Dirichlet to Neumann Maps for Infinite Quantum Graphs

TL;DR

This paper investigates the Dirichlet problem and Dirichlet to Neumann maps on infinite quantum graphs, revealing that for many boundary functions, the map produces Radon measures, advancing understanding of boundary value problems in quantum graph theory.

Contribution

It introduces a detailed analysis of Dirichlet to Neumann maps on infinite quantum graphs, including their measure-valued boundary behavior for a dense set of functions.

Findings

Dirichlet to Neumann map takes values in Radon measures on the boundary.

Analysis applies to a large class of infinite quantum graphs.

Provides new insights into boundary value problems in quantum graph settings.

Abstract

The Dirichlet problem and Dirichlet to Neumann map are analyzed for elliptic equations on a large collection of infinite quantum graphs. For a dense set of continuous functions on the graph boundary, the Dirichlet to Neumann map has values in the Radon measures on the graph boundary.