Positivity properties of the Dirichlet-to-Neumann operator on graphs

TL;DR

This paper investigates the positivity properties of the semigroup generated by the Dirichlet-to-Neumann operator on quantum graphs, revealing how these properties depend on graph topology, edge lengths, and potential parameter.

Contribution

It characterizes how the positivity of the semigroup varies with graph topology, edge lengths, and potential, providing new insights into quantum graph boundary operators.

Findings

Semigroup alternates between positive, eventually positive, and non-positive depending on parameters.

Topology influences positivity behavior, linked to a reduced graph structure.

Results depend on rational independence of edge lengths and potential value.

Abstract

We explore positivity properties of the semigroup generated by the negative of the Dirichlet-to-Neumann operator with real potential $\lambda$, defined on a subset of the vertices of a quantum graph. We show that for rationally independent edge lengths and suitable graph topologies, this semigroup will alternate between being positive, eventually positive without being positive (that is, positive only for sufficiently large times), and not even eventually positive, as $\lambda \to \infty$. For other graph topologies, the semigroup will alternate between being positive and not eventually positive. The topological conditions are related to a reduced graph which is a schematic map of the connections between the vertices on which the Dirichlet-to-Neumann operator acts.