4,121 research outputs found
The eigenvalues of the pseudospectral Fourier approximation to the operator sin (2x) d/dx
It is shown that the eigenvalues Z sub i of the pseudospectral Fourier approximation to the operator sin(2x) curly d/curly dx satisfy (R sub e) (Z sub i) = + or - 1 or (R sub e)(Z sub I) = 0. Whereas this does not prove stability for the Fourier method, applied to the hyperbolic equation U sub t = sin (2x)(U sub x) - pi x pi; it indicates that the growth in time of the numerical solution is essentially the same as that of the solution to the differential equation
Spectral methods in time for hyperbolic equations
A pseudospectral numerical scheme for solving linear, periodic, hyperbolic problems is described. It has infinite accuracy both in time and in space. The high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular, one step explicit scheme. The algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution. The class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator
New, Highly Accurate Propagator for the Linear and Nonlinear Schr\"odinger Equation
A propagation method for the time dependent Schr\"odinger equation was
studied leading to a general scheme of solving ode type equations. Standard
space discretization of time-dependent pde's usually results in system of ode's
of the form u_t -Gu = s where G is a operator (matrix) and u is a
time-dependent solution vector. Highly accurate methods, based on polynomial
approximation of a modified exponential evolution operator, had been developed
already for this type of problems where G is a linear, time independent matrix
and s is a constant vector. In this paper we will describe a new algorithm for
the more general case where s is a time-dependent r.h.s vector. An iterative
version of the new algorithm can be applied to the general case where G depends
on t or u. Numerical results for Schr\"odinger equation with time-dependent
potential and to non-linear Schr\"odinger equation will be presented.Comment: 14 page
Transcription factor search for a DNA promoter in a three-states model
To ensure fast gene activation, Transcription Factors (TF) use a mechanism
known as facilitated diffusion to find their DNA promoter site. Here we analyze
such a process where a TF alternates between 3D and 1D diffusion. In the latter
(TF bound to the DNA), the TF further switches between a fast translocation
state dominated by interaction with the DNA backbone, and a slow examination
state where interaction with DNA base pairs is predominant. We derive a new
formula for the mean search time, and show that it is faster and less sensitive
to the binding energy fluctuations compared to the case of a single sliding
state. We find that for an optimal search, the time spent bound to the DNA is
larger compared to the 3D time in the nucleus, in agreement with recent
experimental data. Our results further suggest that modifying switching via
phosphorylation or methylation of the TF or the DNA can efficiently regulate
transcription.Comment: 4 pages, 3 figure
A pseudospectral Legendre method for hyperbolic equations with an improved stability condition
A new pseudospectral method is introduced for solving hyperbolic partial differential equations. This method uses different grid points than previously used pseudospectral methods: in fact the grid are related to the zeroes of the Legendre polynomials. The main advantage of this method is that the allowable time step is proportional to the inverse of the number of grid points 1/N rather than to 1/n(2) (as in the case of other pseudospectral methods applied to mixed initial boundary value problems). A highly accurate time discretization suitable for these spectral methods is discussed
An efficient scheme for numerical simulations of the spin-bath decoherence
We demonstrate that the Chebyshev expansion method is a very efficient
numerical tool for studying spin-bath decoherence of quantum systems. We
consider two typical problems arising in studying decoherence of quantum
systems consisting of few coupled spins: (i) determining the pointer states of
the system, and (ii) determining the temporal decay of quantum oscillations. As
our results demonstrate, for determining the pointer states, the
Chebyshev-based scheme is at least a factor of 8 faster than existing
algorithms based on the Suzuki-Trotter decomposition. For the problems of
second type, the Chebyshev-based approach has been 3--4 times faster than the
Suzuki-Trotter-based schemes. This conclusion holds qualitatively for a wide
spectrum of systems, with different spin baths and different Hamiltonians.Comment: 8 pages (RevTeX), 3 EPS figure
Adiabatic theorem for non-hermitian time-dependent open systems
In the conventional quantum mechanics (i.e., hermitian QM) the adia- batic
theorem for systems subjected to time periodic fields holds only for bound
systems and not for open ones (where ionization and dissociation take place)
[D. W. Hone, R. Ketzmerik, and W. Kohn, Phys. Rev. A 56, 4045 (1997)]. Here
with the help of the (t,t') formalism combined with the complex scaling method
we derive an adiabatic theorem for open systems and provide an analytical
criteria for the validity of the adiabatic limit. The use of the complex
scaling transformation plays a key role in our derivation. As a numerical
example we apply the adiabatic theorem we derived to a 1D model Hamiltonian of
Xe atom which interacts with strong, monochromatic sine-square laser pulses. We
show that the gener- ation of odd-order harmonics and the absence of
hyper-Raman lines, even when the pulses are extremely short, can be explained
with the help of the adiabatic theorem we derived
Quantum Detection with Unknown States
We address the problem of distinguishing among a finite collection of quantum
states, when the states are not entirely known. For completely specified
states, necessary and sufficient conditions on a quantum measurement minimizing
the probability of a detection error have been derived. In this work, we assume
that each of the states in our collection is a mixture of a known state and an
unknown state. We investigate two criteria for optimality. The first is
minimization of the worst-case probability of a detection error. For the second
we assume a probability distribution on the unknown states, and minimize of the
expected probability of a detection error.
We find that under both criteria, the optimal detectors are equivalent to the
optimal detectors of an ``effective ensemble''. In the worst-case, the
effective ensemble is comprised of the known states with altered prior
probabilities, and in the average case it is made up of altered states with the
original prior probabilities.Comment: Refereed version. Improved numerical examples and figures. A few
typos fixe
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