arXiv subjects
Publications and source records attributed to Vidian Rousse.
In this article we focus on a semiclassical Schrödinger equation with matrix-valued potential presenting a symmetric conjoint crossing of three eigenvalues. The potential we consider is well-known in the chemical literature as a pseudo Jahn-Teller potential. We analyze the energy transfers which occur between the three modes in terms of Wigner measures.
In this article, a class of Fourier Integral Operators which converge to the unitary group of the Schrödinger equation in semiclassical limit $ε\to 0$ is constructed. The convergence is in the uniform operator norm and allows for a error bound $C_Nε^{N+1}$ for any integer $N$ and extends to Ehrenfest timescaleswith bound $C_Nε^{N+1-ρ}$ where $ρ$ can be made arbitrary small. In the chemical literature those approximations are known as simple Initial Value Representations.
A class of Fourier Integral Operators which converge to the unitary group of the Schroedinger equation in semiclassical limit $\eps\to 0$ is constructed. The convergence is in the uniform operator norm and allows for an error bound of order $O(\eps^{1-\rho})$ for Ehrenfest timescales, where $\rho$ can be made arbitrary small. For the shorter times of order O(1), the error can be improved to arbitrary order in $\eps$. In the chemical literature the approximation is known as the Herman-Kluk propagator.
In this work, a class of semiclassical Fourier Integral Operators (FIOs) with complex phase associated to some canonical transformation of the phase space $T^*\R^d$ is constructed. Upon some general boundedness assumptions on the symbol and the canonical transformation, their continuity (as operators) from the Schwartz class into itself and from $L^2$ into itself are proven.