Spectral properties of the nonstationary two-dimensional Dirac operator

1995 ◽  
Vol 47 (5) ◽  
pp. 686-693
Author(s):  
E. I. Amerova ◽  
L. P. Nizhnik
Keyword(s):  
Dirac Operator ◽  
Spectral Properties ◽  
Two Dimensional

scholarly journals Spectra of graphene nanoribbons with armchair and zigzag boundary conditions

2014 ◽  
Vol 26 (10) ◽  
pp. 1450018 ◽  
Author(s):  
Pedro Freitas ◽  
Petr Siegl
Keyword(s):  
Dirac Operator ◽  
Boundary Conditions ◽  
Spectral Properties ◽  
Graphene Nanoribbons ◽  
Continuous Model ◽  
Bounded Domains ◽  
Two Dimensional ◽  
Edge States ◽  
Finite Multiplicity

We study the spectral properties of the two-dimensional Dirac operator on bounded domains together with the appropriate boundary conditions which provide a (continuous) model for graphene nanoribbons. These are of two types, namely, the so-called armchair and zigzag boundary conditions, depending on the line along which the material was cut. In the former case, we show that the spectrum behaves in what might be called a classical way; while in the latter, we prove the existence of a sequence of finite multiplicity eigenvalues converging to zero and which correspond to edge states.

scholarly journals Spectral properties of the equation (∇ + ieA) × u = ±mu

2001 ◽  
Vol 131 (5) ◽  
pp. 1065-1089
Author(s):  
Daniel M. Elton
Keyword(s):  
Spectral Properties ◽  
Time Variable ◽  
Two Dimensional ◽  
Vector Product ◽  
Relativistic Invariance ◽  
Speed Of Light ◽  
Large Parameter ◽  
Real Vector ◽  
Pauli Equation ◽  
Minkowski Metric

We develop a spectral theory for the equation (∇ + ieA) × u = ±mu on Minkowski 3-space (one time variable and two space variables); here, A is a real vector potential and the vector product is defined with respect to the Minkowski metric. This equation was formulated by Elton and Vassiliev, who conjectured that it should have properties similar to those of the two-dimensional Dirac equation. Our equation contains a large parameter c (speed of light), and this motivates the study of the asymptotic behaviour of its spectrum as c → +∞. We show that the essential spectrum of our equation is the same as that of Dirac (theorem 3.1), whereas the discrete spectrum agrees with Dirac to a relative accuracy δλ/mc2 ~ O(c−4) (theorem 3.3). In other words, we show that our equation has the same accuracy as the two-dimensional Pauli equation, its advantage over Pauli being relativistic invariance.

Spectral properties of a two-dimensional resonant metal-dielectric photonic crystal

2012 ◽  
Vol 112 (4) ◽  
pp. 585-593 ◽  
Author(s):  
S. Ya. Vetrov ◽  
N. V. Rudakova ◽  
I. V. Timofeev ◽  
V. P. Timofeev
Keyword(s):  
Photonic Crystal ◽  
Spectral Properties ◽  
Two Dimensional

scholarly journals Dirac operators with Lorentz scalar shell interactions

2018 ◽  
Vol 30 (05) ◽  
pp. 1850013 ◽  
Author(s):  
Markus Holzmann ◽  
Thomas Ourmières-Bonafos ◽  
Konstantin Pankrashkin
Keyword(s):  
Dirac Operator ◽  
Spectral Properties ◽  
Smooth Surface ◽  
Three Dimensional ◽  
The Self ◽  
Eigenvalue Asymptotics ◽  
Yang Mills ◽  
Lorentz Scalar ◽  
Definition Of ◽  
The Individual

This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator, we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues and their total number are governed by an effective Schrödinger operator on the boundary with an external Yang–Mills potential and a curvature-induced potential.

scholarly journals A Special Class of Infinite Dimensional Dirac Operators on the Abstract Boson-Fermion Fock Space

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Asao Arai
Keyword(s):  
Dirac Operator ◽  
Hilbert Spaces ◽  
Special Class ◽  
Spectral Properties ◽  
Fock Space ◽  
Dirac Operators ◽  
Fredholm Property ◽  
Perturbation Parameter ◽  
Coupling Constant ◽  
Infinite Dimensional

Spectral properties of a special class of infinite dimensional Dirac operatorsQ(α)on the abstract boson-fermion Fock spaceℱ(ℋ,𝒦)associated with the pair(ℋ,𝒦)of complex Hilbert spaces are investigated, whereα∈Cis a perturbation parameter (a coupling constant in the context of physics) and the unperturbed operatorQ(0)is taken to be a free infinite dimensional Dirac operator. A variety of the kernel ofQ(α)is shown. It is proved that there are cases where, for all sufficiently large|α|withα<0,Q(α)has infinitely many nonzero eigenvalues even ifQ(0)has no nonzero eigenvalues. Also Fredholm property ofQ(α)restricted to a subspace ofℱ(ℋ,𝒦)is discussed.

scholarly journals Spectral Properties and Length Scales of Two‐dimensional Magnetic Field Models

2007 ◽  
Vol 667 (2) ◽  
pp. 956-962 ◽  
Author(s):  
W. H. Matthaeus ◽  
J. W. Bieber ◽  
D. Ruffolo ◽  
P. Chuychai ◽  
J. Minnie
Keyword(s):  
Magnetic Field ◽  
Spectral Properties ◽  
Two Dimensional ◽  
Length Scales
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