Asymptotic Behavior of Solutions in Finite Difference Schemes

Many problems of numerically solving the Schrödinger equation require that we choose asymptotic distances many times greater than the characteristic size of the region of interaction. If the solution to one-dimensional equations can be immediately chosen in a form that preserves unitarity, the invariance of probability (in the form of, e.g., fulfilling an optical theorem) is a real problem for two-dimensional equations. An addition that does not exceed the discretization error and ensures a high degree of unitarity is proposed as a result of studying the properties of a discrete two-dimensional equation.