Semiclassical Simple Initial Value Representations

In this article, a class of Fourier Integral Operators which converge to the unitary group of the Schr\"odinger equation in semiclassical limit $\epsilon\to 0$ is constructed. The convergence is in the uniform operator norm and allows for a error bound $C_N\epsilon^{N+1}$ for any integer $N$ and extends to Ehrenfest timescaleswith bound $C_N\epsilon^{N+1-\rho}$ where $\rho$ can be made arbitrary small. In the chemical literature those approximations are known as simple Initial Value Representations.