213 research outputs found
Perturbation Theory for Metastable States of the Dirac Equation with Quadratic Vector Interaction
The spectral problem of the Dirac equation in an external quadratic vector
potential is considered using the methods of the perturbation theory. The
problem is singular and the perturbation series is asymptotic, so that the
methods for dealing with divergent series must be used. Among these, the
Distributional Borel Sum appears to be the most well suited tool to give
answers and to describe the spectral properties of the system. A detailed
investigation is made in one and in three space dimensions with a central
potential. We present numerical results for the Dirac equation in one space
dimension: these are obtained by determining the perturbation expansion and
using the Pad\'e approximants for calculating the distributional Borel
transform. A complete agreement is found with previous non-perturbative results
obtained by the numerical solution of the singular boundary value problem and
the determination of the density of the states from the continuous spectrum.Comment: 10 pages, 1 figur
Spectral Statistics for the Dirac Operator on Graphs
We determine conditions for the quantisation of graphs using the Dirac
operator for both two and four component spinors. According to the
Bohigas-Giannoni-Schmit conjecture for such systems with time-reversal symmetry
the energy level statistics are expected, in the semiclassical limit, to
correspond to those of random matrices from the Gaussian symplectic ensemble.
This is confirmed by numerical investigation. The scattering matrix used to
formulate the quantisation condition is found to be independent of the type of
spinor. We derive an exact trace formula for the spectrum and use this to
investigate the form factor in the diagonal approximation
Coherent Schwinger Interaction from Darboux Transformation
The exactly solvable scalar-tensor potential of the four-component Dirac
equation has been obtained by the Darboux transformation method. The
constructed potential has been interpreted in terms of nucleon-nucleon and
Schwinger interactions of neutral particles with lattice sites during their
channeling Hamiltonians of a Schwinger type is obtained by means of the Darboux
transformation chain. The analitic structure of the Lyapunov function of
periodic continuation for each of the Hamiltonians of the family is considered.Comment: 12 pages, Latex, six figures; six sections, one figure adde
Eigenvalue estimates for non-selfadjoint Dirac operators on the real line
We show that the non-embedded eigenvalues of the Dirac operator on the real
line with non-Hermitian potential lie in the disjoint union of two disks in
the right and left half plane, respectively, provided that the of
is bounded from above by the speed of light times the reduced Planck
constant. An analogous result for the Schr\"odinger operator, originally proved
by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For
massless Dirac operators, the condition on implies the absence of nonreal
eigenvalues. Our results are further generalized to potentials with slower
decay at infinity. As an application, we determine bounds on resonances and
embedded eigenvalues of Dirac operators with Hermitian dilation-analytic
potentials
Quantum Effects for the Dirac Field in Reissner-Nordstrom-AdS Black Hole Background
The behavior of a charged massive Dirac field on a Reissner-Nordstrom-AdS
black hole background is investigated. The essential self-adjointness of the
Dirac Hamiltonian is studied. Then, an analysis of the discharge problem is
carried out in analogy with the standard Reissner-Nordstrom black hole case.Comment: 18 pages, 5 figures, Iop styl
Qualitative Properties of the Dirac Equation in a Central Potential
The Dirac equation for a massive spin-1/2 field in a central potential V in
three dimensions is studied without fixing a priori the functional form of V.
The second-order equations for the radial parts of the spinor wave function are
shown to involve a squared Dirac operator for the free case, whose essential
self-adjointness is proved by using the Weyl limit point-limit circle
criterion, and a `perturbation' resulting from the potential. One then finds
that a potential of Coulomb type in the Dirac equation leads to a potential
term in the above second-order equations which is not even infinitesimally
form-bounded with respect to the free operator. Moreover, the conditions
ensuring essential self-adjointness of the second-order operators in the
interacting case are changed with respect to the free case, i.e. they are
expressed by a majorization involving the parameter in the Coulomb potential
and the angular momentum quantum number. The same methods are applied to the
analysis of coupled eigenvalue equations when the anomalous magnetic moment of
the electron is not neglected.Comment: 22 pages, plain Tex. In the final version, a section has been added,
and the presentation has been improve
Hestenes' Tetrad and Spin Connections
Defining a spin connection is necessary for formulating Dirac's bispinor
equation in a curved space-time. Hestenes has shown that a bispinor field is
equivalent to an orthonormal tetrad of vector fields together with a complex
scalar field. In this paper, we show that using Hestenes' tetrad for the spin
connection in a Riemannian space-time leads to a Yang-Mills formulation of the
Dirac Lagrangian in which the bispinor field is mapped to a set of Yang-Mills
gauge potentials and a complex scalar field. This result was previously proved
for a Minkowski space-time using Fierz identities. As an application we derive
several different non-Riemannian spin connections found in the literature
directly from an arbitrary linear connection acting on Hestenes' tetrad and
scalar fields. We also derive spin connections for which Dirac's bispinor
equation is form invariant. Previous work has not considered form invariance of
the Dirac equation as a criterion for defining a general spin connection
Local energy decay of massive Dirac fields in the 5D Myers-Perry metric
We consider massive Dirac fields evolving in the exterior region of a
5-dimensional Myers-Perry black hole and study their propagation properties.
Our main result states that the local energy of such fields decays in a weak
sense at late times. We obtain this result in two steps: first, using the
separability of the Dirac equation, we prove the absence of a pure point
spectrum for the corresponding Dirac operator; second, using a new form of the
equation adapted to the local rotations of the black hole, we show by a Mourre
theory argument that the spectrum is absolutely continuous. This leads directly
to our main result.Comment: 40 page
Anomalous electron trapping by localized magnetic fields
We consider an electron with an anomalous magnetic moment g>2 confined to a
plane and interacting with a nonzero magnetic field B perpendicular to the
plane. We show that if B has compact support and the magnetic flux in the
natural units is F\ge 0, the corresponding Pauli Hamiltonian has at least 1+[F]
bound states, without making any assumptions about the field profile.
Furthermore, in the zero-flux case there is a pair of bound states with
opposite spin orientations. Using a Birman-Schwinger technique, we extend the
last claim to a weak rotationally symmetric field with B(r) = O(r^{-2-\delta})
correcting thus a recent result. Finally, we show that under mild regularity
assumptions the existence can be proved for non-symmetric fields with tails as
well.Comment: A LaTeX file, 12 pages; to appear in J. Phys. A: Math. Ge
Zero Modes of Quantum Graph Laplacians and an Index Theorem
We study zero modes of Laplacians on compact and non-compact metric graphs
with general self-adjoint vertex conditions. In the first part of the paper the
number of zero modes is expressed in terms of the trace of a unitary matrix
that encodes the vertex conditions imposed on functions in the
domain of the Laplacian. In the second part a Dirac operator is defined whose
square is related to the Laplacian. In order to accommodate Laplacians with
negative eigenvalues it is necessary to define the Dirac operator on a suitable
Kre\u{\i}n space. We demonstrate that an arbitrary, self-adjoint quantum graph
Laplacian admits a factorisation into momentum-like operators in a
Kre\u{\i}n-space setting. As a consequence, we establish an index theorem for
the associated Dirac operator and prove that the zero-mode contribution in the
trace formula for the Laplacian can be expressed in terms of the index of the
Dirac operator
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