Perturbations and stability of rotating stars. I. Completeness of normal modes

Abstract

Linear adiabatic perturbations of a differentially rotating, axisymmetric, perfect-fluid stellar model have normal modes described by a quadratic eigenvalue problem of the form where A and C are symmetric operators, B antisymmetric, and £ the Lagrangian displacement vector. We study this problem and the associated time evolution equation. We show that, in the Hilbert space H', whose norm is square-integration weighted by A, the operators A~lB and A~XC are anti-selfadjoint and selfadjoint, respectively, when restricted to vectors £ belonging to a particular but arbitrary axial harmonic. We then find bounds on the spectrum of normal modes and show that any initial data in the domain of C leads to a solution whose growth rate is limited by the spectrum and which can be expressed in a certain weak sense as a linear superposition of the normal modes. The normal modes are defined more precisely in terms of parallel projection operators associated with each isolated part of the spectrum. The quadratic eigenvalue problem can be reformulated in the space H' © ' (initial data space, or phase space) as a linear eigenvalue problem for an operator T, the generator of time evolution. This operator is not selfadjoint in H' © H' but it is selfadjoint in a Krein space (an indefinite inner-product space) formed by equipping H' © H' with the symplectic inner product. The normal modes are its eigenvectors and generalized eigenvectors.