  It is commonly believed that Anderson localized states and extended states do
not coexist at the same energy. Here we propose a simple mechanism to achieve
the coexistence of localized and extended states in a band in a class of
disordered quasi-1D and quasi-2D systems. The systems are partially disordered
in a way that a band of extended states always exists, not affected by the
randomness, whereas the states in all other bands become localized. The
extended states can overlap with the localized states both in energy and in
space, achieving the aforementioned coexistence. We demonstrate such
coexistence in disordered multi-chain and multi-layer systems.
