Asymptotics of Solutions to a Third-Order Equation in a Neighborhood of an Irregular Singular Point

We construct the uniform asymptotics of solutions to a third order equation whose holomorphic coefficients have an arbitrary irregular singularity in the space of functions of exponential growth. In general, the problem of constructing the asymptotics of solutions of differential equations in a neighborhood of an irregular singular point was formulated by Poincaré in his articles devoted to the analytical theory of differential equations. This problem for the equations with degeneracies of arbitrary order, if there are multiple roots, was solved only in some special cases, for example, when the equation has second order. The main method in the case of degenerations of higher order is the requantization based on the Laplace–Borel transform. This method was developed of constructing the asymptotics of solutions of differential equations in a neighborhood of irregular singular points in the case that the main symbol of the differential operator has multiple roots. The problem of constructing the asymptotics of solutions to higher order equations is much more complicated. To solve it, we use the requantization method that was not required when solving a similar problem for second order equations. We will solve some model problem that is an important step towards solving the Poincaré general problem of constructing the asymptotics of solutions in a neighborhood of an irregular singular point for an equation of arbitrary order. The problem of further research is to generalize the method of the present article to equations of arbitrary order.