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 one-dimensional bilinear hyperbolic Hamiltonian, H = pq or equivalently H = 1/2(P-2 - Q(2)). Their semiclassical behaviour is discussed for eigenvalues either near or away from the separatrix energy {H = O}, especially in the phase-space vicinity of the saddle-point (q, p) = (0, 0).