Self-adjointness of the 2D Dirac operator with singular interactions supported on star-graphs

TL;DR

This paper analyzes the self-adjointness of 2D Dirac operators with singular interactions on star-graphs, reducing the problem to eigenvalue calculations of a spin-orbit operator and providing new insights into deficiency indices and self-adjoint extensions.

Contribution

It introduces a decomposition method to analyze Dirac operators with singular interactions on star-graphs, enabling analytical and numerical determination of deficiency indices and characterizing self-adjoint extensions.

Findings

Deficiency indices depend on the number of edges and interaction parameters.

Examples show Lorentz-scalar interaction strength can alter deficiency indices.

The paper characterizes the distinguished self-adjoint extension for certain operators.

Abstract

We consider the two-dimensional Dirac operator with Lorentz-scalar $\delta$-shell interactions on each edge of a star-graph. An orthogonal decomposition is performed which shows such an operator is unitarily equivalent to an orthogonal sum of half-line Dirac operators with off-diagonal Coulomb potentials. This decomposition reduces the computation of the deficiency indices to determining the number of eigenvalues of a one-dimensional spin-orbit operator in the interval $(-1/2,1/2)$. If the number of edges of the star graph is two or three, these deficiency indices can then be analytically determined for a range of parameters. For higher numbers of edges, it is possible to numerically calculate the deficiency indices. Among others, examples are given where the strength of the Lorentz-scalar interactions directly change the deficiency indices while other parameters are all fixed and where the deficiency indices are $(2,2)$, neither of which have been observed in the literature to the best knowledge of the authors. For those Dirac operators which are not already self-adjoint and do not have $0$ in the spectrum of the associated spin-orbit operator, the distinguished self-adjoint extension is also characterized.