The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions

TL;DR

This paper investigates the spectrum of second derivative operators on vector-valued functions with general self-adjoint boundary conditions, introducing new methods that connect spectral properties with graph theory.

Contribution

It presents two approaches to analyze eigenvalues of self-adjoint elliptic problems, including an abstract rank condition and applications to graph-related boundary conditions.

Findings

Eigenvalue spectrum symmetry results

Application of rank condition to boundary problems

Connections between spectrum and graph theory

Abstract

We consider a large class of self-adjoint elliptic problem associated with the second derivative acting on a space of vector-valued functions. We present two different approaches to the study of the associated eigenvalues problems. The first, more general one allows to replace a secular equation (which is well-known in some special cases) by an abstract rank condition. The latter seems to apply particularly well to a specific boundary condition, sometimes dubbed "anti-Kirchhoff" in the literature, that arise in the theory of differential operators on graphs; it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities. In either case our results yield, among other, some results on the symmetry of the spectrum.