  Bearing in mind the application to the theory of core-collapse supernovae, we
performed a global linear analysis on the stability of spherically symmetric
accretion flows through a standing shock wave onto a proto neutron star. As
unperturbed flows, we adopted the spherically symmetric steady solutions to the
Euler equations obtained with realistic equation of state and formulae for
neutrino reaction rates taken into account. Then we solved the equations for
linear perturbations numerically, and obtained the eigen frequencies and eigen
functions. We found (1) the flows are stable for all modes if the neutrino
luminosity is lower than $\sim 1\times 10^{52}$ ergs/s for
$\dot{M}=1.0M_{\odot}/{\rm s}$. (2) For larger luminosities, the non-radial
instabilities are induced, probably via the advection-acoustic cycles.
Interestingly, the modes with $\ell=2$ and 3 become unstable at first for
relatively low neutrino luminosities, e.g. $\gtrsim 2-3\times 10^{52}$ ergs/s
for the same accretion rate, whereas the $\ell=1$ mode is the most unstable for
higher luminosities, $\sim 3-7\times 10^{52}$ ergs/s. These are all oscillatory
modes. (3) For still larger luminosities, $\sim 7\times 10^{52}$ ergs/s for
$\dot{M}=1.0M_{\odot}/{\rm s}$, non-oscillatory modes, both radial and
non-radial, become unstable. These non-radial modes were identified as
convection. We confirmed the results obtained by numerical simulations that the
instabilities induced by the advection-acoustic cycles are more important than
the convection for lower neutrino luminosities.
