  This paper is the continuation of our work with Victor Guillemin; Victor and
I proved that the Taylor expansion of the potential at a generic non degenerate
critical point is determined by the semi-classical spectrum of the associated
Schr\"odinger operator near the corresponding critical value. Here, I show
that, under some genericity assumptions, the potential of the 1D Schroedinger
operator is determined by its semi-classical spectrum. Moreover, there is an
explicit reconstruction. This paper is strongly related to a paper of David
Gurarie (J. Math. Phys. 36:1934--1944 (1995)).
