Asymptotic Expansions of Eigenvalues of the First Boundary-Value Problem for Singularly Perturbed Second-Order Differential Equation with Turning Points

For singularly perturbed second-order equations, the dependence of the eigenvalues of the first boundary-value problem on a small parameter at the highest derivative is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable. This leads to the emergence of so-called turning points. Asymptotic expansions with respect to a small parameter are obtained for all eigenvalues of the considered boundary-value problem. It turns out that the expansions are determined only by the behavior of the coefficients in the neighborhood of the turning points.