Quantum Gross Laplacian and Applications

TL;DR

This paper introduces the quantum Gross Laplacian, explores its relation to the classical version, and provides explicit solutions to quantum white noise differential equations, including the quantum Gross heat equation.

Contribution

It presents the first noncommutative extension of the Gross Laplacian and connects it to classical operators, with applications to solving quantum stochastic differential equations.

Findings

Established the quantum Gross Laplacian and its properties.

Derived explicit solutions for quantum white noise differential equations.

Connected classical and quantum Gross Laplacians through specific operator cases.

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.