  The stability of nonaxisymmetric perturbations in differentially rotating
astrophysical accretion disks is analyzed by fully incorporating the properties
of shear flows. We verify the presence of discrete unstable eigenmodes with
complex and pure imaginary eigenvalues, without any artificial disk edge
boundaries, unlike Ogilvie & Pringle(1996)'s claim. By developing the
mathematical theory of a non-self-adjoint system, we investigate the nonlocal
behavior of eigenmodes in the vicinity of Alfven singularities at
omega_D=omega_A, where omega_D is the Doppler-shifted wave frequency and
omega_A=k_// v_A is the Alfven frequency. The structure of the spectrum of
discrete eigenmodes is discussed and the magnetic field and wavenumber
dependence of the growth rate are obtained. Exponentially growing modes are
present even in a region where the local dispersion relation theory claims to
have stable eigenvalues. The velocity field created by an eigenmode is
obtained, which explains the anomalous angular momentum transport in the
nonlinear stage of this stability.
