  The linearized operator for non-radial oscillations of spherically symmetric
self-gravitating gaseous stars is analyzed in view of the functional analysis.
The evolution of the star is supposed to be governed by the Euler-Poisson
equations under the equation of state of the ideal gas, and the motion is
supposed to be adiabatic. We consider the case of not necessarily isentropic,
that is, not barotropic motions. Basic theory of self-adjoint realization of
the linearized operator is established. Some problems in the investigation of
the concrete properties of the spectrum of the linearized operator are
proposed. The existence of eigenvalues which accumulate to 0 is proved in a
mathematically rigorous fashion.The absence of continuous spectra and the
completeness of eigenfunctions for the operators reduced by spherical harmonics
is discussed.
