The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schr"odinger equation on a semi-infinite strip

We consider an initial-boundary value problem for a generalized 2D time-dependent Schrodinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time $L^2$-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method.

💡 Research Summary

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The paper addresses the numerical solution of a generalized two‑dimensional time‑dependent Schrödinger equation posed on a semi‑infinite strip, where the coefficients (mass density ρ, metric tensor B, and potential V) may vary with both spatial variables. Classical approaches based on the Crank–Nicolson finite‑difference scheme provide second‑order accuracy in time and space but become computationally expensive because each time step requires solving a complex linear system, especially when the potential depends on the transverse coordinate y.

To overcome this difficulty the authors introduce a Strang‑type splitting with respect to the potential. The potential V(x,y) is decomposed into a “background” part eV(x) that is constant for x ≥ X0 and a remainder ΔV(x,y)=V−eV. The time interval