4,935 research outputs found
Semi-spectral Chebyshev method in Quantum Mechanics
Traditionally, finite differences and finite element methods have been by
many regarded as the basic tools for obtaining numerical solutions in a variety
of quantum mechanical problems emerging in atomic, nuclear and particle
physics, astrophysics, quantum chemistry, etc. In recent years, however, an
alternative technique based on the semi-spectral methods has focused
considerable attention. The purpose of this work is first to provide the
necessary tools and subsequently examine the efficiency of this method in
quantum mechanical applications. Restricting our interest to time independent
two-body problems, we obtained the continuous and discrete spectrum solutions
of the underlying Schroedinger or Lippmann-Schwinger equations in both, the
coordinate and momentum space. In all of the numerically studied examples we
had no difficulty in achieving the machine accuracy and the semi-spectral
method showed exponential convergence combined with excellent numerical
stability.Comment: RevTeX, 12 EPS figure
Asymptotic expansions and fast computation of oscillatory Hilbert transforms
In this paper, we study the asymptotics and fast computation of the one-sided
oscillatory Hilbert transforms of the form where the bar indicates the Cauchy principal value and is a
real-valued function with analytic continuation in the first quadrant, except
possibly a branch point of algebraic type at the origin. When , the
integral is interpreted as a Hadamard finite-part integral, provided it is
divergent. Asymptotic expansions in inverse powers of are derived for
each fixed , which clarify the large behavior of this
transform. We then present efficient and affordable approaches for numerical
evaluation of such oscillatory transforms. Depending on the position of , we
classify our discussion into three regimes, namely, or
, and . Numerical experiments show that the convergence
of the proposed methods greatly improve when the frequency increases.
Some extensions to oscillatory Hilbert transforms with Bessel oscillators are
briefly discussed as well.Comment: 32 pages, 6 figures, 4 table
On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions
Integral representations are considered of solutions of the inhomogeneous
Airy differential equation . The solutions of these equations
are also known as Scorer functions. Certain functional relations for these
functions are used to confine the discussion to one function and to a certain
sector in the complex plane. By using steepest descent methods from
asymptotics, the standard integral representations of the Scorer functions are
modified in order to obtain non-oscillating integrals for complex values of
. In this way stable representations for numerical evaluations of the
functions are obtained. The methods are illustrated with numerical results.Comment: 12 pages, 5 figure
Lagrange-mesh calculations in momentum space
The Lagrange-mesh method is a powerful method to solve eigenequations written
in configuration space. It is very easy to implement and very accurate. Using a
Gauss quadrature rule, the method requires only the evaluation of the potential
at some mesh points. The eigenfunctions are expanded in terms of regularized
Lagrange functions which vanish at all mesh points except one. It is shown that
this method can be adapted to solve eigenequations written in momentum space,
keeping the convenience and the accuracy of the original technique. In
particular, the kinetic operator is a diagonal matrix. Observables in both
configuration space and momentum space can also be easily computed with a good
accuracy using only eigenfunctions computed in the momentum space. The method
is tested with Gaussian and Yukawa potentials, requiring respectively a small
or a great mesh to reach convergence.Comment: Extended versio
Static pair free energy and screening masses from correlators of Polyakov loops: continuum extrapolated lattice results at the QCD physical point
We study the correlators of Polyakov loops, and the corresponding gauge
invariant free energy of a static quark-antiquark pair in 2+1 flavor QCD at
finite temperature. Our simulations were carried out on = 6, 8, 10, 12,
16 lattices using Symanzik improved gauge action and a stout improved staggered
action with physical quark masses. The free energies calculated from the
Polyakov loop correlators are extrapolated to the continuum limit. For the free
energies we use a two step renormalization procedure that only uses data at
finite temperature. We also measure correlators with definite Euclidean time
reversal and charge conjugation symmetry to extract two different screening
masses, one in the magnetic, and one in the electric sector, to distinguish two
different correlation lengths in the full Polyakov loop correlator
On damping created by heterogeneous yielding in the numerical analysis of nonlinear reinforced concrete frame elements
In the dynamic analysis of structural engineering systems, it is common
practice to introduce damping models to reproduce experimentally observed
features. These models, for instance Rayleigh damping, account for the damping
sources in the system altogether and often lack physical basis. We report on an
alternative path for reproducing damping coming from material nonlinear
response through the consideration of the heterogeneous character of material
mechanical properties. The parameterization of that heterogeneity is performed
through a stochastic model. It is shown that such a variability creates the
patterns in the concrete cyclic response that are classically regarded as
source of damping
The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries
We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body
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