Neumann Domains on Quantum Graphs

TL;DR

This paper introduces the concept of Neumann domains on quantum graphs, establishing foundational bounds and properties, and develops a probabilistic framework for analyzing spectral functions on these graphs.

Contribution

It defines Neumann points and domains on quantum graphs, proves bounds and properties, and introduces a probabilistic approach for spectral analysis.

Findings

Bounds on the number of Neumann points

Properties of the probability distribution of Neumann points

Bounds and features of Neumann domain properties

Abstract

The Neumann points of an eigenfunction $f$ on a quantum (metric) graph are the interior zeros of $f'$. The Neumann domains of $f$ are the sub-graphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of the well-studied nodal points and nodal domains. We prove some foundational results in this field: bounds on the number of Neumann points and properties of the probability distribution of this number. Two basic properties of Neumann domains are presented: the wavelength capacity and the spectral position. We state and prove bounds on those as well as key features of their probability distributions. To rigorously investigate those probabilities, we establish the notion of random variables for quantum graphs. In particular, we provide conditions for considering spectral functions of quantum graphs as random variables with respect to the natural density on $\mathbb{N}$.