This paper is concerned with the efficient numerical treatment of 1D stationary Schrödinger equations in the semi-classical limit when including a turning point of first order. As such it is an extension of the paper [AN18], where turning points still had to be excluded. For the considered scattering problems we show that the wave function asymptotically blows up at the turning point as the scaled Planck constant ε → 0, which is a key challenge for the analysis. Assuming that the given potential is linear or quadratic in a small neighborhood of the turning point, the problem is analytically solvable on that subinterval in terms of Airy or parabolic cylinder functions, respectively. Away from the turning point, the analytical solution is coupled to a numerical solution that is based on a WKB-marching method -using a coarse grid even for highly oscillatory solutions. We provide an error analysis for the hybrid analytic-numerical problem up to the turning point (where the solution is asymptotically unbounded) and illustrate it in numerical experiments: If the phase of the problem is explicitly computable, the hybrid scheme is asymptotically correct w.r.t. ε. If the phase is obtained with a quadrature rule of, e.g., order 4, then the spatial grid size has the limitation h = O(ε 7/12 ) which is slightly worse than the h = O(ε 1/2 ) restriction in the case without a turning point.

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We consider a model for the wave function of an electron, injected at a fixed energy E into an electronic device with stationary potential V(x). This wave function is the solution of the stationary 1D Schrödinger equation. The scattering problem is modeled on an interval where the potential varies. Moreover, V(x) is assumed constant in the exterior, i.e. in the leads of the device. Here we are interested in including turning points – points x ¯ where the potential and the energy of the particle coincide, i.e. E = V prefix( true x ¯ postfix) . We show that including a turning point lets the wave function blow‐up asymptotically as the scaled Planck constant ε → 0. This is an essential difference to the uniformly bounded wave function if turning points are excluded.

This paper is concerned with an efficient numerical method for solving the 1D stationary Schrödinger equation in the highly oscillatory regime. Being a hybrid, analytical-numerical approach it does not have to resolve each oscillation, in contrast to standard schemes for ODEs. We build upon the WKB-based marching method from [1] and extend it in two ways: By comparing the O(h) and O(h 2 ) methods from [1] we design an adaptive step size controller for the WKB method. While this WKB method is very efficient in the highly oscillatory regime, it cannot be used close to turning points. Hence, we introduce for such regions an automated method coupling, choosing between the WKB method for the oscillatory region and a standard Runge-Kutta-Fehlberg 4(5) method in smooth regions.A similar approach was proposed recently in [2, 3], however, only for a O(h)-method. Hence, we compare our new strategy to their method on two examples (Airy function on the spatial interval [0, 10 8 ] with one turning point at x = 0 and a parabolic cylinder function having two turning points), and illustrate the advantages of the new approach w.r.t. accuracy and efficiency.