Borel-Laplace Transform and Asymptotic Theory

Sternin\, Boris Yu.; Shatalov\, Victor E.

• Part 1 Introduction - resurgent analysis in the theory of differential equations: singular points of ordinary differential equations - classification of singular points, the Borel-Laplace transform, the Euler example; equations on an infinite cylinder - modification of the Borel-Laplace transform, asymptotics of functions of exponential growth, asymptotic expansions of solutions; semi-classical approximations - WKB-expansions (elementary theory), exact WKB-approximation (quantum oscillator), asymptotics at infinity (the airy equation). Part 2 Borel-Laplace transform: enitre functions of exponential type - definitions, the Borel-Laplace transform, examples; hyperfunctions with compact support - definitions, the Borel-Laplace transform, examples; hyperfunctions of exponential growth - definitions, the Borel-Laplace transform, generalized hyperfunctions, examples; microfunctions - endlessly continuable hyperfunctions, microfunctions and their Borel-Laplace transform, microlocalization, examples. Part 3 Resurgent analysis: preliminary remarks; resurgent functions - definition of resurgent functions, the connection homomorphism and the Stokes phenomenon, asymptotic expansions, resurgent functions with simple singularities, generalizations of the notion of a resurgent function - resurgent representation; investigation near focal points - Legendre uniformization; investigation near focal points - connection homomorphism - the monodromy properties of a resurgent function, alient derivatives, alient differential equations, Stokes phenomenon and univaluedness; examples - resurgent functions of the airy type, special functions of higher order, cylinder-parabolic functions. Part 4 Applications: ordinary differential equations - reduction to the Volterra equation, analytic continuation of terms of the Neumann series, convergence of the Neumann series, resurgent solutions to ordinary differential equations; partial differential equations - asymptotic solutions to the Schrodinger equation, general equations with polynomial coefficients, examples; the Saddle Point method - statement of the problem, one-dimensional case, multi-dimensional case. Appendix - integral transforms of ramifying analytic functions: integral transform of homogeneous functions; transforms, associated with the F-transform; the R-transform; the delta/deltas transform.