99-362 Setsuro Fujiie, Thierry Ramond

Exact WKB analysis and the Langer modification, with application to barrier top resonances. (1083K, ps) Sep 29, 99

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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.)

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