The integral representations of the q-Bessel-Macdonald functions

The q-Bessel-Macdonald functions of kinds 1, 2 and 3 are considered. Their representations by classical integral are constructed.Comment: 10 pages, Late

Poisson formula for a family of non-commutative Lobachevsky spaces

We define an analog of the Poisson integral formula for a family of the non-commutative Lobachevsky spaces. The $q$-Fourier transform of the Poisson kernel is expressed through the $q$-Bessel-Macdonald function.Comment: LateX, 15 pages, Contribution in the Fridrich Karpelevich memorial volum

The asymptotic behavior of q-exponentials and q-Bessel functions

The connections between q-Bessel functions of three types and q-exponential of three types are established. The q-exponentials and the q-Bessel functions are represented as the Laurent series. The asymptotic behaviour of the q-exponentials and the q-Bessel functions is investigated

q-convolution and its q-Fourier transform

The functions on a lattice generated by the integer degrees of $q^2$ are considered, 0<q<1. The $q^2$-translation operator is defined. The multiplicators and the $q^2$-convolutors are defined in the functional spaces which are dual with respect to the $q^2$-Fourier transform. The $q^2$-analog of convolution of two $q^2$-distributions is constructed. The $q^2$-analog of an arbitrary (non integer) order derivative is introducedComment: 17 pages, Late

The qq-Fourier transform of qq-distributions

We consider functions on the lattice generated by the integer powers of $q^2$ for $0<q<1$ and construct the $q$-analog of Fourier transform based on the Jackson integral in the space of distributions on the lattice.Comment: 18 pages, latex, typos corrected, added reference

Unitary Representations of Quantum Lorentz Group and Quantum Relativistic Toda Chain

The aim of this paper is to give a group theoretical interpretation of the three types of Bessel-Jackson functions. We consider a family of quantum Lorentz groups and a family of quantum Lobachevsky spaces. For three members of quantum Lobachevsky spaces the Casimir operators give rise to the two-body relativistic open Toda lattice Hamiltonians. Their eigen-functions are the modified Bessel-Jackson functions of three types. We construct the principal series of unitary irreducible representations of the quantum Lorentz groups. Special matrix elements in the irreducible spaces are the Bessel-Macdonald-Jackson functions. They are the wave functions of the two-body relativistic open Toda lattice. We obtain integral representations for these functions.Comment: Latex, 25 page

The Modified q-Bessel Functions and the q-Bessel-Macdonald Functions

We define a q-analog of the modified Bessel and Bessel-Macdonald functions. As for the q-Bessel functions of Jackson there is a couple of functions of the both kind. They are arisen in the Harmonic analysis on quantum symmetric spaces similarly to their classical counterpart. Their definition is based on the power expansions. We derive the recurrence relations, difference equations, q-Wronskians, and an analog of asymptotic expansions which turns out is exact in some domain if $q\neq 1$. Some relations for the basic hypergeometric function which follow from this fact are discussed.Comment: 16 pages, latex, no figure

Jackson Integral Representations of Modified qq-Bessel Functions and qq-Bessel-Macdonald Functions

The $q$ analog of Modified Bessel functions and Bessel-Macdonald functions, were defined in our previous work (q-alg/950913) as general solutions of a second order difference equations. Here we present a collection of their representations by the Jackson q-integral.Comment: Latex, 28 page

Causality relations for materials with strong artificial optical chirality

We demonstrate that the fundamental causality principle being applied to strongly chiral artificial materials yields the generalized Kramers-Kronig relations for the observables -- circular dichroism and optical activity. The relations include the Blaschke terms determined by material-specific features - the zeros of transmission amplitude on the complex frequency plane. By the example of subwavelength arrays of chiral holes in silver films we show that the causality relations can be used not only for a precise verification of experimental data but also for resolving the positions of material anomalies and resonances and quantifying the degree of their chiral splitting.Comment: 5 pages 4 figure

Extreme optical activity and circular dichroism of chiral metal hole arrays

We report extremely strong optical activity and circular dichroism exhibited by subwavelength arrays of four-start-screw holes fabricated with one-pass focused ion beam milling of freely suspended silver films. Having the fourth order rotational symmetry, the structures exhibit the polarization rotation up to 90 degrees and peaks of full circular dichroism and operate as circular polarizers within certain ranges of wavelengths in the visible. We discuss the observations on the basis of general principles (symmetry, reciprocity and reversibility) and conclude that the extreme optical chirality is determined by the chiral localized plasmonic resonances.Comment: 4 pages, 4 figure