Quadratic growth of convergence radii for eigenvalues of two-parameter Sturm-Liouville equations
The paper deals with two-parameter Sturm-Liouville problem \(y'' + (\mu + \lambda r) y = 0\) on a finite interval \([a,b]\) with selfadjoint boundary conditions \(\alpha_1 y(a) + \alpha_2 y'(a) = 0\), \(\beta_1 y(b) + \beta_2 y'(b) = 0\). Here \(\lambda\) and \(\mu\) are parameters and the function \(r\) is continuous and real-valued. The \(n\)-th eigenvalue \(\mu_n (\lambda)\) of such a problem can be expanded in powers of \(\lambda\) for sufficiently small \(|\lambda |\). It is proved that the convergence radii \(\rho_n\) corresponding to these expansions are of order \(O(n^2)\) if \(r\) is analytic on \([a,b]\). Here the Liouville transformation is applied to the given problem. In addition, concrete lower bounds for \(\liminf_{ n \to \infty} (\rho_n/n^2)\) are given. The results apply nicely to the Mathieu equation and the two-parameter Airy equation.
scientific article; zbMATH DE number 895814