Well-posedness for degenerate Schrödinger equations

We consider the initial value problem for Schrodinger type equations $$\frac{1}{i}\partial_tu-a(t)\Delta_xu+\sum_{j=1}^nb_j(t,x)\partial_{x_j}u=0$$ with $a(t)$ vanishing of finite order at $t=0$ proving the well-posedness in Sobolev and Gevrey spaces according to the behavior of the real parts $\Re b_j(t,x)$ as $t\to0$ and $|x|\to\infty$. Moreover, we discuss the application of our approach to the case of a general degeneracy.