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Infinite Dimensional Analysis Quantum Probability and Related Topics

QWN-EULER OPERATOR AND ASSOCIATED CAUCHY PROBLEM

Abdessatar Barhoumi, Habib Ouerdiane, Hafedh M. S. Rguigui

Published Mar 1, 2012• 21 citations
Publisher

Abstract

In this paper the quantum white noise (QWN)-Euler operator [Formula: see text] is defined as the sum [Formula: see text], where [Formula: see text] and NQ stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that [Formula: see text] has an integral representation in terms of the QWN-derivatives [Formula: see text] as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to the QWN-Euler operator is worked out in the basis of the QWN coordinate system.

DOI
10.1142/s021902571250004x

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In this paper the quantum white noise (QWN)-Euler operator [Formula: see text] is defined as the sum [Formula: see text], where [Formula: see text] and NQ stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that [Formula: see text] has an integral representation in terms of the QWN-derivatives [Formula: see text] as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to the QWN-Euler operator is worked out in the basis of the QWN coordinate system.
Abstract In this paper we study the homogeneous Wick differential equation associated to the quantum white noise (𝚀𝚆𝙽) Euler operator Δ E g , Q ΔEg,Q\Delta _E^{g, Q}ΔEg,Q​ acting on generalized operators. Δ E g , Q ΔEg,Q\Delta _E^{g, Q}ΔEg,Q​ is defined as sum of the extension of the 𝚀𝚆𝙽-Gross Laplacian and the 𝚀𝚆𝙽-conservation operator. It is shown that the operator Δ E g , Q ΔEg,Q\Delta _E^{g, Q}ΔEg,Q​ has a representation in terms of the 𝚀𝚆𝙽-derivatives { D c − , D c + : c ∈ N } $\{ D_c^ -, D_c^ +: \;c \in N\} $ . The poisson equation is worked out as a non homogeneous Wick differential equation associated to Δ E g , Q ΔEg,Q\Delta _E^{g, Q}ΔEg,Q​ .
In this paper we introduce the quantum white noise (QWN) conservation operator N Q acting on nuclear algebra of white noise operators) endowed with the Wick product.Similarly to the classical case, we give a useful integral representation in terms of the QWNderivatives {D - t , D + t ; t ∈ R} for the QWN-conservation operator from which it follows that the QWN-conservation operator is a Wick derivation.Via this property, a relation with the Cauchy problem associated to the QWN-conservation operator and the Wick differential equation is worked out.
Regarding a Fock space operator as a function of quantum white noise = ( at;a t ; t2 T ), we introduce its quantum white noise derivatives (qwn- derivatives) as a kind of functional derivatives with respect to at and a t . We prove that every white noise operator is dierentiable and the qwn-derivatives form a gen- eralized white noise operator eld on T. We show a relation between qwn-derivatives and generalized quantum stochastic integrals. We obtain a condition under which qwn-derivatives are dened pointwisely.
A rigorous mathematical formulation of higher powers of quantum white noises is given on the basis of the most recent theory of white noise distributions due to Cochran, Kuo and Sengupta. The renormalized quantum Itô formula due to Accardi, Lu and Volovich is derived from the renormalized product formula based on integral kernel operators on white noise functions. During the discussion, the analytic characterization of operator symbols and the expansion theorem for a white noise operator in terms of integral kernel operators are established.