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arXiv:0804.3938

Quantum Gross Laplacian and Applications

Habib Ouerdiane, Samah Horrigue

Published Apr 24, 2008• 2 citations
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Abstract

In this paper, we introduce and study a noncommutative extension of the Gross Laplacian, called quantum Gross Laplacian. Then, applying the quantum Gross Laplacian to the particular case where the operator is the multiplication operator, we find a relation between classical and quantum Gross Laplacian. As application, we give explicit solution of linear quantum white noise differential equation. In particular, we give a explicit solution of the quantum Gross heat equation.

DOI
10.48550/arxiv.0804.3938

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In this paper, we introduce and study a noncommutative extension of the Gross Laplacian, called quantum Gross Laplacian. Then, applying the quantum Gross Laplacian to the particular case where the operator is the multiplication operator, we find a relation between classical and quantum Gross Laplacian. As application, we give explicit solution of linear quantum white noise differential equation. In particular, we give a explicit solution of the quantum Gross heat equation.
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.
Abstract In this paper, we give a probabilistic representation of the heat equation associated with the quantum K-Gross Laplacian using infinite-dimensional stochastic calculus in two variables. Applying the heat semigroup to the particular case where the operator is the multiplication one, we establish a relation between the classical and the quantum heat semigroup. Finally, using a combination between convolution calculus and the generalized stochastic calculus, we give a generalization of the Feynman–Kac formula.
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 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.