Uniform convergence of the spectral expansion for a differential operator with periodic matrix coefficients.

Veliev, O.A.

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We study a Sturm-Liouville problem containing a spectral parameter in the boundary conditions. We associate to this problem a self-adjoint operator in a Pontryagin space Π₁. Using this operator-theoretic formulation and analytic methods, we study the asymptotic behavior of the eigenvalues under the variation of a large physical parameter in the boundary conditions. The spectral analysis is applied to investigate the well-posedness and stability of the wave equation of a string. ...

Spectral properties of some regular boundary value problems for fourth order differential operators

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In this paper we consider the problem $\begin{matrix} y^{i v} + p_{2} (x) y^{''} + p_{1} (x) y^{'} + p_{0} (x) y = λ y, 0 < x < 1, \\ y^{(s)} (1) - {(- 1)}^{σ} y^{(s)} (0) + \sum_{l = 0}^{s - 1} α_{s, l} y^{(l)} (0) = 0, s = 1, 2, 3, \\ y (1) - {(- 1)}^{σ} y (0) = 0, \end{matrix}$ where λ is a spectral parameter; p j (x) ∈ L 1(0, 1), j = 0, 1, 2, are complex-valued functions; α s;l, s = 1, 2, 3, $l = \bar{0, s - 1}$ , are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α 3,2 + α 1,0 ≠ α 2,1. It is proved that the system of root functions of this...