On the completeness of systems of eigenfunctions and associated functions of differential operators of orders 2 − ε and 1 − ε

Agibalova, Anna V.

doi:10.1007/s10958-011-0309-7

Cited by 7 publications

(5 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In his paper [6] M. M. Dzhrbashian wrote, that "the question about the completeness of the systems of eigenfunctions of the operator A ρ or a finer question about whether these systems compose a basis in L 2 (0, 1), has a certain interest but its solving is apparently associated with significant analytic difficulties". The questions of the completeness of the systems of eigenfunctions and associated functions for similar problems were studied by A. V. Agibalova in [7,8]. Undoubtedly, we shall note the fundamental results of M. M. Malamud and L. L. Oridoroga [9][10][11][12] obtained in this direction.…”

Section: Resultsmentioning

confidence: 96%

On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

Aleroev

2019

Axioms

View full text Add to dashboard Cite

The present paper is devoted to the spectral analysis of operators induced by fractional differential equations and boundary conditions of Sturm-Liouville type. It should be noted that these operators are non-self-adjoint. The spectral structure of such operators has been insufficiently explored. In particular, a study of the completeness of systems of eigenfunctions and associated functions has begun relatively recently. In this paper, the completeness of the system of eigenfunctions and associated functions of one class of non-self-adjoint integral operators corresponding boundary value problems for fractional differential equations is established. The proof is based on the well-known Theorem of M.S. Livshits on the spectral decomposition of linear non-self-adjoint operators, as well as on the sectoriality of the fractional differentiation operator. The results of Dzhrbashian-Nersesian on the asymptotics of the zeros of the Mittag-Leffler function are used.

show abstract

Section: Resultsmentioning

confidence: 96%

On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

Aleroev

2019

Axioms

View full text Add to dashboard Cite

show abstract

“…Therefore the question is often raised in researches. For example, the completeness of the generalized eigenfunctions of the differential operators of fractional orders is investigated in [1]. In [2] is studied the completeness of the set of the eigenfunctions corresponding to a system of two simultaneous Sturm-Liouville problems coupled by means of two different spectral parameters.…”

Section: Introductionmentioning

confidence: 99%

On completeness of the system of eigenfunctions of the Dirichlet problem for Lame equations in the strip

Pleshchinskii¹,

Stekhina²,

Tumakov³

2015

astp

View full text Add to dashboard Cite

show abstract

“…Remark 1. 3 We emphasize that Krein's Theorem is valid when an operator A is a "weak" perturbation of a selfadjoint operator A . In other words, the completeness of the system of the root vectors of A holds if and only if the spectra of the operators A and A have "distributed densities" of the same order (see [10]).…”

Section: Introductionmentioning

confidence: 99%

“…We also mention some very recent papers , , , , . The first paper is devoted to the Riesz basis property of the generalized eigenfunctions of the perturbed harmonic oscillator operator.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the completeness of root vectors generated by systems of coupled hyperbolic equations

Shubov

2014

Mathematische Nachrichten

View full text Add to dashboard Cite

The paper is the second in a set of two papers, which are devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non‐selfadjoint unbounded matrix differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary‐value problem (IBVP) for a non‐homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with a one‐parameter family of dissipative boundary conditions at one end and with a heavy load at the other end; this filament problem is treated for two cases of the boundary parameter: non‐singular and singular; (c) IBVP for a three‐dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping; (d) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (e) IBVP for a coupled Euler‐Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending‐torsion vibration model); (f) IBVP for two coupled Timoshenko beams model, which is currently accepted as an appropriate model describing vibrational behavior of a longer double‐walled carbon nanotube. Problems (a)–(c) have been discussed in the first paper of the aforementioned set. Problems (d)–(f) are discussed in the present paper.

show abstract

On the completeness of systems of eigenfunctions and associated functions of differential operators of orders 2 − ε and 1 − ε

Cited by 7 publications

References 2 publications

On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

On completeness of the system of eigenfunctions of the Dirichlet problem for Lame equations in the strip

On the completeness of root vectors generated by systems of coupled hyperbolic equations

Contact Info

Product

Resources

About