Optimized computation of the voigt and complex probability functions

Methods for computing the complex probability function w(z) = e^?z2 erfc (?iz), which is related to the Voigt spectrum line profiles, are developed. The basic method is a rational approximation, minimizing the relative error of the imaginary part on the real axis. It is complemented by other methods in order to increase efficiency and to overcome the inevitable failure of any rational approximation near the real axis. The procedures enable one to evaluate both real and imaginary parts of w(z) with high relative accuracy. The methods are simple, as demonstrated by a sample FORTRAN program.     

On the location of poles of Ruelle's zeta function

We discuss examples of one-dimensional lattice spin systems of classical statistical mechanics whose generalized zeta function has all its poles and zeros on the real axis. The close relation between certain hyperbolic dynamical systems and these spin systems lets one expect that this is also true for some of the dynamical systems. In fact, we have found several one-dimensional expansive systems, among them the Gauss map whose zeta functions have their zeros, respectively their poles, on the real axis. Such a behaviour is closely related to the spectral properties of the sytems transfer operator which in the cases considered is a positive nuclear operator in a Banach space of holomorphic functions. We formulate a general conjecture concerning the spectrum of this class of operators.     

Dynamic Collision Frequency in Kelbg‐Pseudopotential‐Modelled Plasmas and the Method of Moments with Local Constraints

The dense plasma dynamic collision frequency is modeled by the first two terms of its asymptotic expansion at high frequencies and its values at a few interpolation points on the real axis. This makes the dynamic collision frequency a non‐rational function whose extension onto the upper half‐plane of the complex frequency is holomorphic with a non‐negative imaginary part and with a continuous extension to the real axis. The validity of the suggested analytic form of the latter is tested against the simulation data, where the Kelbg effective potential was used. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simultaneous measurement of retardation and fast axis angle of eighth-wave plate in real time

A simultaneous measurement method for the retardation and the fast axis angle of the eighth-wave plate in real time is proposed. The beam emitted from the laser passes through the circular polarizer and the eighth-wave plate to be measured successively, and then is split by the Dammann grating to form three sub-beams. They are analyzed by an analyzer array and detected by a detector array. Three detection signals are obtained simultaneously to calculate the retardation and the fast axis angle in real time. In experiments, a crystal quartz sample is measured at different fast axis angles. The average and standard deviation of its retardation respectively are 40.9°and 0.5°. The maximum measurement deviation of the fast axis angle is 2.1°. The usefulness of the method is verified.     

On the generation of solitons and breathers in the modified Korteweg-de Vries equation

We consider the evolution of an initial disturbance described by the modified Korteweg-de Vries equation with a positive coefficient of the cubic nonlinear term, so that it can support solitons. Our primary aim is to determine the circumstances which can lead to the formation of solitons and/or breathers. We use the associated scattering problem and determine the discrete spectrum, where real eigenvalues describe solitons and complex eigenvalues describe breathers. For analytical convenience we consider various piecewise-constant initial conditions. We show how complex eigenvalues may be generated by bifurcation from either the real axis, or the imaginary axis; in the former case the bifurcation occurs as the unfolding of a double real eigenvalue. A bifurcation from the real axis describes the transition of a soliton pair with opposite polarities into a breather, while the bifurcation from the imaginary axis describes the generation of a breather from the continuous spectrum. Within the class of initial conditions we consider, a disturbance of one polarity, either positive or negative, will only generate solitons, and the number of solitons depends on the total mass. On the other hand, an initial disturbance with both polarities and very small mass will favor the generation of breathers, and the number of breathers then depends on the total energy. Direct numerical simulations of the modified Korteweg-de Vries equation confirms the analytical results, and show in detail the formation of solitons, breathers, and quasistationary coupled soliton pairs. Being based on spectral theory, our analytical results apply to the entire hierarchy of evolution equations connected with the same eigenvalue problem. (c) 2000 American Institute of Physics.     

Energy Decay for the Damped Wave Equation Under a Pressure Condition

We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This result holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis.     

Rapid computation of the Voigt and complex error functions

A simple and efficient method is proposed for the calculation of the complex error function, including its real part, which is the Voigt spectrum line profile. This procedure is based on the observation by Harstad that a rational approximation to the error function, which is the value of the complex error function on the imaginary axis, analytically continues off this axis upon replacement of y by y-ix. The error involved in this approximation can be minimized by choice of an optimal set of coefficients according to methods developed by Curtis and Osborne. An analysis shows that the results are sufficiently accurate for most applications over the whole complex plane, but with the error increasing toward the real axis. The procedure is simpler and faster than previous methods, requiring only two arithmetic statements for coding in FORTRAN. Comparisons are presented of the time required for various methods on various IBM computer systems which supports this conclusion.     

A set of Lippmann-Schwinger equations: (III). Complete continuity of the projected kernel

The disconnected Lippmann-Schwinger kernel for a three-body system is shown to be completely continuous on the real energy axis, when it is projected and operated on a subspace of continuous functions composed of a finite number of partial waves.     

On the connection between the continuum eigenfunctions and the dispersion function of the vlasov operator

It is shown that a function may be extracted from the continuum eigenmode of the Vlasov operator which is related to the boundary values of the dispersion function on the real axis of the complex frequency plane. The precise content and possible consequences of this relationship are investigated in detail, especially the possibility of reconstructing the dispersion function and establishing stability criteria.     

Mixed quantum mechanical periodic and potential wall problem

We consider the Schrödinger equation with an even-square integrable potential of period one on the negative real axis and a wall potential of heighta > 0 on the positive real axis. The spectrum of this Schrödinger equation is determined and it is proved that bounded solutions never exist if the energyE < a is lying in a gap of the periodic spectrum.     

Convergence of the diagonal operator-valued Padé approximants to the Dyson expansion

The diagonal operator-valued Padé approximants formed from the Dyson expansion to the Schrödinger time-evolution operator are shown to converge everywhere in the complex plane, except on a certain subset of the real axis.     

Some Improvements in the Method of the Weakly Conjugate Operator

We present some improvements in the method of the weakly conjugate operator, one variant of the Mourre theory. When applied to certain two-body Schrödinger operators, this leads to a limiting absorption principle that is uniform on the positive real axis.     

Statistics of the number of zero crossings: from random polynomials to the diffusion equation

We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0, 1] decays as a power law n(-theta(d)) where theta(d)>0 is the exponent associated with the decay of the persistence probability for the diffusion equation with random initial conditions in space dimension d. For n even, the probability that such polynomials have no root on the full real axis decays as n(-2[theta(d)+theta(2)]). For d=1, this connection allows for a physical realization of real random polynomials. We further show that the probability that such polynomials have exactly k real roots in [0, 1] has an unusual scaling form given by n(-phi(k/logn)) where phi(x) is a universal large deviation function.     

Analysis of the perturbed Chandler wobble of the Earth pole

Based on the model of a viscoelastic rigid body for the deformable Earth, we investigate the behavior of the main component of the perturbed Earth pole oscillation process, the Chandler wobble. An amplitude- frequency analysis of the perturbed Chandler wobble of the Earth pole due to the precessional motion of the lunar orbit is provided. The parameters of the complex dynamical process that describes the mutual orientation of the Earth’s instantaneous rotation axis, the axis of its figure, and its angular momentum vector are studied qualitatively. Using a numerical-analytical approach, we consider the possibilities for identifying the Chandler wobble parameters and fitting the developed refined theoretical model to the real trajectory measurements of the Earth pole.     

Upper Bound on the Density of Ruelle Resonances for Anosov Flows

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (−i)V, called Ruelle resonances, close to the real axis and for large real parts.     

Topological Calculation of the Phase of the Determinant of a Non Self-Adjoint Elliptic Operator

We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is real and its sign is determined by the parity of the number of the eigenvalues of the operator, which lie on the positive part of the imaginary axis. It follows that, for many geometrically defined operators, the phase of the determinant is a topological invariant. In numerous examples, coming from geometry and physics, we calculate the phase of the determinants in purely topological terms. Some of those examples were known in physical literature, but no mathematically rigorous proofs and no general theory were available until now.The first author was partially supported by the Alfred P. Sloan foundation.The second author was partially supported by the NSF grant DMS-0204421.     

Exponential convergence of a spectral projection of the KdV equation

It is shown that a spectral approximation of the Korteweg–de Vries equation converges exponentially fast to the true solution if the Fourier basis is used and if the solution is analytic in a fixed strip about the real axis. Computations are carried out which show that the exponential convergence rate can be achieved in practice.     

Application of the schwarz-christoffel mapping to planar gravity: static solutions

Point-like masses on a given initial domain of the complex plane are interacting globally under gravity and as a consequence the initial domain is mapped into a final domain which encodes all effects of localized sources . The mapping between the two domains is a truncated beta function which so far has not been fully investigated. In this paper we propose to know more about the final domain and so we start from the remark that when the masses are put on the real axis of the upper half plane, the mapping in question is the Schwarz-Christoffel mapping (SCM), a popular and useful tool in computational geometry. The linking of planar gravity with the SCM will allow us to use all what we know from this mapping to get answer for certain static solutions in planar gravity. Among the problems we will study we mention: masses distributed on the real axis of the full complex plane, masses distributed on a circle, the dust string either closed or infinite open. Strings with tensions are however not linked to SCM as these mappings do not involve the concept of tension.     

High-order cycles in the logistic map or centers of cardioids in the mandelbrot set

Asymptotic expressions for the positions of the centres (along the real axis) of high-order cycle cardioids closest to the limit point –2 in the Mandelbrot set are obtained by a combination of numerical and analytical methods.     

Delocalization of Slowly Damped Eigenmodes on Anosov Manifolds

We look at the properties of high frequency eigenmodes for the damped wave equation on a compact manifold with an Anosov geodesic flow. We study eigenmodes with damping parameters which are asymptotically close enough to the real axis. We prove that such modes cannot be completely localized on subsets satisfying a condition of negative topological pressure. As an application, one can deduce the existence of a ??strip?? of logarithmic size without eigenvalues below the real axis under this dynamical assumption on the set of undamped trajectories.