 
Summary: Serdica Math. J. 34 (2008), 267310
RESOLVENT AND SCATTERING MATRIX AT THE
MAXIMUM OF THE POTENTIAL
Ivana Alexandrova, JeanFranžcois Bony, Thierry Ramond
Communicated by G. Popov
Dedicated to Vesselin Petkov on the occasion of his 65th birthday
Abstract. We study the microlocal structure of the resolvent of the semi
classical Schršodinger operator with short range potential at an energy which
is a unique nondegenerate global maximum of the potential. We prove that
it is a semiclassical Fourier integral operator quantizing the incoming and
outgoing Lagrangian submanifolds associated to the fixed hyperbolic point.
We then discuss two applications of this result to describing the structure of
the spectral function and the scattering matrix of the Schršodinger operator
at the critical energy.
1. Introduction. We consider the semiclassical Schršodinger operator
(1.1) P = P0 + V, P0 = 
1
2
h2
, 0 < h 1,
