  The purposes of this note are: 1) to propose a direct and "elementary" proof
of the main result proved by Guillemin-Paul-Uribe [GPU], namely that the
semi-classical spectrum near a global minimum of the classical Hamiltonian
determines the whole semi-classical Birkhoff normal form (denoted the BNF) in
the non-resonant case. 2) to present in the completely resonant case a similar
problem.
