Study of random process theory aids digital data processing

Study of techniques for all random process technology, including stationary, nonstationary, and Gaussian bivariate, aids digital data processing. It presents material on digital filtering, correlation function, optimal spectral smoothing, deterministic data processing, and nonstationary spectrum and correlation analyses

The Dn Ruijsenaars-Schneider model

The Lax pair of the Ruijsenaars-Schneider model with interaction potential of trigonometric type based on Dn Lie algebra is presented. We give a general form for the Lax pair and prove partial results for small n. Liouville integrability of the corresponding system follows a series of involutive Hamiltonians generated by the characteristic polynomial of the Lax matrix. The rational case appears as a natural degeneration and the nonrelativistic limit exactly leads to the well-known Calogero-Moser system associated with Dn Lie algebra.Comment: LaTeX2e, 14 pages; more remarks are added in the last sectio

Quantum Calogero-Moser Models: Integrability for all Root Systems

The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the (A) series, i.e. (su(N)) type, root systems.Comment: 45 pages, LaTeX2e, no figure

Exactly solvable potentials of Calogero type for q-deformed Coxeter groups

We establish that by parameterizing the configuration space of a one-dimensional quantum system by polynomial invariants of q-deformed Coxeter groups it is possible to construct exactly solvable models of Calogero type. We adopt the previously introduced notion of solvability which consists of relating the Hamiltonian to finite dimensional representation spaces of a Lie algebra. We present explicitly the $G_2^q$-case for which we construct the potentials by means of suitable gauge transformations.Comment: 22 pages Late

Quantum vs Classical Integrability in Ruijsenaars-Schneider Systems

The relationship (resemblance and/or contrast) between quantum and classical integrability in Ruijsenaars-Schneider systems, which are one parameter deformation of Calogero-Moser systems, is addressed. Many remarkable properties of classical Calogero and Sutherland systems (based on any root system) at equilibrium are reported in a previous paper (Corrigan-Sasaki). For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all "integer valued". In this paper we report that similar features and results hold for the Ruijsenaars-Schneider type of integrable systems based on the classical root systems.Comment: LaTeX2e with amsfonts 15 pages, no figure

Operator Transformations Between Exactly Solvable Potentials and Their Lie Group Generators

One may obtain, using operator transformations, algebraic relations between the Fourier transforms of the causal propagators of different exactly solvable potentials. These relations are derived for the shape invariant potentials. Also, potentials related by real transformation functions are shown to have the same spectrum generating algebra with Hermitian generators related by this operator transformation.Comment: 13 pages with one Postscript figure, uses LaTeX2e with revte

Non-crystallographic reduction of generalized Calogero-Moser models

We apply a recently introduced reduction procedure based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero–Moser systems. For rational potentials the familiar generalized Calogero Hamiltonian is recovered. For the Hamiltonians of trigonometric, hyperbolic and elliptic types, we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio. We explicitly show for the simplest of these non-crystallographic models, how the corresponding classical equations of motion can be derived from a Lie algebraic Lax pair based on the larger, crystallographic Coxeter group

The non-dynamical r-matrices of the degenerate Calogero-Moser models

A complete description of the non-dynamical r-matrices of the degenerate Calogero-Moser models based on $gl_n$ is presented. First the most general momentum independent r-matrices are given for the standard Lax representation of these systems and those r-matrices whose coordinate dependence can be gauged away are selected. Then the constant r-matrices resulting from gauge transformation are determined and are related to well-known r-matrices. In the hyperbolic/trigonometric case a non-dynamical r-matrix equivalent to a real/imaginary multiple of the Cremmer-Gervais classical r-matrix is found. In the rational case the constant r-matrix corresponds to the antisymmetric solution of the classical Yang-Baxter equation associated with the Frobenius subalgebra of $gl_n$ consisting of the matrices with vanishing last row. These claims are consistent with previous results of Hasegawa and others, which imply that Belavin's elliptic r-matrix and its degenerations appear in the Calogero-Moser models. The advantages of our analysis are that it is elementary and also clarifies the extent to which the constant r-matrix is unique in the degenerate cases.Comment: 25 pages, LaTeX; expanded by an appendix detailing the proof of Theorem 1 and a concluding section in version

Turbulent Mixing in the Interstellar Medium -- an application for Lagrangian Tracer Particles

We use 3-dimensional numerical simulations of self-gravitating compressible turbulent gas in combination with Lagrangian tracer particles to investigate the mixing process of molecular hydrogen (H2) in interstellar clouds. Tracer particles are used to represent shock-compressed dense gas, which is associated with H2. We deposit tracer particles in regions of density contrast in excess of ten times the mean density. Following their trajectories and using probability distribution functions, we find an upper limit for the mixing timescale of H2, which is of order 0.3 Myr. This is significantly smaller than the lifetime of molecular clouds, which demonstrates the importance of the turbulent mixing of H2 as a preliminary stage to star formation.Comment: 10 pages, 5 figures, conference proceedings "Turbulent Mixing and Beyond 2007

Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials

The Ruijsenaars-Schneider systems are `discrete' version of the Calogero-Moser (C-M) systems in the sense that the momentum operator p appears in the Hamiltonians as a polynomial in e^{\pm\beta' p} (\beta' is a deformation parameter) instead of an ordinary polynomial in p in the hierarchies of C-M systems. We determine the polynomials describing the equilibrium positions of the rational and trigonometric Ruijsenaars-Schneider systems based on classical root systems. These are deformation of the classical orthogonal polynomials, the Hermite, Laguerre and Jacobi polynomials which describe the equilibrium positions of the corresponding Calogero and Sutherland systems. The orthogonality of the original polynomials is inherited by the deformed ones which satisfy three-term recurrence and certain functional equations. The latter reduce to the celebrated second order differential equations satisfied by the classical orthogonal polynomials.Comment: 45 pages. A few typos in section 6 are correcte