Applications of a class of Herglotz operator pencils

TL;DR

This paper studies a special class of operator pencils with real eigenvalues, proving a new oscillation theorem and applying it to biological models, revealing spectral properties and eigenvalue distributions.

Contribution

It introduces a novel oscillation theorem for a class of operator pencils with polynomial eigenvalue dependence and applies it to biological models.

Findings

Eigenvalues are real and can be partitioned into k disjoint intervals.

Spectrum has finite accumulation points, indicating non-compact resolvents.

Application to epidemic and dispersal models demonstrates practical relevance.

Abstract

We identify a class of operator pencils, arising in a number of applications, which have only real eigenvalues. In the one-dimensional case we prove a novel version of the Sturm oscillation theorem: if the dependence on the eigenvalue parameter is of degree $k$ then the real axis can be partitioned into a union of $k$ disjoint intervals, each of which enjoys a Sturm oscillation theorem: on each interval there is an increasing sequence of eigenvalues that are indexed by the number of roots of the associated eigenfunction. One consequence of this is that it guarantees that the spectrum of these operator pencils has finite accumulation points, implying that the operators do not have compact resolvents. As an application we apply this theory to an epidemic model and several species dispersal models arising in biology.