Harmonic oscillators coupled by springs: discrete solutions as a Wigner Quantum System

We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency $\omega$, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner Quantum System, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton's equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1|M). Then we study the properties and spectra of the physical operators in a class of unitary representations of gl(1|M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain

A comparison between mathematical models of stationary configuration of an undewater towed system with experimental validation

The analysis of underwater towed systems attracted the interest of many researchers because of the recent years utilization of remotely-operated underwater vehicle (ROV) and towed array in offshore and military applications. The purpose of this work is to show, by experimental validation, that towed cable configurations may be computed effectively and accurately by discretizing the towing cable rather than using a continuous modeling approach. Two mathematical models have been developed to predict the stationary configuration of an underwater towed system loaded by hydrodynamic forces. The system is composed of a towed inextensible cable, with no bending stiffness, and a depressor that is fixed at the cable free end. This configuration is currently used for underwater remotely-operated vehicle. This work investigates the comparison between continuous and discrete models of the 2D static equations of the steady-state towing problem in a vertical plane at different towing speeds. The results of the models have been validating using experimental trials. In the first part of this paper, a continuous model is presented, which is based on geometric compatibility relations, equilibrium equation. A set of nonlinear differential equations has been derived and solved using Runge-Kutta iterative procedure. In the second part, a discrete rod model is proposed to determinate the cable shape, which is based on a system of nonlinear algebraic equations that are solved numerically. This two models are both suitable for analyzing an underwater towed system having a known top tension and inclination angle obtained from experiments. The third part of the paper describes the experiments, which have been in a towing tank basin (CNR-INSEAN). In the fourth and last part of this study it is demonstrated the effort and cost of numerically integrating the continuous model do not compare favorably with the relative ease and efficiency of solving the discrete model, which yields the same results

Distribution of the first particle in discrete orthogonal polynomial ensembles

We show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure, if the (difference or q-) log-derivative of the weight function is rational. In a number of classical special cases the recurrence procedure is equivalent to the difference and q-Painleve equations of chao-dyn/9507010, [Sakai]. Our approach is based on the formalism of discrete integrable operators and discrete Riemann--Hilbert problems developed in math.CO/9912093, math-ph/0111008.Comment: 41 page

Discrete Riemannian Geometry

Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calculi and digraphs. Arrows originating from a vertex span its (co)tangent space. If the metric is to measure length and angles at some point, it has to be taken as an element of the left-linear tensor product of the space of 1-forms with itself, and not as an element of the (non-local) tensor product over the algebra of functions. It turns out that linear connections can always be extended to this left tensor product, so that metric compatibility can be defined in the same way as in continuum Riemannian geometry. In particular, in the case of the universal differential calculus on a finite set, the Euclidean geometry of polyhedra is recovered from conditions of metric compatibility and vanishing torsion. In our rather general framework (which also comprises structures which are far away from continuum differential geometry), there is in general nothing like a Ricci tensor or a curvature scalar. Because of the non-locality of tensor products (over the algebra of functions) of forms, corresponding components (with respect to some module basis) turn out to be rather non-local objects. But one can make use of the parallel transport associated with a connection to `localize' such objects and in certain cases there is a distinguished way to achieve this. This leads to covariant components of the curvature tensor which then allow a contraction to a Ricci tensor. In the case of a differential calculus associated with a hypercubic lattice we propose a new discrete analogue of the (vacuum) Einstein equations.Comment: 34 pages, 1 figure (eps), LaTeX, amssymb, epsfi