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Kyushu Journal of Mathematics

A WIENER INTEGRAL APPROACH TO NON-COMMUTATIVE HARMONIC OSCILLATORS

Setsuo Taniguchi

Published Jan 1, 2009• 0 citations
Publisher

Abstract

A Wiener integral approach to the non-commutative harmonic oscillator, i.e. an approach via Brownian motions and matrix-valued stochastic differential equations, will be given. Such an approach is a continuation and an extension of the one made by the author in an earlier paper (Kyushu J. Math. 62 (2008), 63-68).
DOI
10.2206/kyushujm.63.347

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Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.