Spectral and oscillation properties for a linear pencil of fourth-order differential operators

TL;DR

This paper investigates the spectral and oscillation characteristics of a linear pencil of fourth-order differential operators, revealing properties of eigenvalues and eigenfunctions related to their zeros.

Contribution

It provides new insights into the simplicity of negative eigenvalues and the zero distribution of eigenfunctions for a specific class of fourth-order differential operator pencils.

Findings

Negative eigenvalues are simple.

Eigenfunctions corresponding to negative eigenvalues have a specific number of zeros.

The spectral and oscillation properties are characterized for the given differential operators.

Abstract

The present paper deals with the spectral and the oscillation properties of a linear pencil $A-\lambda B$. Here $A$ and $B$ are linear operators generated by the differential expressions $(py")"$ and $-y"+ cry$, respectively. In particular, it is shown that the negative eigenvalues of this problem are simple and the corresponding eigenfunctions $y_{-n}$ have $n-1$ zeros in $(0,1)$.