Study of the eigenvalues of the simplified equation of magnetohydrodynamic balloon plasma instability by the method of two-sided approximation (Q2739970)

This paper deals with the eigenvalue problem for simplified magnetohydrodynamic ballooning non-stability plasma equation \(((1+x^2)y')'-(\lambda+ \gamma^2(1+x^2)-\chi^2(1+x^2))y=0,\;-\infty<x<+\infty\), \(y(x)\to 0\) as \(x\to\pm\infty\), where \(\lambda,\gamma,\chi\) are some parameters. This equation is reduced by the changing of variables \(x=\text{sinh}\xi, \Psi(\xi)=\text{cosh}^{1/2}y(x)\) to the equation \(\Psi''-(q(\xi)-\Lambda)\Psi=0, \xi\in (-\infty,+\infty)\), where \(\Lambda=-\lambda-1/4\), \(q(\xi)=\gamma^2\text{cosh}^2\xi- (\lambda^2-1/4) \text{sinh}^2\xi\). The differential operator of this equation has a discrete spectrum. Convergence of the bilateral approximation for the Sturm-Liouville problem for the considered equation is proved.