669 research outputs found
Pseudospectral Calculation of the Wavefunction of Helium and the Negative Hydrogen Ion
We study the numerical solution of the non-relativistic Schr\"{o}dinger
equation for two-electron atoms in ground and excited S-states using
pseudospectral (PS) methods of calculation. The calculation achieves
convergence rates for the energy, Cauchy error in the wavefunction, and
variance in local energy that are exponentially fast for all practical
purposes. The method requires three separate subdomains to handle the
wavefunction's cusp-like behavior near the two-particle coalescences. The use
of three subdomains is essential to maintaining exponential convergence. A
comparison of several different treatments of the cusps and the semi-infinite
domain suggest that the simplest prescription is sufficient. For many purposes
it proves unnecessary to handle the logarithmic behavior near the
three-particle coalescence in a special way. The PS method has many virtues: no
explicit assumptions need be made about the asymptotic behavior of the
wavefunction near cusps or at large distances, the local energy is exactly
equal to the calculated global energy at all collocation points, local errors
go down everywhere with increasing resolution, the effective basis using
Chebyshev polynomials is complete and simple, and the method is easily
extensible to other bound states. This study serves as a proof-of-principle of
the method for more general two- and possibly three-electron applications.Comment: 23 pages, 20 figures, 2 tables, Final refereed version - Some
references added, some stylistic changes, added paragraph to matrix methods
section, added last sentence to abstract
On perturbations of Dirac operators with variable magnetic field of constant direction
We carry out the spectral analysis of matrix valued perturbations of
3-dimensional Dirac operators with variable magnetic field of constant
direction. Under suitable assumptions on the magnetic field and on the
pertubations, we obtain a limiting absorption principle, we prove the absence
of singular continuous spectrum in certain intervals and state properties of
the point spectrum. Various situations, for example when the magnetic field is
constant, periodic or diverging at infinity, are covered. The importance of an
internal-type operator (a 2-dimensional Dirac operator) is also revealed in our
study. The proofs rely on commutator methods.Comment: 12 page
Homoclinic chaos in the dynamics of a general Bianchi IX model
The dynamics of a general Bianchi IX model with three scale factors is
examined. The matter content of the model is assumed to be comoving dust plus a
positive cosmological constant. The model presents a critical point of
saddle-center-center type in the finite region of phase space. This critical
point engenders in the phase space dynamics the topology of stable and unstable
four dimensional tubes , where is a saddle direction and
is the manifold of unstable periodic orbits in the center-center sector.
A general characteristic of the dynamical flow is an oscillatory mode about
orbits of an invariant plane of the dynamics which contains the critical point
and a Friedmann-Robertson-Walker (FRW) singularity. We show that a pair of
tubes (one stable, one unstable) emerging from the neighborhood of the critical
point towards the FRW singularity have homoclinic transversal crossings. The
homoclinic intersection manifold has topology and is constituted
of homoclinic orbits which are bi-asymptotic to the center-center
manifold. This is an invariant signature of chaos in the model, and produces
chaotic sets in phase space. The model also presents an asymptotic DeSitter
attractor at infinity and initial conditions sets are shown to have fractal
basin boundaries connected to the escape into the DeSitter configuration
(escape into inflation), characterizing the critical point as a chaotic
scatterer.Comment: 11 pages, 6 ps figures. Accepted for publication in Phys. Rev.
Obstruction Results in Quantization Theory
We define the quantization structures for Poisson algebras necessary to
generalise Groenewold and Van Hove's result that there is no consistent
quantization for the Poisson algebra of Euclidean phase space. Recently a
similar obstruction was obtained for the sphere, though surprising enough there
is no obstruction to the quantization of the torus. In this paper we want to
analyze the circumstances under which such obstructions appear. In this context
we review the known results for the Poisson algebras of Euclidean space, the
sphere and the torus.Comment: 34 pages, Latex. To appear in J. Nonlinear Scienc
Covariant gauge fixing and Kuchar decomposition
The symplectic geometry of a broad class of generally covariant models is
studied. The class is restricted so that the gauge group of the models
coincides with the Bergmann-Komar group and the analysis can focus on the
general covariance. A geometrical definition of gauge fixing at the constraint
manifold is given; it is equivalent to a definition of a background (spacetime)
manifold for each topological sector of a model. Every gauge fixing defines a
decomposition of the constraint manifold into the physical phase space and the
space of embeddings of the Cauchy manifold into the background manifold (Kuchar
decomposition). Extensions of every gauge fixing and the associated Kuchar
decomposition to a neighbourhood of the constraint manifold are shown to exist.Comment: Revtex, 35 pages, no figure
Constraints on the mass and abundance of black holes in the Galactic halo: the high mass limit
We establish constraints on the mass and abundance of black holes in the
Galactic halo by determining their impact on globular clusters which are
conventionally considered to be little evolved. Using detailed Monte Carlo
simulations and simple analytic estimates, we conclude that, at Galactocentric
radius R~8 kpc, black holes with masses M_bh >~(1-3) x 10^6 M_sun can comprise
no more than a fraction f_bh ~ 0.025-0.05 of the total halo density. This
constraint significantly improves those based on disk heating and dynamical
friction arguments as well as current lensing results. At smaller radius, the
constraint on f_bh strengthens, while, at larger radius, an increased fraction
of black holes is allowed.Comment: 13 pages, 10 figures, revised version, in press, Monthly Notice
Bosonization method for second super quantization
A bosonic-fermionic correspondence allows an analytic definition of
functional super derivative, in particular, and a bosonic functional calculus,
in general, on Bargmann- Gelfand triples for the second super quantization. A
Feynman integral for the super transformation matrix elements in terms of
bosonic anti-normal Berezin symbols is rigorously constructed.Comment: In memoriam of F. A. Berezin, accepted in Journal of Nonlinear
Mathematical Physics, 15 page
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
Symplectic connections and Fedosov's quantization on supermanifolds
A (biased and incomplete) review of the status of the theory of symplectic
connections on supermanifolds is presented. Also, some comments regarding
Fedosov's technique of quantization are made.Comment: Submitted to J. of Phys. Conf. Se
Spectral properties on a circle with a singularity
We investigate the spectral and symmetry properties of a quantum particle
moving on a circle with a pointlike singularity (or point interaction). We find
that, within the U(2) family of the quantum mechanically allowed distinct
singularities, a U(1) equivalence (of duality-type) exists, and accordingly the
space of distinct spectra is U(1) x [SU(2)/U(1)], topologically a filled torus.
We explore the relationship of special subfamilies of the U(2) family to
corresponding symmetries, and identify the singularities that admit an N = 2
supersymmetry. Subfamilies that are distinguished in the spectral properties or
the WKB exactness are also pointed out. The spectral and symmetry properties
are also studied in the context of the circle with two singularities, which
provides a useful scheme to discuss the symmetry properties on a general basis.Comment: TeX, 26 pages. v2: one reference added and two update
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