Completeness of the Linear, Isentropic Normal Modes

Abstract

The question of the completeness of the linear, isentropic normal modes of a quasi-static star was approached by Eisenfeld for stellar models with a vanishing surface density. Eisenfeld considered the integro-differential operator involved in the system of governing equations after separation of the angular variables. His aim was to prove that the eigenvectors of the operator form a complete orthonormal basis of a Hilbert space, but his treatment has been vitiated by an invalid conclusion. On the other hand, the completeness is closely connected to the question whether the tensorial operator that appears in the vectorial wave equation, is not merely symmetric but admits a self-adjoint extension. Kaniel & Kovetz and Dyson & Schutz pointed to the existence of a lower bound for the operator in its domain. However, an alternative derivation yields, in addition to the lower bound determined before, a term for which the existence of a lower bound remains to be proved. In the supposition that the operator has nevertheless a lower bound, Kaniel & Kovetz and Dyson & Schutz made use of the self-adjointness of the operator. They gave a spectral theorem, which enabled them to formulate an expansion theorem for a general time-dependent, linear, isentropic displacement field in a quasi-static star, in terms of normal modes.