  We consider a semi-classical Schrodinger operator with a degenerate potential
V(x,y) =f(x) g(y) . g is assumed to be a homogeneous positive function of m
variables and f is a strictly positive function of n variables, with a strict
minimum. We give sharp asymptotic behaviour of low eigenvalues bounded by some
power of the parameter h, by improving Born-Oppenheimer approximation.
