A K Common

The symmetric strong moment problem

Sequences of polynomials that occur as denominators in the two point Pade table for two series expansions are considered in the special case when the series coefficients are solutions of a strong symmetric Stieltjes moment problem. The continued fractions whose convergents generate these polynomials as denominators are presented, together with determinant representations for the polynomials and the continued fraction coefficients. The log-normal distribution is used as an example.

Continued-fraction solutions to the Riccati equation and integrable lattice systems

Continued-fraction solutions to the matrix Riccati equation are discussed which are constructed by using the concept of form invariance. It is demonstrated that this technique is related to the AKNS method of deriving integrable nonlinear lattice systems. This gives an explanation why continued-fraction solutions related to the Toda lattice were obtained in a previous work. Continued fractions corresponding to Kac-Van Moerbeke, discrete nonlinear Schrodinger and discrete modified KdV lattice equations are constructed. A method for linearising the Kac-Van Moerbeke lattice equations is rederived and particular solutions are generated. The authors approach demonstrates the crucial role played by the boundary condition at the finite end of the lattice for the existence of this method. These results are extended to the other two lattice systems above in the semi-infinite case and corresponding particular solutions generated in terms of Bessel functions.

Applications of Hermite-Pade approximants to water waves and the harmonic oscillator on a lattice

Three types of Hermite-Pade approximants are considered, known respectively as quadratic, integral and differential Pade approximants. The singularity structure of each type of approximant is described. It is more complicated than that of the standard Pade approximant, and this property may often be used to estimate the types of singularity of a function from its power series expansion, as well as evaluate it on its branch cuts. Two applications in different fields are described which illustrate these properties of the above types of Hermite-Pade approximants. The first concerns the characteristic values of Mathieus equation which are related to the energy eigenvalues of the harmonic oscillator on a lattice. The second concerns the investigation of the singularity structure and values of various physical quantities associated with periodic and solitary water waves.

Properties of states bound by potentials with positive Laplacian

States of two-body systems bound by a spherically symmetric potential whose Laplacian is positive are considered. Constraints are derived on the kinetic energies and radii of states with the lowest energy for a given angular momentum, which improve on previous results of the author. These constraints become relatively very tight at high angular momentum. Inequalities are also derived between the energies of states of differing angular momentum and particle mass.

Two discretisations of the Ermakov-Pinney equation

Abstract We propose two candidates for discrete analogues to the nonlinear Ermakov-Pinney equation. The first one based on an association with a two-dimensional conformal mapping defines a second-degree difference scheme. It possesses the same features as in the continuum: a nonlinear superposition principle relating its general solution to a second-order linear difference equation and by direct linearisation a relationship with a third-order difference equation. The second form, which is new, is obtained from a slight improvement of the superposition principle. It has the advantage of leading to a first degree difference scheme and preserves all the nice properties of its linearisation.

Approximations to large amplitude solitary waves on nonlinear electrical lattices

Abstract In this paper we describe an approximate method to characterise solitary wave solutions of nonlinear lattice equations. It is based upon one and two point Pade approximations to a series of the real exponential travelling wave solutions of the underlying dispersive system. The theory is applied to an example of a lattice system which models an experimental nonlinear transmission line and the results obtained are consistent with numerical simulations even for relatively large amplitude solitary waves. The speed-amplitude relation is investigated and compared with the derived using quasi-continuum methods.

Linearization of the relativistic and discrete-time Toda lattices for particular boundary conditions

The authors considered solutions to the Kac-Van Moerbeke and semi-infinite Toda, discrete modified KdV and nonlinear Schrodinger equations. Using the AKNS approach, solutions of these equations were related to continued-fraction solutions of certain Riccati equations. A method for linearizing the Kac-Van Moerbeke lattice was rederived and extended to all the above lattices. Their approach demonstrated the crucial role played by the boundary condition at the finite end. The study is extended to the relativistic and discrete-time Toda lattices.

Improved constraints on states bound by a class of potentials

Recent work on states bound by a central potential whose Laplacian is positive is extended to cover a wider class of potentials for which energy level ordering theorems can be proved. This class includes power law potentials in particular. Constraints on the moments of the radial distance are improved and similarly for the kinetic energy. These constraints are shown to provide tight bounds on the energies of ground states for power law potentials, and on their variation with angular momentum. They are also used to bound the position of the maximum of the wavefunction for these states.

The Pinney equation and its discretization

The Pinney equation is part of the original Ermakov system which has been the subject of intensive study recently. Here we show that it may be related to a two-dimensional conformal Riccati equation leading to a new method for its linearization. A discrete analogue of the Pinney equation is constructed using the above connection with the conformal group. An alternative discretization is obtained by using a discrete Schwarz derivative. Both of these nonlinear difference equations are linearizable.

The convergence of Legendre Pade approximants to the Coulomb and other scattering amplitudes

The convergence of sequences of Legendre Pade approximants to scattering amplitudes arising in potential scattering is discussed. It is shown that if the amplitude for scattering by the Coulomb potential or the absorptive part of a scattering amplitude whose double spectral function obeys a certain bound are suitably modified, convergent sequences of Legendre Pade approximants may be constructed. Numerical results are presented for the Coulomb scattering amplitude.