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We present efficient algorithms for multi-precision and multi-accuracy calculation of error functions and the Dawson integral, all with complex arguments. These algorithms achieve exceptional accuracies, ranging from 26 significant digits (SD) up to 30 SD depending on the function. In addition to the Faddeyeva or Faddeeva function, or the scaled complementary error function, , the list of functions considered include the error function, erf z ; the complementary error function, erfc z ; the imaginary error function, erfi (z) ; and the Dawson integral Daw (z) . These algorithms are integrated into a modern Fortran module confirming the claimed accuracies and superior efficiency compared to other competitive codes in the literature. Additionally, we highlight an observation regarding the built-in " Erfc _ Scaled (x) " function included in a widely used Fortran compiler, which exhibits a significant decline in accuracy when employed in quadruple-precision arithmetic. [ABSTRACT FROM AUTHOR]

As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of complex random normal matrix models that one finds a complementary error function behavior at the boundary (also called edge) of the droplet as the matrix size increases. Such behavior is seen both in the density of the eigenvalues and the correlation kernel, where the Faddeeva plasma kernel emerges. These results are neatly expressed with the help of the outward unit normal vector on the edge. We prove that such universal behaviors transcend this class of random normal matrices, being also valid in a specific "elliptic" class of determinantal point processes defined on C d , which are higher dimensional generalizations of the determinantal point processes describing the eigenvalues of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble. These models describe a system of particles in C d with mutual repulsion, that are confined to the origin by an external field V (z) = | z | 2 - τ Re (z 1 2 + ... + z d 2) , where 0 ≤ τ < 1 . Their average density of particles converges to a uniform law on a 2d-dimensional ellipsoidal region. It is on the hyperellipsoid bounding this region that we find a complementary error function behavior and the Faddeeva plasma kernel. To the best of our knowledge, this is the first instance of the Faddeeva plasma kernel emerging in a higher dimensional model. The results provide evidence for a possible edge universality theorem for determinantal point processes on C d . [ABSTRACT FROM AUTHOR]

We derive a high-rank separable potential formula of the atom-atom interaction by using the two-body wave function in the coordinate space as inputs. This high-rank separable potential can be utilized to numerically solve the two-body Lippmann-Schwinger equation and three-body Faddeev equation. By analyzing the convenience and stability of numerical calculations for different kinds of the matrix forms of the Lippmann-Schwinger and Faddeev equations, we can find the optimal forms of the kernal matrices in the two- and three-body scattering equations. We calculate the dimer bound energy, two-body scattering phase shift and off-shell t-matrix, the trimer bound energy, atom-dimer scattering length, and three-body recombination rate using the high-rank separable potentials, taking the identical 4He atoms as an application example. All the calculations converge quickly for the rank number N ⩾ 3. The high-rank separable potential is valid for two-body scattering calculation of 4He atoms, but not accurate enough for reproducing the three-body scattering results by using only two-body s-wave interaction and describing the contributions of two-body high partial-waves to the three-body scattering for the 4He3 system. [ABSTRACT FROM AUTHOR]

In this work we develop a new algorithm for the efficient computation of the Voigt/complex error function. In particular, in this approach we propose a two-domain scheme where the number of the interpolation grid-points is dependent on the input parameter y. The error analysis we performed shows that the MATLAB implementation meets the requirements for radiative transfer applications involving the HITRAN molecular spectroscopic database. The run-time test shows that this MATLAB implementation provides rapid computation, especially at smaller ranges of the parameter x. [ABSTRACT FROM AUTHOR]

In the last seven years, Kaniadakis statistics, or κ -statistics, have been applied in reactor physics to obtain generalized nuclear data, which can encompass, for instance, situations that lie outside thermal equilibrium. In this sense, numerical and analytical solutions were developed for the Doppler broadening function using the κ -statistics. However, the accuracy and robustness of the developed solutions contemplating the κ distribution can only be appropriately verified if applied inside an official nuclear data processing code to calculate neutron cross-sections. Hence, the present work inserts an analytical solution for the deformed Doppler broadening cross-section inside the nuclear data processing code FRENDY, developed by the Japan Atomic Energy Agency. To do that, we applied a new computational method called the Faddeeva package, developed by MIT, to calculate error functions present in the analytical function. With this deformed solution inserted in the code, we were able to calculate, for the first time, deformed radiative capture cross-section data for four different nuclides. The usage of the Faddeeva package brought more accurate results when compared to other standard packages, reducing the percentage errors in the tail zone in relation to the numerical solution. The deformed cross-section data agreed with the expected behavior compared to the Maxwell–Boltzmann. [ABSTRACT FROM AUTHOR]

Faddeeva–Voigt broadening (FVB) couples the physical characteristics of both Lorentzian and Gaussian profiles as a combined analytic function shaping the dielectric response. Accurate extraction of the Gaussian and Lorentzian broadening contents in line-shape analysis is essential for reliable optical characterization of semiconductors and dielectrics. By adding the Gaussian-broadening width to each Lorentzian width, we investigate how FVB affects critical-point (CP) analysis. We revisit a selection of earlier work based on classical Lorentz broadening in modulation spectroscopy and spectral ellipsometry. To generalize CP analysis, we derive the FVB's analytical representation in terms of fractional derivatives of the Faddeeva function and apply the twenty-pole Martin–Donoso–Zamudio approximation for its precise and efficient computation of the FVB of model dielectric functions and derivatives. We investigate the FVB of the electroreflectance line shape of HgCdTe for three-dimensional M0 transitions and of the photoreflectance line shape of InP excitonic E0 transitions. Furthermore, we explore how FVB affects the dielectric functions of three-dimensional excitonic and two-dimensional M0 transitions vs Tanguy's analytical two-dimensional exciton E1 and E1+Δ1 fits of GaAs to the second-order derivatives. We use the Akaike information criterion to quantitatively estimate the goodness of fit that statistically penalizes overfitting due to extraneous parameters. By consolidating both Gaussian and Lorentzian broadenings, the FVB significantly affects the CP analysis of modulation-spectroscopy line shapes and second-order derivatives of the dielectric function. [ABSTRACT FROM AUTHOR]

In this paper we propose a method for computing the Faddeeva function w(z):... erfc(-iz) via truncated modified trapezoidal rule approximations to integrals on the real line. Our starting point is the method due to Matta and Reichel (Math. Comp. 25 (1971), pp. 339-344) and Hunter and Regan (Math. Comp. 26 (1972), pp. 339-541). Addressing shortcomings flagged by Weideman (SIAM. J. Numer. Anal. 31 (1994), pp. 1497-1518), we construct approximations which we prove are exponentially convergent as a function of N+1, the number of quadrature points, obtaining error bounds which show that accuracies of 2x10-15 in the computation of w(z) throughout the complex plane are achieved with N=11, this confirmed by computations. These approximations, moreover, provably achieve small relative errors throughout the upper complex half-plane where w(z) is non-zero. Numerical tests suggest that this new method is competitive, in accuracy and computation times, with existing methods for computing w(z) for complex z [ABSTRACT FROM AUTHOR]

The |$\Xi$| single-particle potential obtained in nuclear matter with next-to-leading-order baryon–baryon interactions in chiral effective field theory is applied to finite nuclei by an improved local-density approximation method. As a premise, phase shifts of |$\Xi N$| elastic scattering and the results of Faddeev calculations for the |$\Xi NN$| bound state problem are presented to show the properties of the |$\Xi N$| interactions in the present parametrization. First, the |$\Xi$| states in |$^{14}$| N are revisited because of recent experimental progress, including discussion on the |$\Xi N$| spin–orbit interaction that is relevant to the location of the |$p$| -state. Then the |$\Xi$| levels in |$^{56}$| Fe are calculated. In particular, the level shift which is expected to be measured experimentally in the near future is predicted. The smallness of the imaginary part of the |$\Xi$| single-particle potential is explicitly demonstrated. [ABSTRACT FROM AUTHOR]

In the last seven years, Kaniadakis statistics, or κ-statistics, have been applied in reactor physics to obtain generalized nuclear data, which can encompass, for instance, situations that lie outside thermal equilibrium. In this sense, numerical and analytical solutions were developed for the Doppler broadening function using the κ-statistics. However, the accuracy and robustness of the developed solutions contemplating the κ distribution can only be appropriately verified if applied inside an official nuclear data processing code to calculate neutron cross-sections. Hence, the present work inserts an analytical solution for the deformed Doppler broadening cross-section inside the nuclear data processing code FRENDY, developed by the Japan Atomic Energy Agency. To do that, we applied a new computational method called the Faddeeva package, developed by MIT, to calculate error functions present in the analytical function. With this deformed solution inserted in the code, we were able to calculate, for the first time, deformed radiative capture cross-section data for four different nuclides. The usage of the Faddeeva package brought more accurate results when compared to other standard packages, reducing the percentage errors in the tail zone in relation to the numerical solution. The deformed cross-section data agreed with the expected behavior compared to the Maxwell–Boltzmann.

In this work we develop a new algorithm for the efficient computation of the Voigt/complex error function. In particular, in this approach we propose a two-domain scheme where the number of the interpolation grid-points is dependent on the input parameter y. The error analysis we performed shows that the MATLAB implementation meets the requirements for radiative transfer applications involving the HITRAN molecular spectroscopic database. The run-time test shows that this MATLAB implementation provides rapid computation, especially at smaller ranges of the parameter x.

A new method for approximating fractional derivatives of Faddeeva function, Gaussian function, and Dawson's integral are presented. Unlike previous approaches, which are dominantly based on some discretization of Riemann‐Liouville integral over a finite interval using polynomial or discrete Fourier basis, we take an alternative approach that is based on expressing computation of Riemann‐Liouville definition of the fractional integral for the semi‐infinite axis in terms of a moment problem. As a result, fractional derivatives of Faddeeva function, Gaussian function, and Dawson's integral are expressed as a weighted sum of complex scaled Gaussian and Dawson's integral. Error bounds for the approximations are provided. Another feature of the proposed method compared with the previous approaches is it can be extended to approximate partial derivatives with respect to the order of the fractional derivative that can be used in PDE constraint optimization problems. [ABSTRACT FROM AUTHOR]

In a previous contribution to this journal [H. P. Stormberg, J. Appl. Phys. 51(4), 1963 (1980)], Stormberg presented an analytical expression for the convolution of Lorentz and Levy line profiles, which models atomic radiative transitions in high pressure plasmas. Unfortunately, the derivations are flawed with errors and the final expression, while correct, is accompanied by misguiding comments about the meaning of the symbols used therein, in particular the 'complex error function.' In this paper, we discuss the broadening mechanisms that give rise to Stormberg's model and present a correct derivation of his final result. We will also provide an alternative expression, based on the Faddeeva function, which has decisive computational advantages and emphasizes the real-valuedness of the result. The MATLAB/Octave scripts of our implementation have been made available on the publisher's website for future reference. [ABSTRACT FROM AUTHOR]

We present a steady Euler ow on the round 3-sphere S³ whose velocity vector fleld has the remarkable property of having two independent first integrals, being tangent to the flbres of an almost submersion onto the 2-sphere. This submersion turns out to be a critical point for the quartic Faddeev-Skyrme model with a standard potential. [ABSTRACT FROM AUTHOR]

Abstract The aim of this work is to discuss some aspects of the reduction of order formalism in the context of the Fadeev–Jackiw symplectic formalism, both at the classical and the quantum level. We start by reviewing the symplectic analysis in a regular theory (a higher derivative massless scalar theory), both using the Ostrogradsky prescription and also by reducing the order of the Lagrangian with an auxiliary field, showing the equivalence of these two approaches. The interpretation of the degrees of freedom is discussed in some detail. Finally, we perform the similar analysis in a singular higher derivative gauge theory (the Podolsky electrodynamics), in the reduced order formalism: we claim that this approach have the advantage of clearly separating the symplectic structure of the model into a Maxwell and a Proca (ghost) sector, thus complementing the understanding of the degrees of freedom of the theory and simplifying calculations involving matrices. [ABSTRACT FROM AUTHOR]

Abstract We consider 5d Yang–Mills–Higgs theory with a compact ADE-type gauge group G and one adjoint scalar field on R 3 , 1 × R + , where R + = [ 0 , ∞) is the half-line. The maximally supersymmetric extension of this model, with five adjoint scalars, appears after a reduction of 6d N = (2 , 0) superconformal field theory on R 3 , 1 × R + × S 1 along the circle S 1. We show that in the low-energy limit, when momenta along R 3 , 1 are much smaller than along R + , the 5d Yang–Mills–Higgs theory reduces to a nonlinear sigma model on R 3 , 1 with a coset G / H as its target space. Here H is a closed subgroup of G determined by the Higgs-field asymptotics at infinity. The 4d sigma model describes an infinite tower of interacting fields, and in the infrared it is dominated by the standard two-derivative kinetic term and the four-derivative Skyrme–Faddeev term. [ABSTRACT FROM AUTHOR]

Highlights • Fano line shapes are frequently encountered in atomic spectroscopy. • Line-profile fitting to experimental data requires a convolution with an experimental window function. • In the past, the convolution was carried out by using either slow numerical or inaccurate approximation procedures. • Here an analytic formula for the convolution of a Fano profile with a Gaussian is derived. • The formula involves the Faddeeva function for which efficient and accurate algorithms are available. Abstract Asymmetric Fano line profiles are frequently encountered, e. g., in the photoionization spectra of atoms and ions. For the fitting of spectral line profiles to experimental spectra the line profiles have to be convolved with the experimental window function. The latter is often taken to be a gaussian. It is shown that the convolution can be represented by a rather simple analytic expression involving the Faddeeva function for the evaluation of which efficient and accurate numerical algorithms are available. [ABSTRACT FROM AUTHOR]

The interest in higher derivatives field theories has its origin mainly in their influence concerning the renormalization properties of physical models and to remove ultraviolet divergences. The noncommutative Podolsky theory is a constrained system that cannot be directly quantized by the canonical way. In this work, we have used the Faddeev–Jackiw method in order to obtain the Dirac brackets of the NC space–time Maxwell, Proca and Podolsky theories. [ABSTRACT FROM AUTHOR]

A presentation à la Faddeev-Reshetikhin-Takhtajan (FRT) of the Onsager, augmented Onsager and sl2-invariant Onsager algebras is given, using the framework of the nonstandard classical Yang-Baxter algebras. Associated current algebras are identified, and generating functions of mutually commuting quantities are obtained. [ABSTRACT FROM AUTHOR]

Abstract By solving exactly the Faddeev equations for the bound-state problem of three mesons, we demonstrate that current theoretical predictions pointing to the existence of a deeply-bound doubly bottom axial vector tetraquark lead to the existence of a unique bound state of three B mesons. We find that the B B ⁎ B ⁎ − B ⁎ B ⁎ B ⁎ state with quantum numbers (I) J P = (1 / 2) 2 − , T b b b , is about 90 MeV below any possible three B -meson threshold for the reported binding of the doubly bottom axial vector tetraquark, T b b. [ABSTRACT FROM AUTHOR]

Atom-molecular collisions are studied using the Faddeev differential equations. The results of calculation of the scattering length for the collision of a 4He atom with a 4He7Li dimer are presented. [ABSTRACT FROM AUTHOR]

The symplectic analysis for the four dimensional Pontryagin and Euler invariants is performed within the Faddeev–Jackiw context. The Faddeev–Jackiw constraints and the generalized Faddeev–Jackiw brackets are reported; we show that in spite of the Pontryagin and Euler classes give rise the same equations of motion, its respective symplectic structures are different to each other. In addition, a quantum state that solves the Faddeev–Jackiw constraints is found, and we show that the quantum states for these invariants are different to each other. Finally, we present some remarks and conclusions. [ABSTRACT FROM AUTHOR]

We derive trace formulas of the Buslaev-Faddeev type for quantum star graphs. One of the new ingredients is high energy asymptotics of the perturbation determinant. [ABSTRACT FROM AUTHOR]

We study the negatively T − and positively T + charged trions in bulk materials in the effective mass approximation within the framework of a potential model. The binding energies of trions in various semiconductors are calculated by employing Faddeev equation in configuration space. Results of calculations of the binding energies for T − are consistent with previous computational studies and are in reasonable agreement with experimental measurements, while the T + is unbound for all considered cases. The mechanism of formation of the binding energy of trions is analyzed by comparing contributions of a mass-polarization term related to kinetic energy operators and a term related to the Coulomb repulsion of identical particles. [ABSTRACT FROM AUTHOR]

In this work we show a rational approximation of the Dawson’s integral that can be implemented for high accuracy computation of the complex error function in a rapid algorithm. Specifically, this approach provides accuracy exceeding ∼ 10 − 14 in the domain of practical importance 0 ≤ y < 0.1 ∩ | x + i y | ≤ 8 . A Matlab code for computation of the complex error function with entire coverage of the complex plane is presented. [ABSTRACT FROM AUTHOR]

We present a construction of an improved two-mode model for modeling the dynamics of interacting ultra-cold bosons confined in a one-dimensional double well trap. Unlike in the typically used two-mode model based on the lowest single-particle eigenstates of the external potential, the improved model uses a basis of properly chosen effective wave functions originating in the many-body model. Accuracy of the improved model is examined and it is shown that within a certain limit of inter-particle interaction strength, the model recovers an exact evolution of the wells' populations much more closely than the traditional two-mode model. [ABSTRACT FROM AUTHOR]

The 12 C ( p, γ ) 13 N reaction is the first process in the CNO cycle. Also it is a source of low-energy solar neutrinos in various neutrino experiments. Therefore, it is of high interest to gain data of the astrophysical S-factor in low energies. By applying Faddeev’s method, we calculated wave functions for the bound state of 13 N . Then the cross sections for resonance and non-resonance were calculated through using Breit–Wigner and direct capture cross section formulae, respectively. After that, we calculated the total S-factor and compared it with previous experimental data, revealing a good agreement altogether. Then, we extrapolated the S-factor in zero energy and the result was 1.32  ±  0.19 (keV.b). In the end, we calculated reaction rate and compared it with NACRE data. [ABSTRACT FROM AUTHOR]

We present a general form of the three-nucleon scattering amplitude. Our result is an operator form in which the scattering amplitude is written as a linear combination of scalar functions and operators acting on spin states. Using this form greatly reduces the numerical complexity of the so-called, three dimensional treatment of the Faddeev equations and can potentially lead to more accurate calculations of scattering observables at higher energies. [ABSTRACT FROM AUTHOR]