This chapter deals with the semiclassical analysis of the individual eigenfunctions in a quantum system, especially when the classical dynamics is chaotic and the quantum bound states are considered. The situation is still barely understood, but analytic methods relevant to the problem have been steadily developing [1-4]. On the one hand, quantum maps have emerged as ideal dynamical models for basic studies, with their ability to exhibit classical chaos within a single degree of freedom [5]. On the other hand, phase space techniques have become recognized as extremely powerful for describing quantum states; however, because these techniques concentrate routinely on the density operators (namely, the eigenprojectors in Wigner or Husimi representations [6]), they are still currently a long way from grasping the semiclassical shapes of the wavefunctions as such.

We argue that well-adapted representations of eigenfunctions are essential for semiclassical analysis and that they should incorporate all previous observations. First, the dynamical problem should be considered in the reduced form of a quantum map; then, its eigenstates should be analyzed in phase space; there, however, they should not be displayed as density operators but directly parametrized as wavefunctions.

This chapter essentially reviews an explicit realization of that program in one degree of freedom, in which the crucial ingredient is a phase-space parametrization of 1-d wavefunctions [7]. Every 1-d wavefunction is first expressed in a holomorphic (Bargmann) representation, then factorized over the zeros of its Husimi function, to end up being represented by a pattern of essentially N ∼ h−1 of those zeros in a 1–1 correspondence; at that point the semiclassical regime appears as a thermodynamic (N → ∞) limit.