Eigenvalues of one-dimensional non-self-adjoint Dirac operators and applications

Abstract

We analyze eigenvalues emerging from thresholds of the essential spectrum of one-dimensional Dirac operators perturbed by complex and nonsymmetric potentials. In the general non-self-adjoint setting we establish the existence and asymptotics of weakly coupled eigenvalues and Lieb-Thirring inequalities. As physical applications we investigate the damped wave equation and armchair graphene nanoribbons. IntroductionDirac operators attracted considerable attention in recent years, in particular in the context of non-self-adjoint spectral theory [4,5,6,9,12,24,25], nonlinear Schrödinger equations e.g. [3,21] or as an effective model for graphene [1,7,15,20]. In this paper we analyze eigenvalues emerging from the thresholds of the essential spectrum of the one-dimensional Dirac operator in L 2 (R) perturbed by a general matrix-valued and non-symmetric potential V preserving the essential spectrum.Our main results include the existence and asymptotics of weakly coupled eigenvalues for the one dimensional Dirac operator (Theorem 2.2) and Lieb-Thirring type inequalities (Theorem 2.4) in the massive as well as the massless case. These results complement the eigenvalue estimates in [6] and also show that the latter are optimal in the weak coupling regime, see Remark 2.3.As physical applications, we investigate the damped wave equation in L 2 (R) and to a two-dimensional model of charge carriers in graphene nanoribbons (or waveguides) with so-called armchair boundary conditions. We emphasize here the inherent non-self-adjoint nature of the former caused by the presence of damping. Moreover, our eigenvalue estimates may be converted to resonance estimates via the well-known method of complex scaling, as in [6].The application for the damped wave equation (Theorem 3.1) demonstrates a natural effect from the physical point of view: The integrable damping εa 1 (x) cannot affect the essential spectrum; however, for any ε > 0, it gives rise to a pair of complex conjugated eigenvalues having the tendency to meet at the real axis. The interpretation of the results for the graphene armchair waveguides is more complicated due to the 4 × 4 matrix structure and the PDE nature of the problem. Nonetheless, in the simplest setting of a diagonal potential that is constant in the transverse direction, the quantities entering the eigenvalue asymptotics are expressed in terms of the integral of the trace of V only, see Example 3.7 and Theorem 3.6.The main ingredient in the proofs is the analysis of a Birman-Schwinger operator. Since the problem is not self-adjoint, the existence of eigenvalues in the gap of the

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“…Using the commutativity of H 0 and G 0 , the operator K z admits the familiar form (5) provided that V is bounded. But we insist working under the minimal regularity assumption (6) in this section.…”

Eigenvalue Estimates for Bilayer Graphene

“…We refer to [7,11,8] for a full discussion and progress towards a resolution of the conjecture. We adopt the following notation, in line with [4], [5].…”

Dirac fermions in armchair graphene nanoribbons trapped by electric quantum dots

“…see Lemma 3.4 in [17]. However, there can emerge bound states with energies either in the gap or within the essential spectrum due to the interaction.…”