We construct complete asymptotic expansions of solutions of the one-dimensional
semiclassical Schrödinger equation near transition points. There are three main
novelties:
Transition points of order
(i.e., trapped points — the simple turning point is
,
the simple pole is
)
are handled.
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).
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
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 and Olver could not because the framework of polyhomogeneous
functions on manifolds-with-corners provides more flexibility (two matched
expansions,
possibly with logarithms, in this case) than that employed by Langer and Olver (one
uniform
expansion without logarithms). We work entirely in the
category. No analyticity is ever assumed, nor proven.