Relativistic ideal Fermi gas at zero temperature and preferred frame

We discuss the limit T->0 of the relativistic ideal Fermi gas of luxons (particles moving with the speed of light) and tachyons (hypothetical particles faster than light) based on observations of our recent paper: K. Kowalski, J. Rembielinski and K.A. Smolinski, Phys. Rev. D, 76, 045018 (2007). For bradyons this limit is in fact the nonrelativistic one and therefore it is not studied herein

Universal low-energy properties of three two-dimensional particles

Universal low-energy properties are studied for three identical bosons confined in two dimensions. The short-range pair-wise interaction in the low-energy limit is described by means of the boundary condition model. The wave function is expanded in a set of eigenfunctions on the hypersphere and the system of hyper-radial equations is used to obtain analytical and numerical results. Within the framework of this method, exact analytical expressions are derived for the eigenpotentials and the coupling terms of hyper-radial equations. The derivation of the coupling terms is generally applicable to a variety of three-body problems provided the interaction is described by the boundary condition model. The asymptotic form of the total wave function at a small and a large hyper-radius $\rho$ is studied and the universal logarithmic dependence $\sim \ln^3 \rho$ in the vicinity of the triple-collision point is derived. Precise three-body binding energies and the $2 + 1$ scattering length are calculated.Comment: 30 pages with 13 figure

Coherent states for the hydrogen atom

We construct a system of coherent states for the hydrogen atom that is expressed in terms of elementary functions. Unlike to the previous attempts in this direction, this system possesses the properties equivalent to the most of those for the harmonic oscillator, with modifications due to the character of the problem.Comment: 6 pages, LATEX, using ioplppt.sty and iopfts.sty. v.2: some misprints are corrected. To appear in J.Phys.

Laplace transform of spherical Bessel functions

We provide a simple analytic formula in terms of elementary functions for the Laplace transform j_{l}(p) of the spherical Bessel function than that appearing in the literature, and we show that any such integral transform is a polynomial of order l in the variable p with constant coefficients for the first l-1 powers, and with an inverse tangent function of argument 1/p as the coefficient of the power l. We apply this formula for the Laplace transform of the memory function related to the Langevin equation in a one-dimensional Debye model.Comment: 5 pages LATEX, no figures. Accepted 2002, Physica Script

A holomorphic representation of the Jacobi algebra

A representation of the Jacobi algebra $\mathfrak{h}_1\rtimes \mathfrak{su}(1,1)$ by first order differential operators with polynomial coefficients on the manifold $\mathbb{C}\times \mathcal{D}_1$ is presented. The Hilbert space of holomorphic functions on which the holomorphic first order differential operators with polynomials coefficients act is constructed.Comment: 34 pages, corrected typos in accord with the printed version and the Errata in Rev. Math. Phys. Vol. 24, No. 10 (2012) 1292001 (2 pages) DOI: 10.1142/S0129055X12920018, references update

Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.Comment: 25 pages; added section 5 for some examples of application

Conformal compactification and cycle-preserving symmetries of spacetimes

The cycle-preserving symmetries for the nine two-dimensional real spaces of constant curvature are collectively obtained within a Cayley-Klein framework. This approach affords a unified and global study of the conformal structure of the three classical Riemannian spaces as well as of the six relativistic and non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both Newton-Hooke and Galilean), and gives rise to general expressions holding simultaneously for all of them. Their metric structure and cycles (lines with constant geodesic curvature that include geodesics and circles) are explicitly characterized. The corresponding cyclic (Mobius-like) Lie groups together with the differential realizations of their algebras are then deduced; this derivation is new and much simpler than the usual ones and applies to any homogeneous space in the Cayley-Klein family, whether flat or curved and with any signature. Laplace and wave-type differential equations with conformal algebra symmetry are constructed. Furthermore, the conformal groups are realized as matrix groups acting as globally defined linear transformations in a four-dimensional "conformal ambient space", which in turn leads to an explicit description of the "conformal completion" or compactification of the nine spaces.Comment: 43 pages, LaTe

The Quantum Mellin transform

We uncover a new type of unitary operation for quantum mechanics on the half-line which yields a transformation to ``Hyperbolic phase space''. We show that this new unitary change of basis from the position x on the half line to the Hyperbolic momentum $p_\eta$, transforms the wavefunction via a Mellin transform on to the critial line $s=1/2-ip_\eta$. We utilise this new transform to find quantum wavefunctions whose Hyperbolic momentum representation approximate a class of higher transcendental functions, and in particular, approximate the Riemann Zeta function. We finally give possible physical realisations to perform an indirect measurement of the Hyperbolic momentum of a quantum system on the half-line.Comment: 23 pages, 6 Figure

Exponential beams of electromagnetic radiation

We show that in addition to well known Bessel, Hermite-Gauss, and Laguerre-Gauss beams of electromagnetic radiation, one may also construct exponential beams. These beams are characterized by a fall-off in the transverse direction described by an exponential function of rho. Exponential beams, like Bessel beams, carry definite angular momentum and are periodic along the direction of propagation, but unlike Bessel beams they have a finite energy per unit beam length. The analysis of these beams is greatly simplified by an extensive use of the Riemann-Silberstein vector and the Whittaker representation of the solutions of the Maxwell equations in terms of just one complex function. The connection between the Bessel beams and the exponential beams is made explicit by constructing the exponential beams as wave packets of Bessel beams.Comment: Dedicated to the memory of Edwin Powe

Analytical model for laser-assisted recombination of hydrogenic atoms

We introduce a new method that allows one to obtain an analytical cross section for the laser-assisted electron-ion collision in a closed form. As an example we perform a calculation for the hydrogen laser-assisted recombination. The $S$-matrix element for the process is constructed from an exact electron Coulomb-Volkov wave function and an approximate laser modified hydrogen state. An explicit expression for the field-enhancement coefficient of the process is expressed in terms of the dimensionless parameter $\kappa= |{e\epsilon_{0}}/{q\omega_{0}}|^{2}$, where $e$ and $q$ are the electron charge and momentum respectively, and $\epsilon_{0}$ and $\omega_{0}$ are the amplitude and frequency of the laser field respectively. The simplified version of the cross section of the process is derived and analyzed within a soft photon approximation.Comment: 10 page