Estimates on Complex Eigenvalues for Dirac Operators on the Half-Line

Cuenin, Jean-Claude

doi:10.1007/s00020-014-2146-9

Cited by 25 publications

(21 citation statements)

References 10 publications

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“…acting in the Hilbert space of spinors H := L 2 (R 3 ; C 4 ). Here α := (α 1 , α 2 , α 3 ) with α µ being the usual 4 × 4 Hermitian Dirac matrices satisfying the anticommutation rules α µ α ν + α ν α µ = 2δ µν I C 4 (2) for µ, ν ∈ {1, . .…”

Section: Introductionmentioning

confidence: 99%

“…V is a real scalar multiple of the diagonal matrix), the literature on spectral properties of H V is enormous and we limit ourselves to quoting the classical Thaller's monograph [15]. For V being matrix-valued and possibly non-Hermitian, a systematic study of the spectrum of H V was initiated by the pioneering work of Cuenin, Laptev and Tretter [4] in the one-dimensional setting and followed by [2,7,5]. Some spectral aspects in the present three-dimensional situation are covered by [6,13,3].…”

Section: Introductionmentioning

confidence: 99%

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Location of eigenvalues of three-dimensional non-self-adjoint Dirac operators

Fanelli

Krejčiřı́k

2019

Lett Math Phys

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Location of eigenvalues of three-dimensional non-self-adjoint Dirac operators

Fanelli

Krejčiřı́k

2019

Lett Math Phys

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“…Works concerning non-self-adjoint Pauli operators are much more sparse in the literature, see [36] and references therein. More results are available in the case of nonself-adjoint Dirac operators, see [5][6][7][8][10][11][12]14,35]. The paper [16] represents a first, physically satisfactory, application of the method of multipliers to spectral theory: the authors established sufficient conditions, which guarantee the total absence of eigenvalues of (1.1).…”

Section: Objectives and State Of The Artmentioning

confidence: 99%

Absence of Eigenvalues of Dirac and Pauli Hamiltonians via the Method of Multipliers

Cossetti

Fanelli

Krejčiřı́k

2020

Commun. Math. Phys.

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By developing the method of multipliers, we establish sufficient conditions on the magnetic field and the complex, matrix-valued electric potential, which guarantee that the corresponding system of Schrödinger operators has no point spectrum. In particular, this allows us to prove analogous results for Pauli operators under the same electromagnetic conditions and, in turn, as a consequence of the supersymmetric structure, also for magnetic Dirac operators.

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“…Notice that the assumption that Re U 11 < 0 or Re U 22 > 0 is needed since the allowed region in terms of ζ is Im ζ < 0, see the discussion of [4] after (2.6) there. Finally, we prove that there is indeed a solution ζ + of (2.17) in a neighborhood of ζ 0 + ; the reasoning for ζ − is analogous.…”

Section: Notice Thatmentioning

confidence: 99%

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

Cuenin

Siegl

2018

Lett Math Phys

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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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Estimates on Complex Eigenvalues for Dirac Operators on the Half-Line

Cited by 25 publications

References 10 publications

Location of eigenvalues of three-dimensional non-self-adjoint Dirac operators

Location of eigenvalues of three-dimensional non-self-adjoint Dirac operators

Absence of Eigenvalues of Dirac and Pauli Hamiltonians via the Method of Multipliers

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

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