Narrow Results

This article is concerned with compositions in the context of three standard quantizations in the framework of Fock spaces, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states also known as a Wick product and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for issues on time dynamics (see [29, 13]). Anti-Wick quantization and Segal–Bargmann transforms are implied here for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick and Weyl symbols of the composition of two operators, we obtain an absolutely converging series and for the Weyl symbol, the remainder terms up to any orders of the expansion are controlled, still in the Fock space framework.

In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators ${({\partial }_t-\mathcal{ℒ})}^s$, $0<s<1$, where $\mathcal{ℒ}$ is the infinitesimal generator of a class of symmetric semigroups. As a by-product, we also obtain a similar result for the nonlocal operators ${\left(-\mathcal{ℒ}\right)}^s$. Our focus is on non-Euclidean situations.

We are concerned with a class of generalized coherent states (GCS) attached to Euclidean Landau levels (or to $m$-true-polyanalytic spaces), which can be obtained by displacing a Gaussian–Hermite function as an admissible (or window) function. Precisely, we evaluate the Wehrl entropy for a density operator representing a projector on a Fock state (pure states) and we give an upper bound for this entropy. We also establish an exact formula of this entropy for the heat operator (mixed states) associated with the harmonic oscillator. In this case, the behavior of the entropy with respect to the temperature parameter shows the dependence of its minimum to the Landau level or equivalently to the window function by means of which the GCS involved in the Wehrl entropy were constructed.