A note on the Dirac operator with Kirchoff-type vertex conditions on noncompact metric graphs

Abstract

In this note we present some properties of the Dirac operator on noncompact metric graphs with Kirchoff-type vertex conditions. In particular, we discuss the specific features of the spectrum of the operator and, finally, we give some further details on the associated quadratic form (and on the form domain).

On the nonlinear Dirac equation on noncompact metric graphs

“…In addition, some recent works proposed the NLDE as an effective equation for BEC in branched thin domains and optical fibers, in the presence of relativistic effects (see [51,57]). The self-adjoint realization of the Dirac operator employed in these applications is the so-called Kirchhoff-type one (for details see [19, Appendix A] and [20]), which is defined as follows: Definition 1.2 (Dirac-Kirchhoff). The Dirac operator with Kirchhoff-type vertex conditions, that we denote by D G , is an operator with action…”

“…Remark 1.4. Recall that, since the spectrum of D G is given by [19, Appendix A] and [20]), then (−mc 2 , mc 2 ) = R\σ(D G ). 1.3.…”

Spectral properties of relativistic quantum waveguides

“…Previous studies deal with the quantisation of graphs and spectral statistics for the Dirac operator [4], and self-adjoint extensions and scattering properties for different graph topologies [9]. Rigorous mathematical studies on linear and nonlinear Dirac equations on metric graphs recently appeared [5,6,7]. The result of the present paper can be understood as the first step toward a rigorous justification of the metric graph model as the limit of shrinking branched waveguides.…”

Inverse Dynamic Problem for a 1-D Dirac System on Finite Metric Tree Graphs and a Leaf Peeling Algorithm

“…In recent years, the study of the Dirac operator on metric graphs has generated a growing interest (see, e.g., [1], and the list of references therein). It has been used to build models for electronic properties of graphene [10], propagation of electromagnetic waves in graphene-like photonic crystals (see [11]), ultracold matter in optical lattices and many others (see, e.g., [7], [8], and references therein).…”