Quadratic Growth of Convergence Radii for Eigenvalues of Two-Parameter Sturm–Liouville Equations

Thenth eigenvalueμof the equationy″+(μ+λr(x)) y=0,a x b, subject to self-adjoint boundary conditions admits a power series expansion into powers ofλfor sufficiently small λ. It is proved that the convergence radii of these series grow liken2asntends to infinity providedr(x) is analytic on [a, b]. Applications to the Airy and Mathieu equation are given