Eigenstate structures around a hyperbolic point

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).