Non-self-adjoint graphs

TL;DR

This paper studies non-self-adjoint Laplace operators on finite metric graphs, analyzing their spectral properties, basis of projectors, and conditions for similarity to self-adjoint operators, with practical methods for relating boundary condition matrices.

Contribution

It introduces a simple approach to relate similarity transforms of Laplacians to elementary matrix similarity transforms for certain graph boundary conditions.

Findings

Spectral properties of non-self-adjoint Laplacians are characterized.

Conditions for the existence of a Riesz basis of projectors are identified.

A method to relate similarity transforms to boundary condition matrices is developed.

Abstract

On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity transforms to self-adjoint Laplacians. Among other things, we describe a simple way how to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions.