Lindblad evolution with subelliptic diffusion

TL;DR

This paper extends mathematical results on Lindblad evolution by analyzing subelliptic diffusion, demonstrating that semiclassical derivative estimates hold in this more general setting with global bounds.

Contribution

It introduces a framework for Lindblad evolution with subelliptic diffusion, extending previous elliptic results to a broader class of jump operators and diffusion conditions.

Findings

Semiclassical derivative estimates are valid in subelliptic cases.

Global $L^p$ bounds are established for the Fokker-Planck equation.

The H"ormander condition ensures the applicability of the results.

Abstract

We consider classical/quantum correspondence in Lindblad evolution with jump operators for which the corresponding Fokker--Planck equation is subelliptic. This allows us to consider the physical model proposed by Zurek and Paz, and to extend some of the recent mathematical results of Hernandez, Ranard and Riedel, Galkowski and Zworski, and Li, where the diffusion term in the Fokker-Planck equation was assumed elliptic. We consider the case where the jump operators $\ell_j$ in the Lindbladian are linear functions of $x$, and place an assumption which implies that the H\"ormander condition holds for the resulting Fokker-Planck equation. By constructing a suitable parametrix for this equation we show that the semiclassical derivative estimates established for elliptic diffusion also hold in the subelliptic case, with global bounds in $L^p$ for all $1\le p\le \infty$.