Unbounded quantum graphs with unbounded boundary conditions

TL;DR

This paper characterizes all lower bounded self-adjoint Laplace operators on metric graphs with unbounded boundary conditions, providing a comprehensive description of their boundary conditions and associated quadratic forms.

Contribution

It introduces a complete framework for describing local self-adjoint Laplace operators on metric graphs with unbounded boundary conditions using boundary projections and operators.

Findings

Characterization of all lower bounded self-adjoint Laplacians

Description of boundary conditions via projections and operators

Determination of associated quadratic forms

Abstract

We consider metric graphs with a uniform lower bound on the edge lengths but no further restrictions. We discuss how to describe every local self-adjoint Laplace operator on such graphs by boundary conditions in the vertices given by projections and self-adjoint operators. We then characterize the lower bounded self-adjoint Laplacians and determine their associated quadratic form in terms of the operator families encoding the boundary conditions.