On spectral properties of the fourth order differential operator with singular coefficients

Abstract

A formal fourth order differential operator with a singular coefficient that is a linear combination of the Dirac delta-function and its derivatives is considered. The asymptotic behavior of spectra and eigenfunctions of a family of differential operators with smooth coefficients approximating the singular coefficients is studied. We explore how behavior of eigenvalues and eigenfunctions is influenced by singular coefficients. The limit operator is constructed and is shown to depend on a type of approximation of singular coefficients.