Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths

In this paper we consider the model of $n$ non-intersecting squared Bessel processes with parameter $\alpha$, in the confluent case where all particles start, at time $t=0$, at the same positive value $x=a$, remain positive, and end, at time $T=t$, at the position $x=0$. The positions of the paths have a limiting mean density as $n\to\infty$ which is characterized by a vector equilibrium problem. We show how to obtain this equilibrium problem from different considerations involving the recurrence relations for multiple orthogonal polynomials associated with the modified Bessel functions. We also extend the situation by rescaling the parameter $\alpha$, letting it increase proportionally to $n$ as $n$ increases. In this case we also analyze the recurrence relation and obtain a vector equilibrium problem for it.Comment: 28 pages, 10 figure

Parametric oscillator in a Kerr medium: evolution of coherent states

We study the temporal evolution of a coherent state under the action of a parametric oscillator and a nonlinear Kerr-like medium. We make use of the interaction picture representation and use an exact time evolution operator for the time independent part of the Hamiltonian. We approximate the interaction picture Hamiltonian in such a way as to make it a member of a Lie algebra. The corresponding time evolution operator behaves like a squeezing operator due to the temporal dependence of the oscillator's frequency. We analyze the probability amplitude and the auto correlation function for different Hamiltonian parameters and we find a very good agreement between our approximate results and converged numerical calculations.Comment: 11 pages, 3 figure

Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions

We consider polynomials P_n orthogonal with respect to the weight J_? on [0,?), where J_? is the Bessel function of order ?. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros are complex and accumulate as n?? near the vertical line Rez=??2. We prove this fact for the case 0???1/2 from strong asymptotic formulas that we derive for the polynomials Pn in the complex plane. Our main tool is the Riemann-Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift-Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for ??1/2

Entanglement and SU(n) symmetry in one-dimensional valence bond solid states

Here we evaluate the many-body entanglement properties of a generalized SU(n) valence bond solid state on a chain. Our results follow from a derivation of the transfer matrix of the system which, in combination with symmetry properties, allows for a new, elegant and straightforward evaluation of different entanglement measures. In particular, the geometric entanglement per block, correlation length, von Neumann and R\'enyi entropies of a block, localizable entanglement and entanglement length are obtained in a very simple way. All our results are in agreement with previous derivations for the SU(2) case.Comment: 4 pages, 2 figure

Dynamics of the Formation of Bright Solitary Waves of Bose-Einstein Condensates in Optical Lattices

We present a detailed description of the formation of bright solitary waves in optical lattices. To this end, we have considered a ring lattice geometry with large radius. In this case, the ring shape does not have a relevant effect in the local dynamics of the condensate, while offering a realistic set up to implement experiments with conditions usually not available with linear lattices (in particular, to study collisions). Our numerical results suggest that the condensate radiation is the relevant dissipative process in the relaxation towards a self-trapped solution. We show that the source of dissipation can be attributed to the presence of higher order dispersion terms in the effective mass approach. In addition, we demonstrate that the stability of the solitary solutions is linked with particular values of the width of the wavepacket in the reciprocal space. Our study suggests that these critical widths for stability depend on the geometry of the energy band, but are independent of the condensate parameters (momentum, atom number, etc.). Finally, the non-solitonic nature of the solitary waves is evidenced showing their instability under collisions.Comment: 7 pages, 7 figures, to appear in PR

The mid-infrared extinction law in the darkest cores of the Pipe Nebula

Context. The properties of dust grains, in particular their size distribution, are expected to differ from the interstellar medium to the high-density regions within molecular clouds. Aims. We measure the mid-infrared extinction law produced by dense material in molecular cloud cores. Since the extinction at these wavelengths is caused by dust, the extinction law in cores should depart from that found in low-density environments if the dust grains have different properties. Methods. We use the unbiased LINES method to measure the slope of the reddening vectors in color-color diagrams. We derive the mid-infrared extinction law toward the dense cores B59 and FeSt 1-457 in the Pipe Nebula over a range of visual extinction between 10 and 50 magnitudes, using a combination of Spitzer/IRAC, and ESO NTT/VLT data. Results. The mid-infrared extinction law in both cores departs significantly from a power-law between 3.6 and 8 micron, suggesting that these cores contain dust with a considerable fraction of large dust grains. We find no evidence for a dependence of the extinction law with column density up to 50 magnitudes of visual extinction in these cores, and no evidence for a variation between our result and those for other clouds at lower column densities reported elsewhere in the literature. This suggests that either large grains are present even in low column density regions, or that the existing dust models need to be revised at mid-infrared wavelengths. We find a small but significant difference in the extinction law of the two cores, that we tentatively associate with the onset of star formation in B59.Comment: 8 pages, 6 figures. Accepted to A&