Application of the repeated quantization method to the problem of making asymptotic solutions of equations with holomorphic coefficients

Abstract

In this work, we derive asymptotics of solutions of ordinary differential equations with holomorphic coefficients in the neighborhood of infinity. This problem represents a particular case of the general problem of constructing asymptotics of linear differential equations with irregular singularities, namely the Poincare problem. The case of infinitely distant singular point is an example of irregular singularity and the problem of derivation of asymptotics of its solutions is reduced to the problem of constructing asymptotics of solutions in the neighborhood of zero of linear differential equations with the cusp-type singularity of the second order. If the principal symbol of differential operator has simple roots, then asymptotics of solution of equation in the neighborhood of an irregular singular point can be represented as a classic non-Fuchs asymptotics, which is a familiar fact. In the case of multiple roots, the method of repeated quantization is used. The method is based on the Laplace-Borel transform. Using repeated quantization in this paper we solve the problem of derivation of asymptotics of solutions in the neighborhood of infinity for a model problem whose singularity index has a special form. The derived asymptotics of solutions differ from the classic non-Fuchs asymptotics and represent their generalizations. The method of solution of this model problem in its essential part is transferred to the general case. Thus, this work is one of steps in solving Poincare problem.