Eigenvalues of the Simplified Ideal MHD Ballooning Equation. Pt.2. The Eigenvalue Problem

摘要

The investigation of the spectrum of the simplified differential equation describing the variation of the amplitude of the ideal MHD ballooning instability along magnetic field lines constitutes a multiparameter Schroedinger eigenvalue problem. An exact eigenvalue relation for the discrete part of the spectrum is obtained in terms of the oblate spheroidal functions. The dependence of the eigenvalues lambda on the two free parameters gamma sup 2 and mu sup 2 of the equation is discussed, together with certain analytical approximations in the limits of small and large gamma sup 2 . A brief review of the principal properties of the spheroidal functions is given in an appendix. (ERA citation 11:035224)