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Rene Schott

Publications and source records attributed to Rene Schott.

5 recordsLinked to original sources
arXiv2012

The Poincare algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states

By introducing an unconventional realization of the Poincare algebra alt_1 of special relativity as conformal transformations, we show how it may occur as a dynamical symmetry algebra for ageing systems in non-equilibrium statistical physics and give some applications, such as the computation of two-time correlators. We also discuss infinite-dimensional extensions of alt_1 in this setting. Finally, we construct canonical Appell systems, coherent states and Leibniz functions for alt_1 as a tool for bosonic quantization.

math-ph↗
arXiv2009

Finite-Dimensional Calculus

We discuss topics related to finite-dimensional calculus in the context of finite-dimensional quantum mechanics. The truncated Heisenberg-Weyl algebra is called a TAA algebra after Tekin, Aydin, and Arik who formulated it in terms of orthofermions. It is shown how to use a matrix approach to implement analytic representations of the Heisenberg-Weyl algebra in univariate and multivariate settings. We provide examples for the univariate case. Krawtchouk polynomials are presented in detail, including a review of Krawtchouk polynomials that illustrates some curious properties of the Heisenberg-Weyl algebra, as well as presenting an approach to computing Krawtchouk expansions. From a mathematical perspective, we are providing indications as to how to implement in finite terms Rota's "finite operator calculus".

math-ph↗
arXiv2005

Distributed Algorithms in an Ergodic Markovian Environment

We provide a probabilistic analysis of the banker algorithm when transition probabilities may depend on time and space. The transition probabilities evolve, as time goes by, along the trajectory of an ergodic Markovian environment, whereas the spatial parameter just acts on long runs. Our model appears as a new (small) step towards more general time and space dependent protocols. Our analysis relies on well-known results in stochastic homogenization theory and investigates the asymptotic behaviour of the rescaled algorithm as the total amount of resource available for allocation tends to the infinity. In the two dimensional setting, we manage to exhibit three different possible regimes for the deadlock time of the limit system.

math.PR↗
arXiv2000

Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes

A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space $\eufrak{h}$ and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over $\eufrak{h}$. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For $\eufrak{h}=L^2(\mathbb{R}_+)$, the divergence operator is shown to coincide with the Hudson-Parthasarathy quantum stochastic integral for adapted integrable processes and with the non-causal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.

math.PR↗