Local smoothing for the quantum Liouville equation

TL;DR

This paper investigates the regularity of solutions to the quantum Liouville equation, extending local smoothing estimates from Schrödinger equations to quantum systems, and identifying conditions for fractional derivatives of the density operator.

Contribution

It introduces local smoothing estimates for the quantum Liouville equation solutions, extending previous results from Schrödinger equations to quantum density operators in Schatten spaces.

Findings

Solutions admit locally fractional derivatives under certain Schatten space conditions

Extension of local smoothing estimates to quantum systems

Analysis of regularity properties of quantum density operators

Abstract

We analyze in this work the regularity properties of the density operator solu- tion to the quantum Liouville equation. As was recently done for the Strichartz inequalities, we extend to systems of orthonormal functions the local smoothing estimates satisfied by the solutions to the Schr\"odinger equation. We show in par- ticular that the local density associated to the solution to the free, linear, quantum Liouville equation admits locally fractional derivatives of given order provided the data belong to some Schatten spaces.