Construction, structure and asymptotic approximations of a microdifferential transparent boundary condition for the linear Schrodinger equation

A transparent boundary condition for the two-dimensional linear Schrodinger equation is constructed through a microlocal approximation of the operator associating the Dirichlet data to the Neumann one in a "M-quasi hyperbolic " region. Several quasi-analytic characterization results concerning the as ymptotic expansion of the total symbol of this operator in a subclass of in homogeneous symbols with a quasi-polynomial-like structure a-re stated. In particular, a high-frequency control giving the behavior of these symbols i s precised. It highlights the way of how to derive some consistent asymptot ic artificial boundary conditions involving fractional derivatives with res pect to the time variable by approximating the micro-transparent condition in the high-frequency regime. These approximate conditions are local accord ing to the space variable and should lead to some efficient and accurate nu merical simulations if they are used to truncate the unbounded domain of pr opagation. (C) 2001 Editions scientifiques et medicales Elsevier SAS.