Spectral Properties of Fourth Order Differential Operators with Periodic and Antiperiodic Boundary Conditions (original) (raw)
Related papers
Simon Stevin, 2023
We study a fourth-order differential operator with a sign-alternating weight function with separated boundary conditions. For large values of the spectral parameter the asymptotics of the solutions to the corresponding differential equations is derived. The study of boundary conditions makes it possible to obtain an equation for eigenvalues of the considered differential operator. The indicator diagram of this equation is studied. The asymptotics of eigenvalues in various sectors of the indicator diagram is obtained.
2015
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 by 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. Copyright q 2009 O. A. Veliev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let LP2, P3,..., Pn be the differential operator generated in the space Lm2 −∞,∞ of vector-valued functions by the differential expression −inynx −in−2P2xyn−2x n∑ v3 Pvxyn−vx, 1 where n is an integer greater than 1 and Pkx, for k 2, 3,..., n, is the m × m matrix with the complex-valued summable entries pk,i,jx satisfying pk,i,jx ...
Mathematical Notes, 2009
The paper deals with the Sturm-Liouville operator Ly = −y ′′ + q(x)y, x ∈ [0, 1], generated in the space L 2 = L 2 [0, 1] by periodic or antiperiodic boundary conditions. Several theorems on Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to Sobolev space W p 1 [0, 1] with some integer p ≥ 0 and satisfy the conditions q (k) (0) = q (k) (1) = 0 for 0 ≤ k ≤ s − 1, where s≤ p. Let the functions Q and S be defined by the equalities Q(x) = x 0 q(t) dt, S(x) = Q 2 (x) and let q n , Q n , S n be the Fourier coefficients of q, Q, S with respect to the trigonometric system {e 2πinx } ∞ −∞. Assume that the sequence q 2n − S 2n + 2Q 0 Q 2n decreases not faster than the powers n −s−2. Then the system of eigen and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L 2 [0, 1] (provided that the eigenfunctions are normalized) if and only if the condition q 2n − S 2n + Q 0 Q 2n ≍ q −2n − S −2n + 2Q 0 Q −2n , n > 1, holds.
Spectral properties of some regular boundary value problems for fourth order differential operators
Central European Journal of Mathematics, 2012
In this paper we consider the problem ıv + 2 () + 1 () + 0 () = λ 0 < < 1 () (1) − (−1) σ () (0) + −1 =0 α () (0) = 0 = 1 2 3 (1) − (−1) σ (0) = 0 where λ is a spectral parameter; () ∈ L 1 (0 1), = 0 1 2, are complex-valued functions; α , = 1 2 3, = 0 − 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 spectral problem forms a basis in the space L (0 1), 1 < < ∞, when α 3 2 + α 1 0 = α 2 1 , () ∈ W 1 (0 1), = 1 2, and 0 () ∈ L 1 (0 1); moreover, this basis is unconditional for = 2.