Smoothed Wigner functions: a tool to resolve semiclassical structures

The Wigner and Husimi distributions are the usual phase space representations of a quantum state. The Wigner distribution has structures of order $ \hbar ^{2}$. On the other hand, the Husimi distribution is a Gaussian smearing of the Wigner function on an area of size $ \hbar $ and then, it only displays structures of size $ \hbar $. We have developed a phase space representation which results a Gaussian smearing of the Wigner function on an area of size $ \hbar ^{\sigma }$, with $ \sigma \geq 1$. Within this representation, the Husimi and Wigner functions are recovered when $ \sigma =1 $ and $ \sigma \gtrsim 2 $ respectively. We treat the application of this intermediate representation to explore the semiclassical limit of quantum mechanics. In particular we show how this representation uncover semiclassical hyperbolic structures of chaotic eigenstates