# On Linear Adiabatic Perturbations of Spherically Symmetric Gaseous Stars Governed by the Euler-Poisson Equations

## TL;DR

This paper analyzes the spectral properties of linearized operators governing non-radial oscillations in spherically symmetric gaseous stars, establishing self-adjointness and eigenvalue behavior under adiabatic Euler-Poisson dynamics.

## Contribution

It provides a rigorous mathematical framework for the spectral analysis of non-isentropic stellar oscillations governed by Euler-Poisson equations.

## Key findings

- Existence of eigenvalues accumulating at zero.
- Self-adjoint realization of the linearized operator.
- Absence of continuous spectra and completeness of eigenfunctions.

## Abstract

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.