On the spectrum of second order differential operators with complex coefficients

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

Spectral Expansion for the Non-self-adjoint Differential Operators with the Periodic Matrix Coefficients

In this paper we construct the spectral expansion for the non-self-adjoint differential operators generated in the space of vektor functions by the ordinary differential expression of arbitrary order with the periodic matrix coefficients by using the essential spectral singularities, singular quasimomenta and the series with parenthesis.

关于具有周期矩阵系数的偶数阶微分算子的谱

On the spectrum of the differential operators of even order with periodic matrix coefficients

本文研究具有周期矩阵系数的偶数阶自伴微分算子的谱结构，给出了保证其谱中间隙数量有限的条件。

Spectral data asymptotics for the higher-order differential operators with distribution coefficients

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the case when, for the two boundary value problems, some coefficients of the differential expressions and of the boundary conditions coincide. We estimate the difference of their spectral data in this case. Although the asymptotic behaviour of spectral data is well-studied for differential operators with regular (integrable) coefficients, to the best of the author…

A general trace formula for the differential operator on a segment with the last coefficient perturbed by a finite signed measure

A first order trace formula is obtained for a regular differential operator perturbed by a finite signed measure multiplication operator.

On the Basis Property of the Root Functions of Differential Operators with Matrix Coefficients

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.

Differential equations for the recurrence coefficients limits for multiple orthogonal polynomials from a Nevai class

A limiting property of the nearest-neighbor recurrence coefficients for multiple orthogonal polynomials from a Nevai class is investigated. Namely, assuming that the nearest-neighbor coefficients have a limit along rays of the lattice, we describe it in terms of the solution of a system of partial differential equations. In the case of two orthogonality measures the differential equation becomes ordinary. For Angelesco systems, the result is illustrated numerically.

Linear differential operators with distribution coefficients of various singularity orders

In this paper, the linear differential expression of order $n \ge 2$ with distribution coefficients of various singularity orders is considered. We obtain the associated matrix for the regularization of this expression. Furthermore, we present the new statements of inverse spectral problems that consist in the recovery of differential operators with distribution coefficients from the Weyl matrix on the half-line and on a finite interval. The uniqueness theorems for these inverse problems are proved by developing the method of spectral mappings.

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

The basic purpose of the present paper is the full solutions of the inverse problem (i.e. a finding of necessary and sufficient conditions) for the operator with complex periodic coefficients.

On the self-adjoint differential operator with the periodic matrix coefficients

In this paper we consider the spectrum of the self-adjoint differential operator L generated by the differential expression of order n with the m by m periodic matrix coefficients, where n and m are respectively odd and even integers and n>1. We prove that the number of gaps in the spectrum of L is finite and find explicit estimation in term of coefficients for the number of the gaps. Moreover, we find a condition on the norms of the coefficients for which the spectrum is real axis. Besides we investigate the bands of the spectrum and prove that most of the real axis is overlapped by m bands.

The characterization problem for one class high order ordinary differential operators with periodic coefficients

The main purpose of the present work is solving the characterization problem which consist of identification of necessary and sufficient conditions on the scattering data ensuring that the reconstructed potential belongs to a particular class

On the point spectrum of some perturbed differential operators with periodic coefficients

Finiteness of the point spectrum of linear operators acting in a Banach space is investigated from point of view of perturbation theory. In the first part of the paper we present an abstract result based on analytical continuation of the resolvent function through continuous spectrum. In the second part, the abstract result is applied to differential operators which can be represented as a differential operator with periodic coefficients perturbed by an arbitrary subordinated differential operator.

Spectrum of a linear differential equation with constant coefficients

In this paper we compute the spectrum, in the sense of Berkovich, of an ultrametric linear differential equation with constant coefficients, defined over an affinoid domain of the analytic affine line $A_k^{1,an}$. We show that it is a finite union of either closed disks or topological closures of open disks and that it satisfies a continuity property.

On a differential operator with absent spectrum

In this paper, we consider spectral problem for the nth order ordinary differential operator with degenerate boundary conditions. We construct a nontrivial example of boundary value problem which has no eigenvalues.

The Second Regularized Trace of Even Order Differential Operators with Operator Coefficient

In this paper, we investigate the spectrum of the self adjoint differential operator with operator coefficitent in a separable Hilbert space. We also derive asymptotic formulas for the sum of eigenvalues of this operator.

Absence of eigenvalues for integro-differential operators with periodic coefficients

Applying perturbation theory methods, the absence of the point spectrum for some nonselfadjoint integro-differential operators is investigated. The considered differential operators are of arbitrary order and act in either $\mathbf{L}_{p}(\mathbb{R}_{+})$ or $\mathbf{L}_{p}(\mathbb{R}) (1\leq p<\infty)$. As an application of general results, new spectral properties of the perturbed Hill operator are derived.

Optimal semiclassical spectral asymptotics for differential operators with non-smooth coefficients

We consider differential operators defined as Friedrichs extensions of quadratic forms with non-smooth coefficients. We prove a two term optimal asymptotic for the Riesz means of these operators and thereby also reprove an optimal Weyl law under certain regularity conditions. The methods used are then extended to consider more general admissible operators perturbed by a rough differential operator and obtaining optimal spectral asymptotics again under certain regularity conditions. For the Weyl law we assume the coefficients are differentiable with Hölder continuous derivatives and for the Rie…

On commuting ordinary differential operators with polynomial coefficients corresponding to spectral curves of genus two

The group of automorphisms of the first Weyl algebra acts on commuting ordinary differential operators with polynomial coefficient. In this paper we prove that for fixed generic spectral curve of genus two the set of orbits is infinite.

Inverse Indefinite Spectral Problem for Second Order Differential Operator with Complex Periodoc Coefficients

The inverse problem for the Sturm- Liouville operator with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is studied. We give formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.

Fokas diagonalization of piecewise constant coefficient linear differential operators on finite intervals and networks

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, includi…

On the essential spectrum of differential operators over geometrically finite orbifolds

We discuss the essential spectrum of essentially self-adjoint elliptic differential operators of first order and of Laplace type operators on Riemannian vector bundles over geometrically finite orbifolds.

Spectrum of a linear differential equation over a field of formal power series

In this paper we associate to a linear differential equation with coefficients in the field of Laurent formal power series a new geometric object, a spectrum in the sense of Berkovich. We will compute this spectrum and show that it contains interesting informations about the equation.

关于具有PT对称周期矩阵系数的微分算子实谱的研究

On the real spectrum of differential operators with PT-symmetric periodic matrix coefficients

本文研究具有PT对称周期矩阵系数的n阶微分算子T的谱，针对n为奇数且m为偶数、n为偶数且m为任意正整数两种剩余情况，给出了使谱包含大实数和正数的系数条件。

Multiplicative matrix-valued functionals and the continuity properties of semigroups correspondings to partial differential operators with matrix-valued coefficients

We define and examine certain matrix-valued multiplicative functionals with local Kato potential terms and use probabilistic techniques to prove that the semigroups of the corresponding partial differential operators with matrix-valued coefficients are spatially continuous and have a jointly continuous integral kernel. These partial differential operators include Yang-Mills type Hamiltonians and Pauli type Hamiltonians, with "electrical" potentials that are elements of the matrix-valued local Kato class.