The Dirichlet-to-Neumann operator for quantum graphs

TL;DR

This paper demonstrates that for certain quantum graphs, any symmetric real matrix can be realized as the Dirichlet-to-Neumann operator, expanding understanding of boundary value problems on metric graphs.

Contribution

It establishes a realization theorem showing the flexibility of the Dirichlet-to-Neumann operator on quantum graphs with boundary.

Findings

Any symmetric real matrix can be realized as the Dirichlet-to-Neumann operator.

The result applies to compact, connected metric graphs with boundary.

It advances the spectral theory of quantum graphs.

Abstract

For a compact, connected metric graphs with a boundary that consists of $k$ vertices, we prove that an arbitrary symmetric $k\times k$ matrix with real entries can be realized as the Dirichlet-to-Neumann operator for the Laplacian plus a constant.