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

TL;DR

This paper investigates self-adjoint extensions of Sturm-Liouville operators on graphs, linking their properties to Hamiltonian dynamics, and establishes a Weyl's law for spectral asymptotics, advancing understanding of quantum graph operators.

Contribution

It introduces a novel connection between self-adjointness and Hamiltonian trajectories, and formulates Kirchhoff conditions for singular vertices, extending spectral analysis on quantum graphs.

Findings

Self-adjointness relates to non-complete Hamiltonian trajectories.

Kirchhoff conditions ensure self-adjoint extensions at vertices.

Weyl's law is established for the spectrum of these operators.

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.