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Mathematica Slovaca

Wick differential and Poisson equations associated to the πš€πš†π™½-Euler operator acting on generalized operators

Hafedh M. S. Rguigui

Published Dec 1, 2016β€’ 12 citations
Publisher

Abstract

Abstract In this paper we study the homogeneous Wick differential equation associated to the quantum white noise (πš€πš†π™½) Euler operator Ξ” E g , Q $\Delta _E^{g, Q}$ acting on generalized operators. Ξ” E g , Q $\Delta _E^{g, Q}$ is defined as sum of the extension of the πš€πš†π™½-Gross Laplacian and the πš€πš†π™½-conservation operator. It is shown that the operator Ξ” E g , Q $\Delta _E^{g, Q}$ has a representation in terms of the πš€πš†π™½-derivatives { D c βˆ’ , 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 Ξ” E g , Q $\Delta _E^{g, Q}$ .

DOI
10.1515/ms-2016-0238

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Abstract In this paper we study the homogeneous Wick differential equation associated to the quantum white noise (πš€πš†π™½) Euler operator Ξ” E g , Q Ξ”Eg,Q\Delta _E^{g, Q}Ξ”Eg,Q​ acting on generalized operators. Ξ” E g , Q Ξ”Eg,Q\Delta _E^{g, Q}Ξ”Eg,Q​ is defined as sum of the extension of the πš€πš†π™½-Gross Laplacian and the πš€πš†π™½-conservation operator. It is shown that the operator Ξ” E g , Q Ξ”Eg,Q\Delta _E^{g, Q}Ξ”Eg,Q​ has a representation in terms of the πš€πš†π™½-derivatives { D c βˆ’ , D c + : c ∈ N } {Dcβˆ’,Dc+:β€…β€Šc∈N}\{ D_c^ -, D_c^ +: \;c \in N\} {Dcβˆ’β€‹,Dc+​:c∈N} . The poisson equation is worked out as a non homogeneous Wick differential equation associated to Ξ” E g , Q Ξ”Eg,Q\Delta _E^{g, Q}Ξ”Eg,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.
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.
By adapting the white noise theory, the quantum analogues of the (classical) Gross Laplacian and L ́evy Laplacian, so called the quantum Gross Laplacian and quantum L ́evy Laplacian, respectively, are introduced as the Laplacians acting on the spaces of generalized operators. Then the integral representations of the quantum Laplacians in terms of quantum white noise derivatives are studied. Correspondences of the classical Laplacians and quantum Laplacians are studied. The solutions of heat equations associated with the quantum Laplacians are obtained from a normal-ordered white noise differential equation.