Oscillation method in the spectral problem for a fourth order differential operator with a self-similar weight (Q2786979)

The paper is aimed to investigate the selfadjoint problem NEWLINENEWLINE\[NEWLINE y^{(4)}-\lambda \rho y=0,NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINE \left(U-1\right)y^{\vee}+i\left(U+1\right)y^{\wedge}=0,NEWLINE\]NEWLINE NEWLINEwhere \(\rho \in W^{-1}_{2}[0,1]\) is a nonnegative generalized weight function, \(U\) is a \(4\times4\) unitary matrix of boundary conditions and \(y^{\vee}\) and \(y^{\wedge}\) are numerical vectors NEWLINENEWLINE\[NEWLINEy^{\wedge}=\left(y(0), y'(0), y(1), y'(1)\right)^{T},NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINEy^{\vee}=\left(-y'''(0), y''(0), y'''(1), -y''(1) \right)^{T}.NEWLINE\]NEWLINE NEWLINEIn the paper, the author presents some information about the oscillation of eigenfunctions of the problems under consideration. It is also considered that the spectral periodicity phenomenon and the properties of spectral asymptotics implied by it. Moreover, some results of numerical experiments that illustrate the theoretical results obtained are presented.