On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems and applications

Abstract

Abstract. The paper is concerned with the completeness property of root functions of general boundary value problems for n × n first order systems of ordinary differential equations on a finite interval. In comparison with the recent paper [45] we substantially relax the assumptions on boundary conditions guarantying the completeness of root vectors, allowing them to be nonweakly regular and even degenerate. Emphasize that in this case the completeness property substantially depends on the values of a potential matrix at the endpoints of the interval.It is also shown that the system of root vectors of the general n × n Dirac type system subject to certain boundary conditions forms a Riesz basis with parentheses. We also show that arbitrary complete dissipative boundary value problem for Dirac type operator with a summable potential matrix admits the spectral synthesis in L 2 ([0, 1]; C n ). Finally, we apply our results to investigate completeness and the Riesz basis property of the dynamic generator of spatially non-homogenous damped Timoshenko beam model.

Asymptotics of determinants of 4‐th order operators at zero

“…There are many results about the determinants for different classes of operators, see, e.g., Gesztesy and Makarov [10], Lunyov and Malamud [21], Malamud and Neidhardt [23], see also the references therein. The fundamental matrix uniquely determines coefficients of the first order differential systems, see Malamud [22].…”

Completeness Property of One-Dimensional Perturbations of Normal and Spectral Operators Generated by First Order Systems

“…Proof. (i) Due to [26,Proposition 5.9] the system of root functions of the operator L forms a Riesz basis with parentheses in L 2 ([0, 1]; C 2 ), where each block is constituted by the root subspaces corresponding to the eigenvalues of L that are mutually ε-close with respect to the sequence Ψ := {−ϕ 1 , −ϕ 2 , π − ϕ 1 , π − ϕ 2 }. Here ϕ j = arg b j , j ∈ {1, 2}, and ε > 0 is sufficiently small.…”

“…In the recent paper [26] it was established the Riesz basis property with parentheses for system (1.1) subject to various classes of boundary conditions with a potential Q(·) ∈ L ∞ ([0, 1]; C n×n ).…”

The Completeness of System of Eigenfunctions of 1D Dirac Operators

“…The recent publications [17]- [20] devoted to the questions of completeness and spectral synthesis for general nn  first order systems of ODE (see also references therein). In [17], [18], [20] it was shown that the completeness property for some classes of boundary conditions essentially depends on boundary values of the potential matrix and explicit conditions of the completeness were found. In particular, in [20], an example of incomplete dissipative 22  Dirac operator was constructed.…”

“…In particular, in [20], an example of incomplete dissipative 22  Dirac operator was constructed. It was shown in [18], [19] that the resolvent of any complete dissipative Dirac type operator admits the spectral synthesis. Moreover, explicit conditions of the dissipativity and completeness of such operators were found.…”