Semiclassical limit for mixed states with singular and rough potentials.

; this assumption on the potential guarantees the well posedness of the Liouville equation in the space of bounded integrable solutions. We find sufficient conditions on the initial data to ensure that the quantum dynamics converges to the classical one. More precisely, we consider the Husimi functions of the solution of the Heisenberg-von Neumann equation, and under suitable assumptions on the initial data we prove that they converge, as

\e \to 0

, to the unique bounded solution of the Liouville equation (locally uniformly in time).

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BV

\e \to 0

, to the unique bounded solution of the Liouville equation (locally uniformly in time).

BV

We will show that Liouville and quantum Liouville operators

L

and

L_\hbar

generate two

C_0

-groups

e^{tL}

and

e^{tL_h}

of isometries in

L^2(\mathbb{R}^{2n})

and

e^{tL_h}

converges ultraweakly to

e^{tL}

. As a consequence we show that the Gaussian mollifier of the Wigner function, called Husimi function, converges in

L^1(\mathbb{R}^{2n})

to the solution of the Liouville equation.