Splitting in Potential Finite-Difference Schemes with Discrete Transparent Boundary Conditions for the Time-Dependent Schrödinger Equation
The time-dependent Schrödinger equation is the key one in many fields. It should be often solved in unbounded space domains. Several approaches are known to deal with such problems using approximate transparent boundary conditions (TBCs) on the artificial boundaries. Among them, there exist the so-called discrete TBCs whose advantages are the complete absence of spurious reflections, reliable computational stability, clear mathematical background and the corresponding rigorous stability theory. In this paper, the Strang-type splitting with respect to the potential is applied to three two-level schemes with different discretizations in space having the approximation order $O(\tau^2+|h|^k)$, $k=2$ or 4. Explicit forms of the discrete TBCs are given and results on existence, uniqueness and uniform in time $L^2$-stability of solutions are stated in a unified manner. Due to splitting, an effective direct algorithm to implement the schemes is presented for general potential.