Remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrodinger equation on the half-axis

摘要

We consider the generalized time-dependent Schrodinger equation on the half-axis and a broad family of finite-difference schemes with the discrete transparent boundary conditions (TBCs) to solve it. We first rewrite the discrete TBCs in a simplified form explicit in space step h. Next, for a selected scheme of the family, we discover that the discrete convolution in time in the discrete TBC does not depend on h and, moreover, it coincides with the corresponding convolution in the semi-discrete TBC rewritten similarly. Both moments allow us to prove the bound for the difference between the kernels of the discrete convolutions in the discrete and semi-discrete TBCs that is the first result of such kind. Numerical experiments on replacing the discrete TBC convolutions by the semi-discrete one exhibit truly small absolute errors though not relative ones in general. The suitable discretization in space of the semi-discrete TBC for the higher-order Numerov scheme is also discussed.