Absence of eigenvalues of Dirac operators with potentials diverging at in‐finity

Kalf, Hubert; Okaji, Takashi; Yamada, Osanobu

doi:10.1002/mana.200310092

Cited by 14 publications

(13 citation statements)

References 7 publications

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“…Assume now that S = ∅, then it is easy to see from (16) that 0 ∈ S and hence 2B 0 n ∈ S, n ∈ N 0 . Note also that no other points can belong to S. Hence, using (15) we get that σ ess (dd * ) = 2B 0 n, n ∈ N.…”

Section: The Spectrum Of D Amentioning

confidence: 95%

“…One of the striking facts about graphene is that the dynamics of its low-energy excitations (the charge carriers) can be described by massless two-dimensional Dirac operators. An interesting feature of Dirac fermions is the lack of localization under the influence of an external electric potential [30,15]. This fact, related to Klein's paradox [4], is due to the peculiar cone-like gapless structure of the spectrum of massless free Dirac operators.…”

Section: Introductionmentioning

confidence: 99%

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Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot

Könenberg¹,

Stockmeyer²

2012

J. Spectr. Theory

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Abstract. We consider a two-dimensional massless Dirac operator H in the presence of a perturbed homogeneous magnetic field B = B 0 + b and a scalar, both decaying at infinity, we show that states in the discrete spectrum of H are superexponentially localized. We establish the existence of such states between the zeroth and the first Landau level assuming that V = 0. In addition, under the condition that b is rotationally symmetric and that V satisfies certain analyticity condition on the angular variable, we show that states belonging to the discrete spectrum of H are Gaussian-like localized.

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Section: The Spectrum Of D Amentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot

Könenberg¹,

Stockmeyer²

2012

J. Spectr. Theory

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“…We recall that the study about the absence of eigenvalues of H embedded in the absolutely continuous spectrum was made in [10], [19], [35], [42], and the references quoted there. In particular, there are no eigenvalues in the absolutely continuous spectrum if…”

Section: Remark 22mentioning

confidence: 99%

Time Delay for the Dirac Equation

Naumkin

Weder

2016

Lett Math Phys

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We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operatorHere H 0 is the free Dirac operator and ζ (t) is such that ζ (t) = 1 for 0 ≤ t ≤ 1 and ζ (t) = 0 for t > 1. This approach allows us to obtain the time delay operator δT (f ) for initial states f in H 3/2+ε 2 R 3 ; C 4 , ε > 0, the Sobolev space of order 3/2 + ε and weight 2. The relation between the time delay operator δT (f ) and the Eisenbud-Wigner time delay operator is given. Also, the relation between the averaged time delay and the spectral shift function is presented.

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“…1(b)). However, this is impossible due to a result of [12] which, in particular, implies that a compactly supported potential cannot generate eigenvalues (see [12], Ex. 6.1).…”

Section: Introductionmentioning

confidence: 99%

Schnol’s Theorem and Spectral Properties of Massless Dirac Operators with Scalar Potentials

Schmidt

Umeda²

2015

Lett Math Phys

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The spectra of massless Dirac operators are of essential interest e.g. for the electronic properties of graphene, but fundamental questions such as the existence of spectral gaps remain open. We show that the eigenvalues of massless Dirac operators with suitable real-valued potentials lie inside small sets easily characterised in terms of properties of the potentials, and we prove a Schnol'-type theorem relating spectral points to polynomial boundedness of solutions of the Dirac equation. Moreover, we show that, under minimal hypotheses which leave the potential essentially unrestrained in large parts of space, the spectrum of the massless Dirac operator covers the whole real line; in particular, this will be the case if the potential is nearly constant in a sequence of regions.

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Absence of eigenvalues of Dirac operators with potentials diverging at in‐finity

Cited by 14 publications

References 7 publications

Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot

Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot

Time Delay for the Dirac Equation

Schnol’s Theorem and Spectral Properties of Massless Dirac Operators with Scalar Potentials

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