Poincaré Husimi representation of eigenstates in quantum billiards

For the representation of eigenstates on a Poincaré section at the boundary of a billiard different variants have been proposed. We compare these Poincaré Husimi functions, discuss their properties, and based on this select one particularly suited definition. For the mean behavior of these Poincaré Husimi functions an asymptotic expression is derived, including a uniform approximation. We establish the relation between the Poincaré Husimi functions and the Husimi function in phase space from which a direct physical interpretation follows. Using this, a quantum ergodicity theorem for the Poincaré Husimi functions in the case of ergodic systems is shown.