Localization properties of groups of eigenstates in chaotic systems

Abstract

In this paper we study in detail the localized wave functions defined in Phys. Rev. Lett. 76, 1613Lett. 76, (1994, in connection with the scarring effect of unstable periodic orbits in highly chaotic Hamiltonian system. These functions appear highly localized not only along periodic orbits but also on the associated manifolds. Moreover, they show in phase space the hyperbolic structure in the vicinity of the orbit, something which translates in configuration space into the structure induced by the corresponding self-focal points. On the other hand, the quantum dynamics of these functions are also studied. Our results indicate that the probability density first evolves along the unstable manifold emanating from the periodic orbit, and localizes temporarily afterwards on only a few, short related periodic orbits. We believe that this type of studies can provide some keys to disentangle the complexity associated to the quantum mechanics of these kind of systems, which permits the construction of a simple explanation in terms of the dynamics of a few classical structures.

Short periodic orbit approach to resonances and the fractal Weyl law

“…Scar functions are special wavefunctions constructed by taking into account classical information in the neighborhood of a periodic orbit [23][24][25][26][27][28][29]. They have been developed for closed systems and are the building blocks of the semiclassical theory of short periodic orbits, by means of which one can find eigenvalues and eigenfunctions of a quantum system starting from purely classical quantities.…”

Wavefunction statistics using scar states

“…Away from the scarred region, that is for θ = 0, the scar states should correspond to universal test states [6,15], or quasimodes [16], although we have not written explicit forms for them in a harmonic-oscillator representation. Note however that one can find explicit expression for the Husimi functions in [13].…”

Semiclassical matrix elements for a chaotic propagator in the scar function basis

“…In this way, the semiclassical approximation of the scar function's matrix elements involves uniquely the action of the classical orbit S X 2 , the scalar product C for the symplectic basis of vectors and the monodromy matrices M t γ . From this former, we obtain it Cayley representation B t through equation (15), after what the complex matrix V t is obtained with (19) and (24) expresses its exponential form, while the real matrices C t and B t defined in (23) allows to obtain D t and E t though (36).…”