On the discrete spectrum of the Dirac operator on bent chain quantum graph

Michail Belov,Igor Popov,Irina Blinova,N Bardis,J Quartieri,N Doukas,C Guarnaccia

doi:10.1051/itmconf/20170901007

Abstract

We study Dirac operators on an infinite quantum graph of a bent chain form which consists of identical rings connected at the touching points by δ-couplings with a parameter α ∈ ℝ. We are interested in the discrete spectrum of the corresponding Hamiltonian. It can be non-empty due to a local (geometrical perturbation of the corresponding infinite chain of rings. The quantum graph of analogous geometry with the Schrodinger operator on the edges was considered by Duclos, Exner and Turek in 2008. They showed that the absence of δ-couplings at vertices (i.e. the Kirchhoff condition at the vertices) lead to the absence of eigenvalues. We consider the relativistic particle (the Dirac operator instead of the Schrodinger one) but the result is analogous. Quantum graphs of such type are suitable for description of grapheme-based nanostructures. It is established that the negativity of α is the necessary and sufficient condition for the existence of eigenvalues of the Dirac operator (i.e. the discrete spectrum of the Hamiltonian in this case is not empty). The continuous spectrum of the Hamiltonian for bent chain graph coincides with that for the corresponding straight infinite chain. Conditions for appearance of more than one eigenvalue are obtained. It is related to the bending angle. The investigation is based on the transfer-matrix approach. It allows one to reduce the problem to an algebraic task. δ-couplings was introduced by the operator extensions theory method.

Highlights

Investigation of the physical properties of various compounds of half-crystals and nanotubes becomes more and more interesting
The paper deals with the spectrum description of the Dirac operator for a quantum graph having the form of bent chain of rings
As for the continuous spectrum for the bending chain, it is not so interesting because it coincides with that for the corresponding straight infinite chain

Summary

Introduction

Investigation of the physical properties of various compounds of half-crystals and nanotubes becomes more and more interesting. Many physical properties are related with the spectral characteristics of the system Hamiltonian. Quantum graph is one of prospective models It is simple and, at the same time, describes properties of physical systems [5,6]. The Schrödinger operator is considered on the graph edges. As for a relativistic particle, i.e. the Dirac operator, the corresponding graph models are not so widely used. Due to the remarkable progress of graphene nanostructures (described better by the Dirac operator than by the Schrödinger one), the relativistic model found new applications. A few years ago, the graph of the same geometry with the Schrödinger operator was studied. The mathematical background for introducing of the pointlike potentials is given by the theory of self-adjoint extensions of symmetric operators [11,12,13]. 1D point-like potential for the Dirac operator was described in many works

Infinite periodic chain

Main result

Findings

Concluding remarks