3,900 research outputs found
Higher-order splitting algorithms for solving the nonlinear Schr\"odinger equation and their instabilities
Since the kinetic and the potential energy term of the real time nonlinear
Schr\"odinger equation can each be solved exactly, the entire equation can be
solved to any order via splitting algorithms. We verified the fourth-order
convergence of some well known algorithms by solving the Gross-Pitaevskii
equation numerically. All such splitting algorithms suffer from a latent
numerical instability even when the total energy is very well conserved. A
detail error analysis reveals that the noise, or elementary excitations of the
nonlinear Schr\"odinger, obeys the Bogoliubov spectrum and the instability is
due to the exponential growth of high wave number noises caused by the
splitting process. For a continuum wave function, this instability is
unavoidable no matter how small the time step. For a discrete wave function,
the instability can be avoided only for \dt k_{max}^2{<\atop\sim}2 \pi, where
.Comment: 10 pages, 8 figures, submitted to Phys. Rev.
Any-order propagation of the nonlinear Schroedinger equation
We derive an exact propagation scheme for nonlinear Schroedinger equations.
This scheme is entirely analogous to the propagation of linear Schroedinger
equations. We accomplish this by defining a special operator whose algebraic
properties ensure the correct propagation. As applications, we provide a simple
proof of a recent conjecture regarding higher-order integrators for the
Gross-Pitaevskii equation, extend it to multi-component equations, and to a new
class of integrators.Comment: 10 pages, no figures, submitted to Phys. Rev.
Dimension-adaptive bounds on compressive FLD Classification
Efficient dimensionality reduction by random projections (RP) gains popularity, hence the learning guarantees achievable in RP spaces are of great interest. In finite dimensional setting, it has been shown for the compressive Fisher Linear Discriminant (FLD) classifier that forgood generalisation the required target dimension grows only as the log of the number of classes and is not adversely affected by the number of projected data points. However these bounds depend on the dimensionality d of the original data space. In this paper we give further guarantees that remove d from the bounds under certain conditions of regularity on the data density structure. In particular, if the data density does not fill the ambient space then the error of compressive FLD is independent of the ambient dimension and depends only on a notion of âintrinsic dimension'
Hooke's law correlation in two-electron systems
We study the properties of the Hooke's law correlation energy (\Ec),
defined as the correlation energy when two electrons interact {\em via} a
harmonic potential in a -dimensional space. More precisely, we investigate
the ground state properties of two model systems: the Moshinsky atom (in
which the electrons move in a quadratic potential) and the spherium model (in
which they move on the surface of a sphere). A comparison with their Coulombic
counterparts is made, which highlights the main differences of the \Ec in
both the weakly and strongly correlated limits. Moreover, we show that the
Schr\"odinger equation of the spherium model is exactly solvable for two values
of the dimension (), and that the exact wave function is
based on Mathieu functions.Comment: 7 pages, 5 figure
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