  We study the aspects of quantum localization in the stadium billiard, which
is a classically chaotic ergodic system, but in the regime of slightly
distorted circle billiard the diffusion in the momentum space is very slow. In
quantum systems with discrete energy spectrum the Heisenberg time $t_H =2\pi
\hbar/\Delta E$, where $\Delta E$ is the mean level spacing (inverse energy
level density), is an important time scale. The classical transport time scale
$t_T$ (diffusion time) in relation to the Heisenberg time scale $t_H$ (their
ratio is the parameter $\alpha=t_H/t_T$) determines the degree of localization
of the chaotic eigenstates, whose measure $A$ is based on the information
entropy. The localization of chaotic eigenstates is reflected also in the
fractional power-law repulsion between the nearest energy levels in the sense
that the probability density (level spacing distribution) to find successive
levels on a distance $S$ goes like $\propto S^\beta$ for small $S$, where
$0\leq\beta\leq1$, and $\beta=1$ corresponds to completely extended states. We
show that the level repulsion exponent $\beta$ is a unique rational function of
$\alpha$, and $A$ is a unique rational function of $\alpha$. $\beta$ goes from
$0$ to $1$ when $\alpha$ goes from $0$ to $\infty$. Also, $\beta$ is a linear
function of $A$, which is similar as in the quantum kicked rotator, but
different from a mixed type billiard.
