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Some results about dissipativity of Kolmogorov operators

Giuseppe Da Prato, Luciano Tubaro (2001)

Czechoslovak Mathematical Journal

Given a Hilbert space H with a Borel probability measure ν , we prove the m -dissipativity in L 1 ( H , ν ) of a Kolmogorov operator K that is a perturbation, not necessarily of gradient type, of an Ornstein-Uhlenbeck operator.

Spectral theory of damped quantum chaotic systems

Stéphane Nonnenmacher (2011)

Journées Équations aux dérivées partielles

We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on X and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined...

Spectrum generating functions for oscillators in Wigner's quantization

Stijn Lievens, Joris Van der Jeugt (2011)

Banach Center Publications

The n-dimensional (isotropic and non-isotropic) harmonic oscillator is studied as a Wigner quantum system. In particular, we focus on the energy spectrum of such systems. We show how to solve the compatibility conditions in terms of 𝔬𝔰𝔭(1|2n) generators, and also recall the solution in terms of 𝔤𝔩(1|n) generators. A method is described for determining a spectrum generating function for an element of the Cartan subalgebra when working with a representation of any Lie (super)algebra. Here, the...

Splitting the conservation process into creation and annihilation parts

Nicolas Privault (1998)

Banach Center Publications

The aim of this paper is the study of a non-commutative decomposition of the conservation process in quantum stochastic calculus. The probabilistic interpretation of this decomposition uses time changes, in contrast to the spatial shifts used in the interpretation of the creation and annihilation operators on Fock space.

Stationary Quantum Markov processes as solutions of stochastic differential equations

Jürgen Hellmich, Claus Köstler, Burkhard Kümmerer (1998)

Banach Center Publications

From the operator algebraic approach to stationary (quantum) Markov processes there has emerged an axiomatic definition of quantum white noise. The role of Brownian motion is played by an additive cocycle with respect to its time evolution. In this report we describe some recent work, showing that this general structure already allows a rich theory of stochastic integration and stochastic differential equations. In particular, if a quantum Markov process is represented by a unitary cocycle, we can...

Stratonovich-Weyl correspondence for discrete series representations

Benjamin Cahen (2011)

Archivum Mathematicum

Let M = G / K be a Hermitian symmetric space of the noncompact type and let π be a discrete series representation of G holomorphically induced from a unitary character of K . Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple ( G , π , M ) by a suitable modification of the Berezin calculus on M . We extend the corresponding Berezin transform to a class of functions on M which contains the Berezin symbol of d π ( X ) for X in the Lie algebra 𝔤 of G . This allows...

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