  The discrete spectrum of complex banded matrices which are compact
perturbations of the standard banded matrix of order $p$ is under
consideration. The rate of stabilization for the matrix entries sharp in the
sense of order which provides finiteness of the discrete spectrum is found. The
$p$-banded matrix with the discrete spectrum having exactly $p$ preassigned
points on the interval $(-2,2)$ is constructed. The results are applied to the
study of the discrete spectrum of asymptotically periodic complex Jacobi
matrices.
