Applications of a Class of Herglotz Operator Pencils

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 spectra of these operator pencils have 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.