We consider an initial-boundary value problem for a generalized 2D time-dependentSchrödinger equation (with variable coefficients) on a semi-infinite strip. For theCrank–Nicolson-type finite-difference scheme with approximate or discrete transparentboundary conditions (TBCs), the Strang-type splitting with respect to the potential isapplied. For the resulting method, the unconditional uniform in time L2-stability isproved. Due to the splitting, an effective direct algorithm using FFT is developed now toimplement the method with the discrete TBC for general potential. Numerical results on thetunnel effect for rectangular barriers are included together with the detailed practicalerror analysis confirming nice properties of the method.

We study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.