Self-adjoint extensions of singular Sturm-Liouville operators on graphs and Weyl's law

TL;DR

This work develops a unified framework for self-adjoint extensions of singular Sturm-Liouville operators on graphs by linking quantum self-adjointness to the completeness of the Hamiltonian flow of the principal symbol $s(x,\xi)=-p(x)\xi^2$. It extends the classical von Neumann theory to graphs, showing that singular vertices act as effective potentials and that local Kirchhoff-type boundary conditions yield a broad class of self-adjoint realizations. A Weyl-type law is established for the eigenvalue counting function $N(\lambda,P)$ in graphs with positive principal symbol, incorporating contributions from regular and LC edges as well as LC endpoints. The results unify and generalize known spectral laws for regular quantum graphs and provide explicit boundary-condition constructions via GK/Naimark theory, with practical implications for spectral problems on networks and quantum graphs with singularities.

Abstract

We study self-adjoint extensions of a second order differential operator of Sturm-Liouville type on a graph. We relate self-adjointness of the operator to the existence of non-complete trajectories of the Hamiltonian vector field defined by its principal symbol outside the vertices. We define Kirchhoff conditions at the vertices which guarantee a self-adjoint extension analogous to the case of quantum graphs. The singular vertices may be interpreted as introducing a singular potential at those points. We also establish a Weyl's law for the spectrum asymptotics.

Theorems & Definitions (31)

• Theorem 2.1
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• Definition 3.4
• Proposition 3.5
• Lemma 4.1
• Theorem 4.2
• Definition 4.1: Lagrangian form
• Lemma 4.1