We present a space–time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal h-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.

中文翻译：

---

变势薛定谔方程的时空DG方法

我们提出了具有变势的线性薛定谔方程的时空超弱不连续伽辽金离散化。对于非常一般的离散空间，所提出的方法在网格相关范数中是适定的且准最优的。当选择测试和试验空间作为分段多项式或新的拟Trefftz 多项式空间时，可以导出该方法的最佳 h收敛误差估计。后者允许自由度数量的大幅减少并允许分段平滑势。多次数值实验验证了该方法的准确性和优点。