20 research outputs found
PT-symmetric operators and metastable states of the 1D relativistic oscillators
We consider the one-dimensional Dirac equation for the harmonic oscillator
and the associated second order separated operators giving the resonances of
the problem by complex dilation. The same operators have unique extensions as
closed PT-symmetric operators defining infinite positive energy levels
converging to the Schroedinger ones as c tends to infinity. Such energy levels
and their eigenfunctions give directly a definite choice of metastable states
of the problem. Precise numerical computations shows that these levels coincide
with the positions of the resonances up to the order of the width. Similar
results are found for the Klein-Gordon oscillators, and in this case there is
an infinite number of dynamics and the eigenvalues and eigenvectors of the
PT-symmetric operators give metastable states for each dynamics.Comment: 13 pages, 2 figure
Effective Hamiltonian with holomorphic variables
The pure-quantum self-consistent harmonic approximation (PQSCHA) permits to
study a quantum system by means of an effective classical Hamiltonian. In this
work the PQSCHA is reformulated in terms of the holomorphic variables connected
to a set of bosonic operators. The holomorphic formulation, based on the
olomorphic path integral for the Weyl symbol of the density matrix, makes it
possible to directly approach general Hamiltonians given in terms of bosonic
creation and annihilation operators.Comment: Proceedings of the Conference "Path Integrals from peV to TeV - 50
Years from Feynman's paper" (Florence, August 1998) -- 2 pages, ReVTe
Effective Hamiltonian with holomorphic variables
The pure-quantum self-consistent harmonic approximation (PQSCHA) permits to
study a quantum system by means of an effective classical Hamiltonian -
depending on quantum coupling and temperature - and classical-like expressions
for the averages of observables. In this work the PQSCHA is derived in terms of
the holomorphic variables connected to a set of bosonic operators. The
holomorphic formulation, based on the path integral for the Weyl symbol of the
density matrix, makes it possible to approach directly general Hamiltonians
given in terms of bosonic creation and annihilation operators.Comment: 11 pages, no figures (2nd version: few mistakes fixed in Sects. IV-V
PAVEL: Decorative Patterns with Packed Volumetric Elements
Many real-world hand-crafted objects are decorated with elements that are
packed onto the object's surface and deformed to cover it as much as possible.
Examples are artisanal ceramics and metal jewelry. Inspired by these objects,
we present a method to enrich surfaces with packed volumetric decorations. Our
algorithm works by first determining the locations in which to add the
decorative elements and then removing the non-physical overlap between them
while preserving the decoration volume. For the placement, we support several
strategies depending on the desired overall motif. To remove the overlap, we
use an approach based on implicit deformable models creating the qualitative
effect of plastic warping while avoiding expensive and hard-to-control physical
simulations. Our decorative elements can be used to enhance virtual surfaces,
as well as 3D-printed pieces, by assembling the decorations onto real-surfaces
to obtain tangible reproductions.Comment: 11 page
States of the Dirac equation in confining potentials
We study the Dirac equation in confining potentials with pure vector
coupling, proving the existence of metastable states with longer and longer
lifetimes as the non-relativistic limit is approached and eventually merging
with continuity into the Schr\"odinger bound states. We believe that the
existence of these states could be relevant in high energy model construction
and in understanding possible resonant scattering effects in systems like
Graphene. We present numerical results for the linear and the harmonic cases
and we show that the the density of the states of the continuous spectrum is
well described by a sum of Breit-Wigner lines. The width of the line with
lowest positive energy, as expected, reproduces very well the Schwinger pair
production rate for a linear potential: we thus suggest a different way of
obtaining informations on the pair production in unbounded, non uniform
electric fields, where very little is known.Comment: 4 page
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