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 [3], 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 \(\varepsilon \rightarrow 0\), 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. \(\varepsilon \). If the phase is obtained with a quadrature rule of, e.g., order 4, then the spatial grid size has the limitation \(h=\mathcal{O}(\varepsilon ^{7/12})\) which is slightly worse than the \(h=\mathcal{O}(\varepsilon ^{1/2})\) restriction in the case without a turning point.

中文翻译：

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半经典极限中的平稳Schrödinger方程：基于WKB的方案与转折点耦合

当涉及一阶转折点时，本文涉及一维平稳Schrödinger方程在半经典极限中的有效数值处理。因此，它是论文[3]的扩展，其中转折点仍然必须排除在外。对于考虑的散射问题，我们表明，随着比例缩放的普朗克常数\（\ varepsilon \ rightarrow 0 \），波函数在转折点处渐近爆炸。，这是分析的关键挑战。假设给定电势在转折点的小范围内是线性或二次方的，则该问题在该子区间上可以分别通过艾里或抛物柱面函数解析地解决。远离转折点，分析解决方案与基于WKB前进方法的数值解决方案耦合-即使对于高度振荡的解决方案，也使用粗网格。我们为混合解析数字问题提供了一个误差分析，直到转折点为止（其中解是渐近无界的），并在数值实验中进行了说明：如果问题的相位是可明确计算的，则该混合方案在渐近正确。 （\ varepsilon \）。如果使用例如4阶的正交规则获得相位，则空间网格大小的限制\（h = \ mathcal {O}（\ varepsilon ^ {7/12}）\）稍差于在没有拐点的情况下，\（h = \ mathcal {O}（\ varepsilon ^ {1/2}）\）限制。