We present a space--time ultra-weak discontinuous Galerkin discretization of the linear Schr\"odinger 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.

翻译：本文针对带变势的线性薛定谔方程，提出了一种时空超弱间断伽辽金离散格式。该格式在网格相关范数下对非常一般的离散空间具有适定性和拟最优性。当检验空间与试验空间选择为分片多项式空间或新型拟特雷夫多项式空间时，导出了该方法的优化$h$收敛误差估计。后者允许大幅减少自由度数目，并适用于分片光滑势能函数。多项数值实验验证了所提方法的精度与优势。