In the chaotic quantum map, the overlap distributions between the
eigenstates 
and a probe state 
are well described by the
random matrix theory (RMT) in most cases. However, if the probe state
is localized on a fixed point, the overlap distribution shows a strong
deviation from the RMT expectation, which is a result of the
scarring phenomena in chaotic eigenstates [1].
In our recent paper [2],
a joint-probability distribution has been developed for the overlaps
with the eigenstates of harmonic oscillator which act as the
probe states. This is a kind of scarred RMT, and has explained
successfully the scarring effect in tunneling rate
distributions. This joint-probability distribution contains all
information of wavefunction statistics near the fixed point, and has
an interesting structure which is directly related to the
corresponding classical dynamics. We find that one linear combination
of 's is an optimized probe state. Following Vergini et
al [3] we call this the scar state, and the statistics of its
overlap with chaotic states shows maximum deviation from RMT. This
scar state shows that the scarring appears along the stable and
unstable manifolds. We show that this joint distribution works very
well not only in the perturbed cat map, but also in the boundary
function of billiard. This means that the chaotic wavefunction
statistics are characterized by the Lyapunov exponent and the angle
between the stable and unstable manifolds of the fixed point. In the
billiard case, we investigate the relation between the statistics of
the boundary functions and those of the real eigenfunctions. This
study gives the way to see which eigenstate should be scarred or not
in a statistical way.
References
[1] L. Kaplan, Phys. Rev. Lett. 80, 2582 (1998);
L. Kaplan and E.J. Heller, Ann. Phys. 264, 171 (1998).
[2] S.C. Creagh, S.-Y. Lee, and N.D. Whelan, Ann. Phys. 295, 194 (2002).
[3] E.G. Vergini and G.G. Carlo, J. Phys. A: Math. Gen. 34, 4525 (2001).