SCATTERING MATRIX AND RESONANCES ASSOCIATED WITH A HETEROCLINE ORBIT

We study the semiclassical resonances for 1 dimensional Schrodinger eq uation when the potential presents exactly two maxima of the same leve l on the real axis. We compute the asymptotic expansion of the scatter ing matrix in the semiclassical regime, for energies E which may depen d on h in a complex vicinity of the critical level. Quantization condi tions are obtained for the resonances, and we describe very precisely their location. In particular we show that the density of resonances g oes up like O(log(1/h)) in any neighbourhood of size O(h) of the criti cal level as opposed to O(1) in the case of a unique maximum. (C) Else vier, Paris.