We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semiinfinite strip. For the Numerov-Crank-Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time L 2 -stability (in particular, L 2 -conservativeness) are proved. Due to the splitting, an effective direct algorithm using FFT in the direction perpendicular to the strip is developed to implement the splitting method for general potential. Numerical results on the tunnel effect for smooth and rectangular barriers together with the practical error analysis on refining meshes are included as well.Keywords: the time-dependent Schrödinger equation, the Numerov discretization in space, the Crank-Nicolson discretization in time, the Strang splitting, discrete transparent boundary conditions, uniqueness, stability, tunnel effect, practical error analysis MSC[2010] 65M06, 65M12, 35Q41. IntroductionThe time-dependent Schrödinger equation with several variables is important in quantum mechanics, atomic and nuclear physics, wave physics, microelectronics and nanotechnologies, etc. Often it should be solved in unbounded space domains.Several approaches were developed and studied to deal with problems of such kind, in particular, see [1,2,3,6,18,20]. One of them exploits the so-called discrete transparent boundary conditions (TBCs) at artificial boundaries [3,11]. Its advantages are the complete absence of spurious reflections, reliable computational stability, clear mathematical background and corresponding rigorous stability theory.Concerning finite-difference schemes in several space variables with the discrete TBCs, the standard Crank-Nicolson scheme in the case of an infinite (or semi-infinite) strip was studied in detail in [3,7,8]; the higher order Numerov-Crank-Nicolson scheme was considered in [19], and a general family of schemes was treated in [23,24]. But all the schemes are implicit, and solving of specific complex systems of linear algebraic equations is required at each time level. Efficient methods to solve similar systems are well developed by the moment in real situation but not the complex one.The splitting technique is widely used to simplify numerical solving of the time-dependent Schrödinger and related equations, in particular, see [4,5], [13]-[17], [21]. The Strang-type splitting with respect to the *

An initial-boundary value problem for the n-dimensional (n 2) time-dependent Schrödinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for n 3. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting method, the uniqueness of solution and the unconditional uniform in time L 2 -stability (in particular, L 2 -conservativeness) are proved. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applicable for general potential. Numerical results on the 2D tunnel effect for a Pöschl-Teller-like potential-barrier and a rectangular potential-well are also included.MSC[2010] classification: 65M06, 65M12, 35Q40. Keywords: the time-dependent Schrödinger equation, the Crank-Nicolson finite-difference scheme, higher-order scheme, the Strang splitting, discrete transparent boundary conditions, stability, tunnel effect IntroductionThe time-dependent Schrödinger equation with several space variables is crucial in quantum mechanics and electronics, nuclear and atomic physics, wave physics, etc. Often it should be solved in unbounded space domains.Several approaches were developed and investigated for solving problems of such kind, in particular, see [1,2,3,6,16,18]. One of them exploits the so-called discrete transparent boundary conditions (TBCs) at artificial boundaries [3,11]. Its advantages are the complete absence of spurious reflections in practice as well as the rigorous mathematical background and stability results in theory.The discrete TBCs for the Crank-Nicolson finite-difference scheme, the higher order Numerov-Crank-Nicolson scheme and a general family of schemes on an infinite or semiinfinite strip were constructed and studied respectively in [3,7,8], [17] and [21,22]. All these schemes are implicit, so to implement them, solving of specific complex systems of linear algebraic equations is required at each time level.The splitting technique is widely used to simplify numerical solving of the timedependent Schrödinger and related equations, in particular, see [4,5,13,14,15,19]. The known Strang-type splitting with respect to the potential has been recently applied to the Crank-Nicolson and the Numerov-Crank-Nicolson scheme with the discrete TBCs in 2D case in [10,20].Higher order methods are important due to their ability to reduce computational costs essentially, and the Numerov-Crank-Nicolson scheme can be written in n-dimensional case as well. But we show that, for n 3, the Numerov space operators lose their important spectral properties existing for n = 2 so that the scheme becomes impractical.In this paper, in the spirit of [21,22], we first split these operators (in space) and recover the properties without reducing...

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