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：本文提出了一种针对含变势线性薛定谔方程的时空超弱间断Galerkin离散格式。对于非常一般的离散空间，该方法在网格依赖范数下具有适定性和拟最优性。当测试空间和试验空间选为分片多项式或新型拟Trefftz多项式空间时，推导出了该方法的最优h-收敛误差估计。后者可大幅减少自由度数，并允许分片光滑势函数。多个数值实验验证了所提方法的精确性和优势。