Setsuro Fujiie, Thierry Ramond
Exact WKB analysis and the Langer modification, with application to barrier top resonances.
(1083K, ps)

ABSTRACT.  The usual first order WKB ansatz fails to describe 
the behaviour at the origin of the solutions of the radial Schr\"odinger 
equation, due to the presence of a double pole at the origin for the 
centrifugal potential. 
Following an idea proposed by Langer in 1937, we study 
the behaviour of these solutions in a fixed complex vicinity of the origin as $h$ (the Plank constant) goes to 0. Working in the framework of the exact WKB method, we justify in particular rigorously the so-called Langer modification. 
As an application we compute the scattering matrix for the radial 
Schr\"odinger equation for energies close to a local quadratic maximum of the potential, and give a quantization rule for the associated resonances. 
(This paper will appear in the Conference Proceedings 
``Toward the exact WKB analysis of differential equations, linear or 
non-linear", T.Kawai and Y.Takei eds.)