spectrum for such matrices is given

具有分段常数偏差变元的非自治线性周期微分方程的 Floquet-Lyapunov 理论

A Floquet-Lyapunov Theory for nonautonomous linear periodic differential equations with piecewise constant deviating arguments

本文将经典 Floquet-Lyapunov 定理推广到含广义分段常数变元的 ω-周期非自治线性脉冲及非脉冲微分方程，证明其与自治线性常微分方程运动学相似，并用示例验证结果有效性。

Periodic points of quasianalytic Hamiltonian billiards

We study absolutely periodic points and trajectories of Hamiltonian systems. Our main result is a necessary and sufficient for a Hamiltonian system to have the following property: if there exists one absolutely periodic trajectory then all trajectories are periodic with the same period.

On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials

The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schrödinger operator with a periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials satisfy this condition. Separable periodic potentials satisfy it, and hence in dimensions two and three Schrödinger…

Spectral and resonance problem for perturbations of periodic Jacobi operators

Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range or exponentially decaying perturbation of a periodic Jacobi operator. As a corollary we can fully solve the inverse resonance problem: given resonances and eigenvalues we can recover the spectral measure of the Jacobi operator; we provide necessary and sufficient conditions under which such an operator exists and is unique; and we show that the inverse resonance problem is stable under small perturbations.