We consider a semi-classical Schrödinger operator -h2Δ + V 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 behavior of low eigenvalues bounded by some power of the parameter h, by improving Born-Oppenheimer approximation.