A space-time Trefftz discontinuous Galerkin method for the Schr\"odinger equation with piecewise-constant potential is proposed and analyzed. Following the spirit of Trefftz methods, trial and test spaces are spanned by non-polynomial complex wave functions that satisfy the Schr\"odinger equation locally on each element of the space-time mesh. This allows to significantly reduce the number of degrees of freedom in comparison with full polynomial spaces. We prove well-posedness and stability of the method, and, for the one- and two-dimensional cases, optimal, high-order, $h$-convergence error estimates in a skeleton norm. Some numerical experiments validate the theoretical results presented.

翻译：根据Trefftz方法的精神,试验和试验空间由非球状复合波的函数跨越,这些函数在时空网格的每个元素上都能满足Schr\'odinger等式的局部等式。这样可以大大降低与全多球空间相比的自由度。我们证明这种方法的稳妥性和稳定性,对于一维和二维案例,在骨骼规范中,最理想的、高度的、合金的误差估计值。一些数字实验证实了所提出的理论结果。