Full semiclassical asymptotics near transition points

We construct complete asymptotic expansions of solutions of the 1D semiclassical Schr\"odinger equation near transition points. There are three main novelties: (1) transition points of order $\kappa\geq 2$ (i.e.\ trapped points -- the simple turning point is $\kappa=1$, the simple pole is $\kappa=-1$) are handled, (2) various terms in the operator are allowed to have controlled singularities of a form compatible with the geometric structure of the problem (some applications are given in the text), and (3) the term-by-term differentiability of the expansions with respect to the semiclassical parameter is included. We prove that any solution to the semiclassical ODE with initial data of exponential type is of exponential-polyhomogeneous type on a suitable manifold-with-corners compactifying the $h\to 0^+$ regime. Consequently, such a solution has an atlas of full asymptotic expansions in terms of elementary functions, and these expansions are well-behaved. The Airy and Bessel functions show up in the expected way, as the asymptotic data at one boundary edge. We are able to handle cases that Langer--Olver could not because the framework of polyhomogeneous functions on manifolds-with-corners provides more flexibility (two matched $h\to 0^+$ expansions, possibly with logarithms, in this case) than that employed by Langer--Olver (one uniform $h\to 0^+$ expansion without logarithms). We work entirely in the $C^\infty$ category. No analyticity is ever assumed, nor proven.