QWN-Euler Operator and Associated Cauchy Problem

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.

Quantum λ-potentials associated to quantum Ornstein–Uhlenbeck semigroups

“…The development of the later along with the results can be found in Refs. [1,[3][4][5][6][8][9][10][11]16,19].…”

QWN-conservation operator and associated Wick differential equation

“…Note that, from [3] and the above discussion, for z ∈ N , the QWN-derivatives D ± z and (D ± z ) * are continuous linear operators from L(F * θ (N ′ ), F θ (N ′ )) into itself and from L(…”

“…Then, in view of Theorem 2.1 and using the same technic of calculus used in [3], we have the following Lemma 3.1. Let be given z ∈ N .…”

Quantum white noise convolution operators with application to differential equations

“…We recall that the QWN-( 1 , 2 )-Gross Laplacian Δ ( 1 , 2 ) can be de ned as in [2], via the symbol map, by…”

“…In in nite-dimensional complex analysis [6], a convolution operator on the test space F is a continuous linear operator from F into itself which commutes with all the translation operators, where the translation operator is de ned by (see [1][2][3][4] It is well known that − is a continuous linear operator from F into itself. The convolution product of a distribution Φ ∈ F * with a test function ∈ F is de ned by (Φ * )( ) = ⟨⟨Φ, − ⟩⟩, ∈ ὔ .…”