Accurate Asymptotic Formulas for Eigenvalues and Eigenfunctions of a Boundary-Value Problem of Fourth Order

Abstract

In the present paper, we consider a nonself-adjoint fourth-order differential operator with the periodic boundary conditions. We compute new accurate asymptotic expression of the fundamental solutions of the given equation. Then, we obtain new accurate asymptotic formulas for eigenvalues and eigenfunctions.

Spectral asymptotics and basis properties of fourth order differential operators with regular boundary conditions

“…According to (14), e w 1 tends exponentially to zero and e w 4 tends exponentially to infinity. Consequently, by (21) and (29), the following equalities are valid:…”

“…It is proved in that the system of root functions of the differential operator forms an unconditional basis of the space L 2 (0,1), where q ( x ) ∈ L 1 (0,1) is an arbitrary complex‐valued function, γ is an arbitrary nonzero complex constant and σ = 0,1. Under the condition γ = 0 (periodic and antiperiodic boundary conditions) in and , necessary and sufficient conditions of unconditional basicity in L 2 (0,1) of the system of root functions of the earlier differential operator were obtained in terms of the Fourier coefficients of the potential q (x) (see also ). Some other interesting results about Riesz basicity of root functions of such operators with trigonometric polynomial potentials are obtained in .…”

“…forms an unconditional basis of the space L 2 .0, 1/, where q.x/ 2 L 1 .0, 1/ is an arbitrary complex-valued function, is an arbitrary nonzero complex constant and D 0, 1. Under the condition D 0 (periodic and antiperiodic boundary conditions) in [14] and [9], necessary and sufficient conditions of unconditional basicity in L 2 .0, 1/ of the system of root functions of the earlier differential operator were obtained in terms of the Fourier coefficients of the potential q .x/ (see also [17][18][19][20][21][22]). Some other interesting results about Riesz basicity of root functions of such operators with trigonometric polynomial potentials are obtained in [14,15].…”

Asymptotic Behavior of Eigenvalue and Eigenfunction of a Six Order Boundary Value Problem

“…It is well known that many researchers have investigated the spectral properties of the Sturm-Liouville operator generated by the separate boundary condition (Aigunov, 1996) and many researchers have found asymptotic formula for the Sturm-Liouville operator's eigenvalues and functions in the case of periodic andanti-periodic boundary conditions (Menken, 2010;Naimark, 1967;Moller and Zinsou, 2012;Jwamer and Aigounv, 2010;Aigounov and Tamila, 2009;Aigunov, 1996 andTamarkin, 1928). Many researchers have been interested in the ongoing Sturm-Liouville issue in recent years as we see N.B.…”

Refinement Asymptotic Formulas of Eigenvalues and Eigenfunctions of a Fourth Order Linear Differential Operator with Transmission Condition and Discontinuous Weight Function

“…We evaluate asymptotic formulas for the eigenfunctions. Hence, the characteristic equation W(λ) is significant for estimating eigenvalues and eigenfunctions of the problem (1)-(9) (see [14] (Theorem 4.1)).…”