Perturbation theory and Padé approximants in realistic large-matrix models of the nuclear effective interaction

Realistic extended shell-model calculations are used to construct exact effective Hamiltonians, the perturbation series for the effective Hamiltonian to any order, and the [N+1, N] Padé approximants to the series. It is found that the Padé approximants give reliable results even when the series diverge, but that for both convergent and divergent series reasonably accurate results can be obtained only in fifth, or even seventh order. In addition, the poles of the low-order Padé approximants are not always reliable indicators of singularities of the perturbation series. The perturbation series and Padé approximants for the Q-box (energy-dependent effective Hamiltonian) are no more accurate in low orders than those for the usual effective Hamiltonian. Explicit formulas for the matrix Padé approximants are given in an appendix.     

Calculation of vibrational energy levels of triatomic molecules with the C2v and Cs symmetries by summing divergent series of the Rayleigh-Schr?dinger perturbation theory

The Rayleigh-Schr?dinger perturbation theory is applied to calculation of vibrational energy levels of triatomic molecules with the C _2v and C _s symmetries: SO₂, H₂S, F₂O, HOF, HOCl, and DOCl. Particular attention is given to the states coupled by anharmonic resonances; for such states, the perturbation theory series diverge. To sum these series, the known methods of Padé, Padé-Borel, and Padé-Hermite and the method of power moments are used. For low-lying levels, all the summation methods give satisfactory results, while the method of quadratic Padé-Hermite approximants appears to be more efficient for high-excited states. Using these approximants, the structure of singularities of the vibrational energy, as a function in the complex plane, is studied.     

Effective conductivity for densely packed highly conducting cylinders

We study the effective heat conductivity λ₃ of a periodic square array of nearly touching cylinders of conductivity h, embedded in a matrix material of conductivity 1. We construct a sequence of two-point Padé approximants for the effective conductivity. As the basis for the construction we use the coefficients of the expansions of λ_e at h=1 and h=∞. The two-point Padé approximants form a sequence of rapidly converging upper and lower bounds on the effective conductivity.     

Variational Matrix Padé Approximants in Two Body Scattering

The convergence and bounding properties of the variational matrix Padé approximants are investigated for non relativistic two body interactions. Selecting L – 1 discrete values qi, i = 1, …, L – 1 and the physical momentum q₀ the off shell scattering amplitudes are L X L matrices. The [N/N] Padé approximants to the Born series of these matrices are the variational solution of the Schwinger principle and the corresponding physical amplitude has variational properties in the off shell momenta. For positive interactions the best approximants to the phase shift is an absolute minimum on the q_i and monotonic convergence to the exact result for N → ∞ or L → ∞ ca be proved. Similar properties are shown for the bound states using the Ritz variational principle. The required mathematical background is extensively worked out, the extensions to non positive, singular and long range potentials are considered and some numerical examples are presented.     

Padé approximation methods applied to the intermolecular force series

The H(1s)-H⁺, He-He, H(1s)-H(1s) and H(1s)-H(2s) interactions are considered as model systems for investigating the use of the Padé approximation method in summing the R ^-1 intermolecular force series. Various Padé approximants and partial sums of the R ^-1 expansions of the second-order Coulomb interaction energies are compared with the corresponding non-expanded results for each interaction. The computations are based on Unsöld's average energy approximation and on exact results for the H(1s)-H interaction. The results indicate that the Padé approximation method is a simple, useful way to remove some of the difficulties associated with the slow rate of convergence of the R ^-1 force series but that it does not alleviate the problems associated with the asymptotic divergent nature of the series. The results for the H(1s)-H⁺ interaction illustrate a possible difficulty in using Unsöld's method in the calculation of interaction energies.     

Perturbation expansions and series acceleration procedures. Part I. ε-convergence and critical parameters

A simple acceleration of convergence technique known as the ‘ε-convergence algorithm’ (ea) is applied to determine the critical temperatures and exponents. Several illustrations involving well-known series expansions appropriate to two- and three-dimensional Ising models, three-dimensional Heisenberg models, etc., are given. Apart from this, a few recently studied ferrimagnetic systems have also been analysed to emphasise the generality of the approach. Where exact solutions are available, our estimates obtained from this procedure are in excellent agreement. In the case of other models, the critical parameters we have obtained are consistent with other estimates such as those of the Padé approximants and group theoretic methods. The same procedure is applied to the partial virial series for hard spheres and hard discs and it is demonstrated that the divergence of pressure occurs when the close-packing density is reached. The asymptotic form for the virial equation of state is found to beP/ρkT ∼ (1 −ρ/ρ _c ⁻¹ for hard spheres and hard discs. Apart from the estimation of ‘critical parameters’, we have applied theea and the parametrised Euler transformation to sum the partial, truncated virial series for hard spheres and hard discs. The resulting values of pressure so obtained, compare favourably with the molecular dynamics results.     

Padé approximants and the effective interaction

Analytic properties of the effective interaction allow us to indicate the positions of the poles of low-order Padé approximants and the domain of convergence of the series of Padé approximants. All evidence favors the conjecture that the Padé approximants will converge to that branch of the effective interaction which reproduces the model space states, if the series converges.     

A method for summation of perturbation series

A method for summation of perturbation series is developed. It consists of a proper rearrangement of the power series through an appropriate redefinition of the perturbation parameter. Well-known divergent power series appearing in a φ⁴-scalar field theory or in the treatment of the linear confining potential model and the Stark effect in hydrogen are used as illustrative examples. A comparison between present results and accurate, numerical ones shows that our series converge even faster than sequences of Padé or Borel-Padé approximants.     

Convergence properties of methods for effective operator calculations studied in a simple many-body model

Several methods for effective interaction and operator calculations based on the Rayleigh-Schrödinger and Brillouin-Wigner perturbation expansions are studied. Special emphasis is given to the use of Padé approximants for effective operator calculations. The convergence properties of the methods are studied numerically in a Lipkin many-body model. Among the various methods using Padé-approximants, the method based on a variational approach in the BW scheme is found to give most encouraging results for the present model.     

Comparative Studies of the Exchange-Correlation Potential Formulas for Coulomb Systems

The exchange-correlation part (xc) to the free energy is numerical evaluated in the RPA at arbitrary degree of degeneracy. The results are compared with numerical data of easy-to-use analytic fit-formulas or Padé approximants of the xc-term. All together results show very high accuracy at extremly high densities (r_s ≈ 1). The agreements disappear between the several formulas for increasing Brueckner parameter r_s. Numerical results for the xc-potentials (pressure and chemical potential) at finite temperatures for an electron-ion system are given. The xc-part of the ground state energy of our electron-ion model is compared with the ground state energy for metallic hydrogen and with Monte-Carlo calculations.     

Continuum Limit of Susceptibility from Strong Coupling Expansion

On the basis of a strong-coupling expansion, we reinvestigate the scaling behavior of the susceptibility ?? of the two-dimensional O(N) sigma model on the square lattice with Padé?CBorel approximants. To exploit the Borel transform, we express the bare coupling g in a series expansion in ??. For large N, the Padé?CBorel approximants exhibit scaling behavior at the four-loop level. We estimate the nonperturbative constant associated with the susceptibility for N????3 and compare the results with previous analytica l results and Monte Carlo data.     

Expressions for the computer-evaluation of the four kernel functions for line formation with Doppler and Lorentz profiles

Rational approximations for the kernel functions K₁(τ), K₂(τ), M₁(τ) and M₂(τ) that describe the transfer of radiation scattered with complete redistribution over Doppler and Lorentz profiles have been obtained from their series and asymptotic expressions by the techniques of Padé approximants with a maximum relative error of less than 10^-4.     

Maximum power output of a class of irreversible non-regeneration heat engines with a non-uniform working fluid and linear phenomenological heat transfer law

Maximum power output of a class of irreversible non-regeneration heat engines with non-uniform working fluid, in which heat transfers between the working fluid and the heat reservoirs obey the linear phenomenological heat transfer law [q ∝ Δ(T ⁻¹)], are studied in this paper. Optimal control theory is used to determine the upper bounds of power of the heat engine for the lumped-parameter model and the distributed-parameter model, respectively. The results show that the maximum power output of the heat engine in the distributed-parameter model is less than or equal to that in the lumped-parameter model, which could provide more realistic guidelines for real heat engines. Analytical solutions of the maximum power output are obtained for the irreversible heat engines working between constant temperature reservoirs. For the irreversible heat engine operating between variable temperature reservoirs, a numerical example for the lumped-parameter model is provided by numerical calculation. The effects of changes of reservoir’s temperature on the maximum power of the heat engine are analyzed. The obtained results are, in addition, compared with those obtained with Newtonian heat transfer law [q ∝ Δ(T)].     

Radiation transfer by a finite-emitting inhomogeneous medium

A functional relation is obtained between radiative transfer in an inhomogeneous finite planar layer with an internal energy source and diffuse reflection. The intensity is derived for the emerging radiation of a polynomial energy source. We use Padé approximants to calculate the emitted intensity for a linear energy source when the single scattering albedo decreases exponentially with optical depth. Numerical results are given for both homogeneous and inhomogeneous media.     

Universal effective coupling constants for the generalized Heisenberg model

The aim of this study is to find universal critical values of the effective dimensionless coupling constant g ₆ and refined universal values g ₄ for Heisenberg ferromagnets with n-component order parameters. These constants appear in the equation of state and determine the nonlinear susceptibilities χ ₄ and χ ₆ in the critical region. Calculations are made of the first three terms of the expansion of g ₆ in powers of g ₄ in the limits of O(n) symmetry three-dimensional λϕ ⁴ theory, the resultant series is resummed by the Padé-Borel method, and then by substituting the fixed point coordinates g ₄ ^* in the resultant expression, numerical values of g ₆ ^* are obtained for different n. These numbers g ₄ ^* for n>3 were determined from a six-loop expansion for the β-function resummed using the Padé-Borel-Leroy technique. An analysis of the accuracy of these g ₆ ^* values showed that they may differ from the true values by no more than 1.6%. These values of g ₆ ^* were compared with those obtained by the 1/n expansion method which allowed the level of accuracy of this method to be assessed. Fiz. Tverd. Tela (St. Petersburg) 40, 1284–1290 (July 1998)     

Strong coupling analysis of the SO(3) lattice gauge theory at different space-time dimensionalities

High order strong coupling series for free energy, internal energy and specific heat are constructed to study the SO(3) lattice gauge theory at space-time dimensionalities from d = 2 to 7. Padé approximants are used to look for singularities. First-order phase transitions for d = 4, 5, 6 and 7 are clearly detected. A second-order phase transition seems to be present in the three-dimensional case.     

Interpolation parameter and expansion for a three-dimensional nontrivial scalar infrared fixed point

We compute a nontrivial infraredϕ ₃ ⁴ -fixed point by means of an interpolation expansion in fixed dimension. The expansion is formulated for an infinitesimal momentum-space renormalization group. We choose a coordinate representation for the fixed-point interaction in derivative expansion, and compute its coordinates to high orders by means of computer algebra. We compute the series for the critical exponentv up to order 25 of interpolation expansion in this representation, and evaluate it using Padé, Borel-Padé, Borel-conformal-Padé, andD log -Padé resummation. The resummation returns 0.6262(13) as the value ofv. Our renormalization group uses canonical resealing, for whichη = 0     

Efficient beam-propagation method based on Padé approximants in the propagation direction

We propose a novel beam-propagation method (BPM) based on the higher-order Padé approximants in both the transverse and the longitudinal directions. Extending the Padé approximation to the propagation direction and adopting the multistep method increase the programming effort by only a small amount. Compared with the conventional BPM's, this method is more accurate and efficient. The accuracy and the propagation step size are well predicted by a phase analysis.     

Optimal configuration of a class of endoreversible heat engines for maximum efficiency with radiative heat transfer law

Optimal configuration of a class of endoreversible heat engines with fixed duration, input energy and radiative heat transfer law (q ∝ Δ(T ⁴)) is determined. The optimal cycle that maximizes the efficiency of the heat engine is obtained by using optimal-control theory, and the differential equations are solved by the Taylor series expansion. It is shown that the optimal cycle has eight branches including two isothermal branches, four maximum-efficiency branches, and two adiabatic branches. The interval of each branch is obtained, as well as the solutions of the temperatures of the heat reservoirs and the working fluid. A numerical example is given. The obtained results are compared with those obtained with the Newton’s heat transfer law for the maximum efficiency objective, those with linear phenomenological heat transfer law for the maximum efficiency objective, and those with radiative heat transfer law for the maximum power output objective.