Quantum chaotic resonances from short periodic orbits

We present an approach to calculating the quantum resonances and resonance wave functions of chaotic scattering systems, based on the construction of states localized on classical periodic orbits and adapted to the dynamics. Typically only a few such states are necessary for constructing a resonance. Using only short orbits (with periods up to the Ehrenfest time), we obtain approximations to the longest-living states, avoiding computation of the background of short living states. This makes our approach considerably more efficient than previous ones. The number of long-lived states produced within our formulation is in agreement with the fractal Weyl law conjectured recently in this setting. We confirm the accuracy of the approximations using the open quantum baker map as an example.

Figure 1

Husimi representation of symmetrized right scar functions corresponding to a period 3 orbit of the triadic baker map, at (a) $N=81$ and (b) $N=243$. In panel (b) white crosses show the location of periodic points of the trajectory, and reflection by the diagonal produces a symmetric partner. Absolute value grows from white to black.

Figure 2

(Color online) In (a) we show the spectrum of the open triadic baker map for $N=81$ (circles) and the spectrum of the scar matrix (crosses). Their moduli (ordered by decreasing value) are displayed in panel (b).

Figure 3

Husimi representation of two exact right resonances, for the triadic baker map at $N=81$.

Figure 4

Husimi representation of two symmetrized right scar functions corresponding to the same orbit of period 5, at $N=81$. White crosses show the periodic points of the orbit, and reflection through the diagonal produces a symmetric partner.