  Using coherent-state representations of quantum mechanics (Bargmann, Husimi,
and stellar representations), we describe analytically the phase-space
structure of the general eigenstates corresponding to a 1-dimensional bilinear
hyperbolic Hamiltonian, H=pq or equivalently H=1/2(P^2-Q^2). Their
semi-classical behaviour is discussed for eigenvalues either near or away from
the separatrix energy {H=0}, especially in the phase-space vicinity of the
saddle-point (q,p)=(0,0).
