The eigenvalues of the simplified ideal MHD ballooning equation

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Abstract
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 Schrödinger 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 λ on the two free parameters γ2 and μ2 of the equation is discussed, together with certain analytical approximations in the limits of small and large γ2. A brief review of the principal properties of the spheroidal functions is given in an appendix.

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