On the exact WKB method from a viewpoint of microlocal analysis

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We study Stokes pheonmenon for a higher order linear differential equation with a large parameter, and we also study the same problems for a non-linear differential equation such as a Painlev'e hierarchy.A Stokes geometry for a higher order linear differential equation is quite different from one for a 2nd order linear differential equation becuase of existence of virtual turning points and new Sotkes curves. It is very complicated, and thus, possibility to succesively obtain a Stokes coefficient on each Stokes curve is quite uncertain. By using so called the depth function, in this study, we have shown that it is always possible to have all the Stokes coefficients succesively.We also have succeeded in constructing an instanton-type solution for the first Painlev'e hierarchy (PI)_m. This result is quite important becuase it contains sufficiently many free parametners, and hence, we can take a family of these solutions as a basis of solutions for a connection problem.