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arXiv · 2510.18922

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

Elisha Falbel
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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.

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Elisha Falbel. 2025-10-21. Self-adjoint extensions of singular Sturm-Liouville operators on graphs and Weyl's law. https://arxiv.org/abs/2510.18922

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