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.     

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.     

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.     

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.     

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.     

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.     

Solitons and poles in the nonlinearity parameter

The KdVN-soliton solutions are analysed in terms of the perturbation parameter λ which governs the nonlinearity. They are generated by rational Stieltjes functionsQ ^(N) (λ), each pole of which can be associated with a soliton. The asymptotic emergence of the separate solitons follows at once from the motion of the poles along the negative real λ-axis. Successive diagonal Padé approximants ofQ ^(N) (λ) are considered. They provide a class of approximate solutions with a striking semisoliton like behaviour.     

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.     

An adaptable summation algorithm

A simple generalization of Wynn's ϵ algorithm is reported. The algorithm includes a parameter K, and yields Padé approximants at K = 1 and repeated Aitken transforms at K = 0. Test on several perturbation series show that the choice K = −1 leads to accurate estimates of perturbation sums.     

Evaluation of dipole and quadrupolar contributions to the asymptotic atom-surface interaction

The position of the reference plane z₀ as well as the dipole and quadrupolar contributions to the long-range atom-surface interaction are evaluated within a simple model. The model treats the substrate as a bulk material and a selvedge, each of which has constant electronic density. The required polarizabilities of the atom are treated as one-term Padé approximants. It is then straightforward to calculate the electronic susceptibility and evaluate expressions for C₃, C′₅, and z₀ that have been recently derived.     

Estimating eigenvalues for lattice hamiltonian models

A straightforward technique is presented for estimating the eigenvalues of a quantum hamiltonian on a lattice. It produces a more reliable and more convergent sequence of estimates than previous techniques based on Padé approximants.     

Forbidden vibrational-rotational transitions and Herman-Wallis factors in fundamental IR bands of symmetric-top molecules

Within the theory of coupled schemes of ordering of vibrational-rotational interactions, the operator of the effective dipole moment of single-quantum vibrational transitions is represented in the form of an infinite series in vibrational (normal coordinates and conjugate momenta) or rotational variables (components of the total angular momentum). Mechanisms of activation of infrared-inactive totally symmetric vibrations in molecules of the D _2a , D _3h , C _3h , D _n (n ≥ 3), S ₄, T, T _a , and O symmetries and forbidden vibrational-rotational transitions in IR bands of active vibrations have been studied. The group-theoretic analysis of tensor parameters in higher-order effective dipole moments of single-quantum vibrational transitions in axially symmetric molecules has been performed. The strengths of allowed transitions and forbidden transitions in fundamental and hot IR bands of axially symmetric molecules are calculated with allowance for the Herman-Wallis factors. For effective dipole moments of multiquantum transitions in molecules, models are developed in the form of infinite series in rotational variables and in the form of Padé approximants.     

Low-energy kaon-nucleon scattering in the I = 0 state

The KN system in the I = 0 state is studied with the method of Padé approximants. The treatment follows the lines of the previous work of the same authors on the I = 1 interaction. From the same Lagrangian, the unitary [1,1] Padé approximants for the scattering functions in the $\text{s, p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{and p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ waves are calculated at low energies in terms of a single coupling constant. The model predicts negative and rather strong s-wave scattering lengths and phase shifts, while the two p-waves are small and of opposite signs. The differential and total cross sections for elastic and charge-exchange processes in K⁺ scattering by neutrons and the differential polarization in K⁺n → K⁰p are compared with data extracted from experiments on K⁺ scattering by deuterons. The model predicts that the cross section in the I = 0 state decreases slowly with energy.     

Derivation of effective spin models from a three band model for CuO-planes

The derivation of effective spin models describing the low energy magnetic properties of undoped CuO₂-planes is reinvestigated. Our study aims at a quantitative determination of the parameters of effective spin models from those of a multi-band model and is supposed to be relevant to the analysis of recent improved experimental data on the spin wave spectrum of La₂CuO₄. Starting from a conventional three-band model we determine the exchange couplings for the nearest and next-nearest neighbor Heisenberg exchange as well as for 4- and 6-spin exchange terms via a direct perturbation expansion up to 12th (14th for the 4-spin term) order with respect to the copper-oxygen hopping t_pd. Our results demonstrate that this perturbation expansion does not converge for hopping parameters of the relevant size. Well behaved extrapolations of the couplings are derived, however, in terms of Padé approximants. In order to check the significance of these results from the direct perturbation expansion we employ the Zhang-Rice reformulation of the three band model in terms of hybridizing oxygen Wannier orbitals centered at copper ion sites. In the Wannier notation the perturbation expansion is reorganized by an exact treatment of the strong site-diagonal hybridization. The perturbation expansion with respect to the weak intersite hybridizations is calculated up to 4th order for the Heisenberg coupling and up to 6th order for the 4-spin coupling. It shows excellent convergence and the results are in agreement with the Padé approximants of the direct expansion. The relevance of the 4-spin coupling as the leading correction to the nearest neighbor Heisenberg model is emphasized. Received 8 June 2001 / Received in final form 28 May 2002 Published online 19 July 2002     

Properties of Padé approximants to the effective interaction

Some properties of Padé approximants to effective interactions in nuclei are investigated. The behaviour of higher order approximants is discussed in model calculations. In the case of [2, l]-approximants a criterion is derived for the occurrence of spurious poles which are not related to level crossings.     

A strong-coupling expansion for fermion field theories and the large-N limit of the gross-neveu model

An expansion in the fermion propagator is formulated for the N-species Gross-Neveu model in the large-N limit. Different regularisation schemes may be adopted and we compare two. We find that a continuum momentum cut-off is easiest to work with and automatically avoids spurious fermionic states which afflict a naive lattice formulation. Chiral symmetry is broken at zeroth order and the resulting expansion is inverse powers of g²N simplifies considerably for large N. In this limit the strong-coupling expansion may be summed to all orders. Extrapolation techniques, like Padé approximants, are not needed. Using a momentum cut-off we recover all the exact results previously derived by summing weak-coupling expansions.     

Large q expansions for q-state gauge-matter Potts models in lagrangian form

We consider the lagrangian form of a q-state generalization of Ising gauge theories with matter fields in d = 3 and 4 dimensions. The theory is exactly soluble in the limit q → ∞ and corrections are easily calculable in power series in $\text{1}\text{q}{}^{\mathrm{<mtext>1</mtext><mtext>d</mtext>}}$. Extrapolating the series for the free energies and latent heats by the method of Padé approximants, we have constructed the phase diagrams for all values of q. Our results agree well with known results for pure spin systems and, for the case q = 2, with Ising Monte Carlo data.     

Series solutions of stagnation slip flow and heat transfer by the homotopy analysis method

An analytical approximation for the similarity solutions of the two- and three-dimensional stagnation slip flow and heat transfer is obtained by using the homotopy analysis method. This method is a series expansion method, but it is different from the perturbation technique, because it is independent of small physical parameters at all. Instead, it is based on a continuous mapping in topology so that it is applicable for not only weakly but also strongly nonlinear flow phenomena. Convergent [m,m] homotopy Padé approximants are obtained and compared with the numerical results and the asymptotic approximations. It is found that the homotopy Padé approximants agree well with the numerical results. The effects of the slip length ℓ and the thermal slip constant β on the heat transfer characteristics are investigated and discussed. Supported by the National Natural Science Foundation of China (Grant No. 10872129)     

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.