Factorisation of Macdonald polynomials

We discuss the problem of factorisation of the symmetric Macdonald polynomials and present the obtained results for the cases of 2 and 3 variables.Comment: 13 pages, LaTex, no figure

η\eta photoproduction on the quasi-free nucleons in the chiral quark model

A chiral quark-model approach is adopted to study the $\eta$ photoproduction off the quasi-free neutron and proton from a deuteron target. Good descriptions of the differential cross sections, total cross sections and beam asymmetries for these two processes are obtained in the low energy region. For $\gamma p\rightarrow \eta p$, the dominant resonances are $S_{11}(1535)$, $S_{11}(1650)$, $D_{13}(1520)$, $D_{13}(1700)$ and $P_{13}(1720)$. While for the $\gamma n\rightarrow \eta n$ process, the dominant resonances are $S_{11}(1535)$, $S_{11}(1650)$, $D_{13}(1520)$, $D_{15}(1675)$ and $P_{13}(1720)$. Furthermore, the $u$ channel backgrounds have significant contributions to the $\eta$ photoproduction processes. The configuration mixings in the $S_{11}(1535,1650)$ and $D_{13}(1520,1700)$ can be extracted, i.e. $\theta_S\simeq 26^\circ$ and $\theta_D\simeq 21^\circ$. It shows that the narrow bump-like structure around $W= 1.68$ GeV observed in $\gamma n\rightarrow \eta n$ can be naturally explained by the constructive interferences between $S_{11}(1535)$ and $S_{11}(1650)$. In contrast, the destructive interference between $S_{11}(1535)$ and $S_{11}(1650)$ produces the shallow dip around $W= 1.67$ GeV in $\gamma p\rightarrow \eta p$. The $S$ wave interfering behaviors in the proton and neutron reactions are correlated with each other in the quark model framework, and no new exotic nucleon resonances are needed in these two reactions.Comment: 12 pages, 11 figures, helicity amplitudes are added, to be published in PR

Eigenproblem for Jacobi matrices: hypergeometric series solution

We study the perturbative power-series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The(small) expansion parameters are being the entries of the two diagonals of length d-1 sandwiching the principal diagonal, which gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series of 3d-5 variables in the generic case, or 2d-3 variables for the eigenvalue growing from a corner matrix element. To derive the result, we first rewrite the spectral problem for a Jacobi matrix as an equivalent system of cubic equations, which are then resolved by the application of the multivariable Lagrange inversion formula. The corresponding Jacobi determinant is calculated explicitly. Explicit formulae are also found for any monomial composed of eigenvector's components.Comment: Latex, 20 pages; v2: corrected typos, added section with example

Topological defect formation in quenched ferromagnetic Bose-Einstein condensates

We study the dynamics of the quantum phase transition of a ferromagnetic spin-1 Bose-Einstein condensate from the polar phase to the broken-axisymmetry phase by changing magnetic field, and find the spontaneous formation of spinor domain walls followed by the creation of polar-core spin vortices. We also find that the spin textures depend very sensitively on the initial noise distribution, and that an anisotropic and colored initial noise is needed to reproduce the Berkeley experiment [Sadler et al., Nature 443, 312 (2006)]. The dynamics of vortex nucleation and the number of created vortices depend also on the manner in which the magnetic field is changed. We point out an analogy between the formation of spin vortices from domain walls in a spinor BEC and that of vortex-antivortex pairs from dark solitons in a scalar BEC.Comment: 10 pages, 11 figure

Random copolymer: Gaussian variational approach

We study the phase transitions of a random copolymer chain with quenched disorder. We apply a replica variational approach based on a Gaussian trial Hamiltonian in terms of the correlation functions of monomer Fourier coordinates. This allows us to study collapse, phase separation and freezing transitions within the same mean field theory. The effective free energy of the system is derived analytically and analysed numerically. Such quantities as the radius of gyration or the average value of the overlap between different replicas are treated as observables and evaluated by introducing appropriate external fields to the Hamiltonian. We obtain the phase diagram and show that this system exhibits a scale dependent freezing transition. The correlations between replicas appear at different length scales as the temperature decreases. This indicates the existence of the topological frustration.Comment: 15 pages, 4 Postscript figure

Dark matter-wave solitons in the dimensionality crossover

We consider the statics and dynamics of dark matter-wave solitons in the dimensionality crossover regime from 3D to 1D. There, using the nonpolynomial Schr\"{o}dinger mean-field model, we find that the anomalous mode of the Bogoliubov spectrum has an eigenfrequency which coincides with the soliton oscillation frequency obtained by the 3D Gross-Pitaevskii model. We show that substantial deviations (of order of 10% or more) from the characteristic frequency $\omega_{z}/\sqrt{2}$ ($\omega_{z}$ being the longitudinal trap frequency) are possible even in the purely 1D regime.Comment: Phys. Rev. A, in pres

Separation of variables for the Ruijsenaars system

We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variables is verified with the use of r-matrix technique. The explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n=2 and n=3. Taking nonrelativistic limit we also construct a separation of variables for the elliptic Calogero-Moser system.Comment: 26 pages, LaTex, no figure