### About one problem that occurs when applying the repeated quantization method to linear differential equation with holomorphic coefficients

M. V. Korovina, V.Y. Smirnov

#### Abstract

The present paper deals with asymptotic expansions for solution degenerate elliptic differential equations. The technique for the interpretation and construction of asymptotic expansions on the basis of the Laplace-Borel transform is referred to as resurgent analysis. The inverse Laplace-Borel transform provides a regular method for the summation of the series. In this paper, we solve the problem of constructing asymptotic expansions the inverse-transform Laplace-Borel of one exponential type functions. This transform is necessary for construction of asymptotic of solutions of differential equations with degeneration in the coefficients with repeated quantization method. This method is used to study the asymptotic of solutions of equations with holomorphic coefficients. By using the repeated quantization method, we will consider an example fourth order  differential equation and used  inverse-transform Laplace-Borel of one exponential type functions, which  receive in the paper, we construct asymptotic expansions of solution of this equation.

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