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

TL;DR

This paper investigates the spectral properties of a fourth order differential operator with singular coefficients involving delta functions, analyzing how eigenvalues and eigenfunctions behave under approximation of these singularities.

Contribution

It introduces a framework for understanding the spectral behavior of operators with singular coefficients and constructs a limit operator based on approximation methods.

Findings

Eigenvalues and eigenfunctions exhibit specific asymptotic behaviors influenced by singular coefficients.

The limit operator depends on the approximation scheme of the singular coefficients.

Spectral properties are characterized for a family of smooth approximations to the singular operator.

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.