On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrodinger equation. Part II

摘要

We continue to consider initial-boundary value problems for a generalized time-dependent Schrodinger equation in 1D on the semi-axis and in 2D on a semi-bounded strip. The Crank-Nicolson finite-difference schemes with general approximate transparent boundary conditions (TBCs), including the discrete TBCs, are investigated. We prove unconditional stability in L-2 and in the energy norm with respect to initial data and free terms in the equation and the approximate TBC, in general non-uniform in time, under new suitable conditions (inequalities) on a non-local operator S of the approximate TBC. These inequalities are valid for the operators S-ref of the discrete TBCs. The results are applied to derive collections of stability bounds with respect to the perturbation S-ref - S. For the discrete convolution-type operators S (in particular, for S-ref), we present necessary and sufficient conditions for the stability inequalities to be valid, together with bounds for norms of S-ref - S, in terms of reproducing functions of the convolution kernels. The analysis exploits the Hardy spaces of analytic functions.