Spectral Properties of the Differential Operators of the Fourth-Order with Eigenvalue Parameter Dependent Boundary Condition

Abstract

We consider the fourth-order spectral problem 𝑦(4)(𝑥)−(𝑞(𝑥)𝑦′(𝑥))=𝜆𝑦(𝑥),𝑥∈(0,𝑙) with spectral parameter in the boundary condition. We associate this problem with a selfadjoint operator in Hilbert or Pontryagin space. Using this operator-theoretic formulation and analytic methods, we investigate locations (in complex plane) and multiplicities of the eigenvalues, the oscillation properties of the eigenfunctions, the basis properties in 𝐿𝑝(0,𝑙), 𝑝∈(1,∞), of the system of root functions of this problem.