  The phenomenon of quantum localization in classically chaotic eigenstates is
one of the main issues in quantum chaos (or wave chaos), and thus plays an
important role in general quantum mechanics or even in general wave mechanics.
In this work we propose two different localization measures characterizing the
degree of quantum localization, and study their relation to another fundamental
aspect of quantum chaos, namely the (energy) spectral statistics. Our approach
and method is quite general, and we apply it to billiard systems. One of the
signatures of the localization of chaotic eigenstates is a fractional power-law
repulsion between the nearest energy levels in the sense that the probability
density to find successive levels on a distance $S$ goes like $\propto S^\beta$
for small $S$, where $0 \leq \beta \leq 1$, and $\beta = 1$ corresponds to
completely extended states. We show that there is a clear functional relation
between the exponent {\beta} and the two different localization measures. One
is based on the information entropy and the other one on the correlation
properties of the Husimi functions. We show that the two definitions are
surprisingly linearly equivalent. The approach is applied in the case of a
mixed-type billiard system (Robnik 1983), in which the separation of regular
and chaotic eigenstates is performed.
