  We provide the foundation of the spectral analysis of the Laplacian on the
orbital Schreier graphs of the Basilica group, the iterated monodromy group of
the quadratic polynomial $z^2-1$. This group is an important example in the
class of self-similar amenable but not elementary amenable finite automata
groups studied by Grigorchuk, \.Zuk, \v Suni\'c, Bartholdi, Vir\'ag,
Nekrashevych, Kaimanovich, Nagnibeda et al. We prove that the spectrum of the
Laplacian has infinitely many gaps and that the support of the KNS Spectral
Measure is a Cantor set. Moreover, on a generic blowup, the spectrum coincides
with this Cantor set, and is pure point with localized eigenfunctions and
eigenvalues located at the endpoints of the gaps.
