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 . Their semiclassical behaviour is discussed for eigenvalues either near or away from the separatrix energy , especially in the phase-space vicinity of the saddle-point (q,p) = (0,0).