Barnabás Apagyi

Interlacing of positive real zeros of Bessel functions

We unify the three distinct inequality sequences (Abramowitz and Stegun (1972) [1, 9.5.2]) of positive real zeros of Bessel functions into a single one.

Effect of scattering lengths on the dynamics of a two-component Bose-Einstein condensate

We examine the effect of the intra- and interspecies scattering lengths on the dynamics of a two-component Bose-Einstein condensate, particularly focusing on the existence and stability of solitonic excitations. For each type of possible soliton pairs, stability ranges are presented in tabulated form. We also compare the numerically established stability of bright-bright, bright-dark, and dark-dark solitons with our analytical prediction and with that of Painleve analysis of the dynamical equation. We demonstrate that tuning the interspecies scattering length away from the predicted value (keeping the intraspecies coupling fixed) breaks the stability of the soliton pairs.

On nonsingular potentials of Cox–Thompson inversion scheme

We establish a condition for obtaining nonsingular potentials using the Cox–Thompson inverse scattering method with one phase shift. The anomalous singularities of the potentials are avoided by maintaining unique solutions of the underlying Regge–Newton integral equation for the transformation kernel. As a by-product, new inequality sequences of zeros of Bessel functions are discovered.

Painlevé test of coupled Gross-Pitaevskii equations

A Painleve (P) test of the coupled Gross-Pitaevskii equations has been carried out with the result that the coupled equations pass the P test only if a special relation containing system parameters (masses and scattering lengths) is satisfied. Computer algebra is applied to evaluate the j = 4 compatibility condition for admissible external potentials. The appearance of an arbitrary real potential embedded in the external potentials is shown to be the consequence of the coupling. The connection with recent experiments related to the stability of two-component Bose-Einstein condensates of Rb atoms is discussed.

Solution of the Cox-Thompson inverse scattering problem using finite set of phase shifts

As ystem ofnonlinear equations is presented for the solution of the Cox– Thompson inverse scattering problem (1970 J. Math. Phys. 11 805) at fixed energy. From a given finite set of phase shifts for physical angular momenta, the nonlinear equations determine related sets of asymptotic normalization constants and nonphysical (shifted) angular momenta from which all quantities of interest, including the inversion potential itself, can be calculated. As a first application of the method we use input data consisting of a finite set of phase shifts calculated from Woods–Saxon and box potentials representing interactions with diffuse or sharp surfaces, respectively. The results for the inversion potentials, their first moments and asymptotic properties are compared with those provided by the Newton–Sabatier quantum inversion procedure. It is found that in order to achieve inversion potentials of similar quality, the Cox–Thompson method requires a smaller set of phase shifts than th eN ewton–Sabatier procedure.

Inversion of real and complex phase shifts to potentials by the generalized Cox?Thompson inverse scattering method at fixed energy

The Cox?Thompson inverse scattering method at fixed energy has been generalized to treat complex phase shifts derived from experiments. New formulae for relating phase shifts to shifted angular momenta are derived. The method is applied to phase shifts of known potentials in order to test its quality and stability and, further, it is used to invert experimental n-? and n-12C phase shifts.

A two-parameter model for the screening of a proton in an electron gas

A simple two-parameter model for the electron density fluctuation induced by a slow proton intruder is examined. The model satisfies both Friedels sum rule and Katos cusp condition. The results obtained for the electron density fluctuation at the proton site and the transport collision frequency are in fair agreement with those given by self-consistent calculations.

SOLUTION OF THE INVERSE SCATTERING PROBLEM AT FIXED ENERGY FOR POTENTIALS BEING ZERO BEYOND A FIXED RADIUS

Based on the relation between the m-function and the spectral function we construct an inverse quantum scattering procedure at fixed energy which can be applied to spherical radial potentials vanishing beyond a fixed radius a. To solve the Gelfand–Levitan–Marchenko integral equation for the transformation kernel, we determine the input symmetrical kernel by using a minimum norm method with moments defined by the input set of scattering phase shifts. The method applied to the box and Gauss potentials needs further practical developments regarding the treatment of bound states.

Effective Rb–Rb inter-atomic potential from ultracold Bose-gas collisions

Scattering phase shifts obtained from 87Rb Bose-gas collision experiments are used to reconstruct effective potentials resulting, self-consistently, in the same scattering events observed in the experiments at a particular energy. We have found that the interaction strength close to the origin suddenly changes from repulsion to attraction when the collision energy crosses, from below, the l = 2 shape resonance position at E ≈ 275 µK. This observation may be utilized in outlining future Bose-gas collision experiments.

Assessment of interspecies scattering lengths a12 from stability of two-component Bose-Einstein condensates

Abstract.A stability method is used to assess possible values of interspecies scattering lengths a12 in two-component Bose-Einstein condensates described within the Gross-Pitaevskii approximation. The technique, based on a recent stability analysis of solitonic excitations in two-component Bose-Einstein condensates, is applied to ninety combinations of atomic alkali pairs with given singlet and triplet intraspecies scattering lengths as input parameters. Results obtained for values of a12 are in a reasonable agreement with the few ones available in the literature and with those obtained from a Painlevé analysis of the coupled Gross-Pitaevskii equations.