Analytic properties of moments matrices

Associated with each matrix element of the Generalized Moments Expansion, GMX(n,m) there is a unique expansion for the ground state energy in terms of the “connected moments” Ik of the Hamiltonian. That is, for any set {n,m} a polynomial in the Ik's may be generated to any desired order L, which is dependent upon the highest moment calculated. Here we wish to study the eigenvectors and eigenvalues of the GMX matrix itself. Furthermore we investigate the interplay between the set {n,m} and the order L of the matrix in determining which combination {n,m,L} yields the “best” (i.e. most convergent) result for the ground state energy.     

HCIZ integral and 2D Toda lattice hierarchy

The expression of the large N Harish Chandra–Itzykson–Zuber (HCIZ) integral in terms of the moments of the two matrices is investigated using an auxiliary unitary two-matrix model, the associated biorthogonal polynomials and integrable hierarchy. We find that the large N HCIZ integral is governed by the dispersionless Toda lattice hierarchy and derive its string equation. We use this to obtain various exact results on its expansion in powers of the moments.     

Connection between light cone singularities and the asymptotic behaviour of the moments of the structure functions

On the basis of the Dyson-Jost-Lehmann representation a light-cone type expansion of the matrix elements of the current commutator can be justified. Suitable moments of the structure functions being defined, the connection between the asymptotic behaviour of the moments as Q² → ∞ and the behaviour of the expansion coefficients as x² → 0 is investigated. To solve this problem a restriction of the allowed class of generalized functions is needed. Then a one-to-one correspondence between both limits can be established.     

Fixed angular momentum moments and level densities

Approximate fixed-J energy moments are evaluated by orthogonal polynomial expansion and parametric derivative methods. Comparisons with exact values, both directly in terms of moments and in terms of level densities calculated from energy moments, show that sufficiently accurate values can be calculated to produce good fixed-J level densities.     

An approach to NLO QCD analysis of the semi-inclusive DIS data with the modified Jacobi polynomial expansion method

The modification of the Jacobi polynomial expansion method (MJEM) is proposed on the basis of the application of the truncated moments instead of the full ones. This allows us to reconstruct the local quark helicity distributions with high precision even for the narrow Bjorken x region accessible for measurement, using as an input only the four first moments extracted from the data in the next to leading order QCD. The variational (extrapolation) procedure is also proposed allowing us to reconstruct the distributions outside the accessible Bjorken x region using the distributions obtained with MJEM in the accessible region. The numerical calculations encourage one that the proposed variational (extrapolation) procedure could be applied to estimate the full first (especially important) quark moments.     

Ground-state energy of a two level system with phonon coupling

The coupling of a two-level system to quantized boson modes has been the focus of many researchers for a number of years. Applications to exciton motion, molecular polaron formation, chaos in quantum systems as well as a number of other effects in condensed matter physics have also been studied. Here we investigate the interaction of bosonic modes with a two-level fermionic system. This quantum system is used as a testing ground for a recently developed Generalized Moments Expansion, GMX(m,n), of which the well-known Connected Moments Expansion (CMX) and Alternate Moments Expansion (AMX) are special cases. The convergence and viability of this scheme are discussed and comparisons are made with a related Canonical Sequence Method (CSM) as well as a Lanczos tridiagonal truncation scheme.     

xF 3 structure function based on the associated Jacobi polynomials

We present the results of the Next-to-leading order (NLO) non-singlet QCD analysis of the experimental data of the CCFR collaboration for the xF ₃ structure function of the deep-inelastic scattering of neutrinos on the nucleon in based on the associated Jacobi polynomials expansion of the structure functions. The structure function is reconstructed from its moments by using the expansion in terms of orthogonal associated Jacobi polynomials. Our results of valence quark distributions are in good agreement with the available theoretical models.     

The inverse problem in the case of bound states

We investigate the inverse problem for bound states in the D = 3 dimensional space. The potential is assumed to be local and spherically symmetric. The present method is based on relationships connecting the moments of the ground state density to the lowest energy of each state of angular momentum ?. The reconstruction of the density ρ(r) from its moments is achieved by means of the series expansion of its Fourier transform F(q). The large q-behavior is described by Padé approximants. The accuracy of the solution depends on the number of known moments. The uniqueness is achieved if this number is infinite. In practice, however, an accuracy better than 1% is obtained with a set of about 15 levels.The method is tested on a simple example, and applied to three different spectra.     

Model independent relations for baryons as solitons in mesonic theories

We derive model-independent relations for SU(2) and SU(3) chiral solitons. These relations depend only on the soliton picture of baryons and therefore are a test of the 1/N expansion. In the two-flavor case we discuss the magnetic moment and the electric quadrupole transition of the Δ. In SU(3) we discuss magnetic moments, including SU(3) breaking effects and 1/N corrections.     

Thermal expansion and lattice dynamics of RB66 compounds at low temperatures

Thermal characteristics of the phonon and magnon subsystems of icosahedral borides RB₆₆ (R = Gd, Tb, Dy, Ho, Eu, or Lu) have been studied based on the obtained experimental data on the thermal expansion of the borides and the earlier results on their heat capacity in the range of 2–300 K. The contribution to the expansion of borides containing paramagnetic R ³⁺ ions, which is characteristic of transition to the spin-glass state, has been revealed. The phonon spectrum moments of RB₆₆ compounds and the Grüneisen parameters have been calculated.     

Determination of the central density on the basis of its moments

The value of the central density is of key importance for annihilation processes. For the ground state we discuss its determination from the moments of the ground state density. We first review the way of reaching the moments from the spectrum. In particular we show how to get the lowest moments in D = 3, namely 〈r⁻²〉 and 〈r⁻¹〉 from the series expansion of the Laplace transform of the density. We then recall a method to obtain the central density based on the Stieltjes moment problem. If the number of known moments is finite, this technique yields a lower bound. We investigate the possibilities to estimate the accuracy of the bound and the corresponding asymptotic value. An application to the muonic ²⁰⁸Pb atom is presented.     

Adequate representation of the dipole moment function for diatomic molecules and the matrices of vibration-rotation transition moments

An exponential representation of perturbations is used as a basis of the perturbed Morse oscillator approach which is applied, in a matrix form, for calculating the radial matrix elements for diatomic molecules. An analytic procedure is developed to deduce an exponential-power series expansion for the dipole moment function M(r) from experimental spectral intensities. It is shown that for real anharmonic molecules, the series expansion in powers of e^ar (α being the Morse parameter) is an adequate form for representing transition operators, just as the usual series expansion in powers of internuclear distance r is adequate for the case of a harmonic oscillator, and it is equivalent to a series expansion in vibrational wavefunctions. An exponential-power series expansion is derived as well for a model dipole moment function which has a correct long-range dependence and limit. To exemplify the accuracy and efficiency of the technique proposed, the (40 × 40) matrices of vibration-rotation transition moments 〈vJ|M(r)|v′J′〉(v, v′ = 0, 1, …, 39) have been calculated for the ground state of CO. Typical results of these computations are presented (up to v = 35, J = 100, and v′ ? v = 1–4) to illustrate the dependence of vibration-rotation interaction functions on the vibrational and rotational quantum numbers.     

On the probability density of the envelope of a sum of random vectors

A most efficient way to calculate the probability density for the projection of the sum of n sinusoidal waves with differing amplitudes and phases that are uniform in (0, 2π) is to expand it in a Fourier series. The corresponding series for the envelope of such a series is known to be a Fourier-Bessel series. It is difficult, however, to calculate moments of the envelope from this series. It is shown that by relating the probability density of the envelope to that of the projection one can find moments of the envelope in an easily computable form. It is also possible to calculate an approximate form for the density function for the envelope amplitude near its maximum value. A generalized expansion is developed, valid when the probability density function depends on both angle and radial distance.     

Evaluation of ensemble averages for simple Hamiltonians perturbed by a GOE interaction

Using an expansion in powers of N^?1, where N is the dimension of the Hamiltonian matrix, we evaluate ensemble averages of the resolvent, of products involving several resolvents, and of the moments of the Hamiltonian H₀ + λV. Here, H₀ is arbitrary but fixed, and V is a GOE ensemble. The nature of the N^?1 expansion is also discussed.     

Evolution of the truncated Mellin moments of the parton distributions in QCD analysis

We review evolution equations for the truncated Mellin moments of the parton distributions and some their applications in QCD analysis. The main finding of the presented approach is that the nth truncated moment of the parton distribution obeys also the DGLAP equation but with a rescaled splitting function P′(z) = z ⁿ P(z). This allows one to avoid the problem of dealing with the experimentally unexplored Bjorken-x region. The evolution equations for truncated moments are universal—they are valid in each order of perturbation expansion and can be useful additional tool in analysis of unpolarized as well as polarized nucleon structure functions.     

FixedJ spectral distribution in large shell model spaces

A method is develloped to exactly calculate the fixedJ configuration centroid energies and widths. The resulting approximate level densities for²⁰⁴Pb and⁰²⁰Pb show large departures with respect to the gaussian approximate level densities. The goodness of the polynomial expansion of fixedJ configuration moments is studied and a simple improvement to this approximation is proposed which gives very good results.     

High-order connected moments expansion for the Rabi Hamiltonian

We analyze the convergence properties of the connected moments expansion (CMX) for the Rabi Hamiltonian. To this end we calculate the moments and connected moments of the Hamiltonian operator to a sufficiently large order. Our large-order results suggest that the CMX is not reliable for most practical purposes because the expansion exhibits considerable oscillations.     

Cluster Method Calculation of the Curie Temperature and Exchange Parameters for the Magnetocaloric Compounds MnFeAs x P1 − x (0.25 ≤ x ≤ 0.65) and Hexagonal MnFeAs

A wealth of experimental and theoretical data on the crystallographic and magnetic properties of the magnetocaloric compounds MnFeAs _x P_1???x (0.25?≤?x?≤?0.65) and MnFeAs has become available in the last decade. By analyzing the data and treating the spin interactions with Callen’s cluster expansion method, we extrapolate first-principle results for the exchange-coupling constants of MnFeAs to the P-substituted compounds and find Curie temperatures that agree, within 5 % deviation, with experiment. Simulations with different coupling parameters show that T _c is weakly dependent on the Fe–Fe interactions. Analysis of lattice expansion as a function of composition shows that changes in the lattice parameters a and c have opposite effects upon the strength of the magnetic interactions between ions. The results indicate that the cluster expansion method provides reliable estimates of magnetic properties, even for metallic compounds characterized by multiple interactions among ions with distinct magnetic moments.     

Two-loop QCD corrections to the coefficient functions of the deep inelastic structure functionsF 2 andF L

We present the analytic two-loop perturbative QCD corrections in the leading twist approximation to the coefficient functions of the operator product expansion for the second to tenth moments of the nonsinglet and singlet deep inelastic structure functionsF ₂ andF _L .     

Link-wise artificial compressibility method

The artificial compressibility method (ACM) for the incompressible Navier–Stokes equations is (link-wise) reformulated (referred to as LW-ACM) by a finite set of discrete directions (links) on a regular Cartesian mesh, in analogy with the lattice Boltzmann method (LBM). The main advantage is the possibility of exploiting well established technologies originally developed for LBM and classical computational fluid dynamics, with special emphasis on finite differences (at least in the present paper), at the cost of minor changes. For instance, wall boundaries not aligned with the background Cartesian mesh can be taken into account by tracing the intersections of each link with the wall (analogously to LBM technology). LW-ACM requires no high-order moments beyond hydrodynamics (often referred to as ghost moments) and no kinetic expansion. Like finite difference schemes, only standard Taylor expansion is needed for analyzing consistency. Preliminary efforts towards optimal implementations have shown that LW-ACM is capable of similar computational speed as optimized (BGK-) LBM. In addition, the memory demand is significantly smaller than (BGK-) LBM. Importantly, with an efficient implementation, this algorithm may be among the few which are compute-bound and not memory-bound. Two- and three-dimensional benchmarks are investigated, and an extensive comparative study between the present approach and state of the art methods from the literature is carried out. Numerical evidences suggest that LW-ACM represents an excellent alternative in terms of simplicity, stability and accuracy.