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：本文针对含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方法。该方法在网格依赖范数下具有适定性，并对非常一般的离散空间达到拟最优性。当检验空间与试验空间分别选取为分片多项式空间或新型拟Trefftz多项式空间时，推导了该方法的最优$h$收敛误差估计。后者能显著降低自由度数量，并适用于分片光滑势。多个数值实验验证了所提方法的准确性与优势。