Locally finite extensions and Gesztesy–Šeba realizations for the Dirac operator on a metric graph

Abstract

We study extensions of direct sums of symmetric operators S = ⊕ n∈N S n. In general there is no natural boundary triplet for S * even if there is one for every S * n , n ∈ N. We consider a subclass of extensions of S which can be described in terms of the boundary triplets of S * n and investigate the self-adjointness, the semi-boundedness from below and the discreteness of the spectrum. Sufficient conditions for these properties are obtained from recent results on weighted discrete Laplacians. The results are applied to Dirac operators on metric graphs with point interactions at the vertices. In particular, we allow graphs with arbitrarily small edge length.

Boundary Conditions Matter: On The Spectrum Of Infinite Quantum Graphs

“…If Ω ⊂ R d is a bounded domain with Lipschitz boundary, then its Dirichlet and Neumann Laplacians both have purely discrete spectrum, whereas on unbounded or nonsmooth domains, the situation is more subtle, see, e.g., [1,Chapter 6] or [14]. Likewise, it is known that infinite quantum graphs may or may not have purely discrete spectrum [48,20,29,18]. This paper is organised as follows: Section 2 contains definitions of metric graphs, Sobolev spaces, and extensions of the Laplace operator.…”