287 research outputs found
Dancing with black holes
We describe efforts over the last six years to implement regularization
methods suitable for studying one or more interacting black holes by direct
N-body simulations. Three different methods have been adapted to large-N
systems: (i) Time-Transformed Leapfrog, (ii) Wheel-Spoke, and (iii) Algorithmic
Regularization. These methods have been tried out with some success on
GRAPE-type computers. Special emphasis has also been devoted to including
post-Newtonian terms, with application to moderately massive black holes in
stellar clusters. Some examples of simulations leading to coalescence by
gravitational radiation will be presented to illustrate the practical
usefulness of such methods.Comment: 8 figures, 10 pages, to appear in "Dynamical Evolution of Dense
Stellar Systems", ed. E. Vesperin
Singular solutions to the Seiberg-Witten and Freund equations on flat space from an iterative method
Although it is well known that the Seiberg-Witten equations do not admit
nontrivial solutions in flat space, singular solutions to them have been
previously exhibited -- either in or in the dimensionally reduced spaces
and -- which have physical interest. In this work, we employ an
extension of the Hopf fibration to obtain an iterative procedure to generate
particular singular solutions to the Seiberg-Witten and Freund equations on
flat space. Examples of solutions obtained by such method are presented and
briefly discussed.Comment: 7 pages, minor changes. To appear in J. Math. Phy
Radial Coulomb and Oscillator Systems in Arbitrary Dimensions
A mapping is obtained relating analytical radial Coulomb systems in any
dimension greater than one to analytical radial oscillators in any dimension.
This mapping, involving supersymmetry-based quantum-defect theory, is possible
for dimensions unavailable to conventional mappings. Among the special cases is
an injection from bound states of the three-dimensional radial Coulomb system
into a three-dimensional radial isotropic oscillator where one of the two
systems has an analytical quantum defect. The issue of mapping the continuum
states is briefly considered.Comment: accepted for publication in J. Math. Phy
Unified treatment of the Coulomb and harmonic oscillator potentials in dimensions
Quantum mechanical models and practical calculations often rely on some
exactly solvable models like the Coulomb and the harmonic oscillator
potentials. The dimensional generalized Coulomb potential contains these
potentials as limiting cases, thus it establishes a continuous link between the
Coulomb and harmonic oscillator potentials in various dimensions. We present
results which are necessary for the utilization of this potential as a model
and practical reference problem for quantum mechanical calculations. We define
a Hilbert space basis, the generalized Coulomb-Sturmian basis, and calculate
the Green's operator on this basis and also present an SU(1,1) algebra
associated with it. We formulate the problem for the one-dimensional case too,
and point out that the complications arising due to the singularity of the
one-dimensional Coulomb problem can be avoided with the use of the generalized
Coulomb potential.Comment: 18 pages, 3 ps figures, revte
An Invertible Linearization Map for the Quartic Oscillator
The set of world lines for the non-relativistic quartic oscillator satisfying
Newton's equation of motion for all space and time in 1-1 dimensions with no
constraints other than the "spring" restoring force is shown to be equivalent
(1-1-onto) to the corresponding set for the harmonic oscillator. This is
established via an energy preserving invertible linearization map which
consists of an explicit nonlinear algebraic deformation of coordinates and a
nonlinear deformation of time coordinates involving a quadrature. In the
context stated, the map also explicitly solves Newton's equation for the
quartic oscillator for arbitrary initial data on the real line. This map is
extended to all attractive potentials given by even powers of the space
coordinate. It thus provides classes of new solutions to the initial value
problem for all these potentials
Comment on "Quantum bound states with zero binding energy"
The purpose of this Comment is to show that the solutions to the zero energy
Schr\"odinger equations for monomial central potentials discussed in a recently
published Letter, may also be obtained from the corresponding free particle
solutions in a straight forwardly way, using an algorithm previously devised by
us. New solutions to the zero energy Schr\"odinger equation are also exhibited.Comment: Accepted for publication in PHISICS LETTERS
The efficiency of resonant relaxation around a massive black hole
Resonant relaxation (RR) is a rapid relaxation process that operates in the
nearly-Keplerian potential near a massive black hole (MBH). RR dominates the
dynamics of compact remnants that inspiral into a MBH and emit gravitational
waves (extreme mass ratio inspiral events, EMRIs). RR can either increase the
EMRI rate, or strongly suppress it, depending on its still poorly-determined
efficiency. We use small-scale Newtonian N-body simulations to measure the RR
efficiency and to explore its possible dependence on the stellar number density
profile around the MBH, and the mass-ratio between the MBH and a star (a
single-mass stellar population is assumed). We develop an efficient and robust
procedure for detecting and measuring RR in N-body simulations. We present a
suite of simulations with a range of stellar density profiles and mass-ratios,
and measure the mean RR efficiency in the near-Keplerian limit. We do not find
a statistically significant dependence on the density profile or the
mass-ratio. Our numerical determination of the RR efficiency in the Newtonian,
single-mass population approximations, suggests that RR will likely enhance the
EMRI rate by a factor of a few over the rates predicted assuming only slow
stochastic two-body relaxation.Comment: 5 pp, 6 figs, ApJ submitte
Attosecond time-scale multi-electron collisions in the Coulomb four-body problem: traces in classical probability densities
In the triple ionization of the Li ground state by single photon absorption
the three electrons escape to the continuum mainly through two collision
sequences with individual collisions separated by time intervals on the
attosecond scale. We investigate the traces of these two collision sequences in
the classical probability densities. We show that each collision sequence has
characteristic phase space properties which distinguish it from the other.
Classical probability densities are the closest analog to quantum mechanical
densities allowing our results to be directly compared to quantum mechanical
results.Comment: 9 pages, 10 figure
An analytic solution for weak-field Schwarzschild geodesics
It is well known that the classical gravitational two body problem can be
transformed into a spherical harmonic oscillator by regularization. We find
that a modification of the regularization transformation has a similar result
to leading order in general relativity. In the resulting harmonic oscillator,
the leading-order relativistic perturbation is formally a negative centrifugal
force. The net centrifugal force changes sign at three Schwarzschild radii,
which interestingly mimics the innermost stable circular orbit (ISCO) of the
full Schwarzschild problem. Transforming the harmonic-oscillator solution back
to spatial coordinates yields, for both timelike and null weak-field
Schwarzschild geodesics, a solution for in terms of elementary
functions of a variable that can be interpreted as a generalized eccentric
anomaly. The textbook expressions for relativistic precession and light
deflection are easily recovered. We suggest how this solution could be combined
with additional perturbations into numerical methods suitable for applications
such as relativistic accretion or dynamics of the Galactic-centre stars.Comment: 8 pages; Published in the MNRAS; The definitive version is available
at www.blackwell-synergy.co
Semiclassical initial value calculations of collinear helium atom
Semiclassical calculations using the Herman-Kluk initial value treatment are
performed to determine energy eigenvalues of bound and resonance states of the
collinear helium atom. Both the configuration (where the classical motion
is fully chaotic) and the configuration (where the classical dynamics is
nearly integrable) are treated. The classical motion is regularized to remove
singularities that occur when the electrons collide with the nucleus. Very good
agreement is obtained with quantum energies for bound and resonance states
calculated by the complex rotation method.Comment: 24 pages, 3 figures. Submitted to J. Phys.
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