Let $P$ be a linear partial differential operator with analytic coefficients. We assume that $P$ is of the form “sum of squares”, satisfying Hörmander’s bracket condition. Let $q$ be a characteristic point for $P$. We assume that $q$ lies on a symplectic Poisson stratum of codimension two. General results of Okaji show that $P$ is analytic hypoelliptic at $q$. Hence Okaji has established the validity of Treves’ conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of...

We consider Schrödinger operators $H$ on ${ℝ}^n$ with variable coefficients. Let $H_0=-\frac{1}{2}▵$ be the free Schrödinger operator and we suppose $H$ is a “short-range” perturbation of $H_0$. Then, under the nontrapping condition, we show that the time evolution operator: $e^{-itH}$ can be written as a product of the free evolution operator $e^{-it{H}_0}$ and a Fourier integral operator $W\left(t\right)$ which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators. These results...