Regularized asymptotics of the solution of a singularly perturbed mixed problem on the semiaxis for an equation of Schrodinger type in the presence of a strong turning point for the limit operator

In the proposed work we construct a regularized asymptotics for the solution of a singularly perturbed inhomogeneous mixed problem on the half-axis arising from a semiclassical transition in the Schrodinger equation in the coordinate representation. The potential energy profile chosen in the paper leads to a singularity in the spectrum of the limit operator in the form strong the turning point. Based on the ideas of asymptotic integration of problems with an unstable spectrum by S.A. Lomov and A.G. Eliseev, it is indicated how and from what considerations regularizing functions and additional regularizing operators should be introduced, the formalism of the regularization method for the problem posed is described in detail, and justification of this algorithm and an asymptotic solution of any order with respect to a small parameter is constructed.