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

TL;DR

This work analyzes positivity properties of the semigroup generated by the negative Dirichlet-to-Neumann operator on quantum graphs with dynamic vertex conditions. By reformulating boundary data through the Dirichlet-to-Neumann map $D_{ olinebreak \\lambda,{V_{ ext{o}}}}$ and deriving a Schur-complement matrix representation, the authors connect positivity regimes to the topology of the reduced graph $G_{ ext{o}}$ and to spectral parameters $ olinebreak \\lambda$, aided by limit theorems that approximate $D_{ olinebreak \\lambda,{V_{ ext{o}}}}$ by graph Laplacians when edge lengths are rationally independent. They classify when the semigroup is strongly positive, positive, eventually positive, or fails these properties, with distinct behavior for trees versus graphs containing cycles, and they extend the disk-domain phenomena to quantum graphs. The results illuminate how graph topology and edge-length arithmetics govern positivity, providing a toolkit based on Schur complements and ergodic-type limits to predict positivity regimes across lambda. These insights have implications for inverse problems, spectral partitioning, and semigroup stability on networks.

Abstract

We explore positivity properties of the semigroup generated by the negative of the Dirichlet-to-Neumann operator with real potential $λ$, 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 $λ\to \infty$. 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.

Figures (13)

• Figure 1.1: The graph $G$ and connected components of $G[{V_{\mathop{\mathrm{o}}\nolimits}}]$ (vertices in black) and $G[{V_{\mathop{\mathrm{i}}\nolimits}}]$ (vertices in white) with connection between $G[{V_{\mathop{\mathrm{i}}\nolimits}}]$ and $G[{V_{\mathop{\mathrm{o}}\nolimits}}]$ shown as dashed lines.
• Figure 1.2: The reduced graph $G_{\mathop{\mathrm{o}}\nolimits}$ of the graph $G$ from Figure \ref{['fig:graph-with-subgraphs']} with connections through ${V_{\mathop{\mathrm{i}}\nolimits}}$ shown as dashed lines. Note that $v_1$ and $v_2$ are considered to be connected by a single edge, likewise $v_4$ and $v_5$.
• Figure 2.1: Spectrum on an interval. Dashed lines indicate where the semigroup is (strongly) positive.
• Figure 2.2: Simple graph with two edges and outer boundary of two nodes.
• Figure 2.3: Spectrum of $D_{\lambda,{V_{\mathop{\mathrm{o}}\nolimits}}}$ corresponding to the graph in Figure \ref{['fig:nw-3-2']}.

Theorems & Definitions (52)

• Definition 1.1: Reduced graph
• Proposition 1.2
• Theorem 1.3
• Theorem 1.5
• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Definition 3.1: Maximal Laplace operator
• Definition 3.2: Trace map