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We present a space-time ultra-weak discontinuous Galerkin discretization ofthe linear Schr\"odinger equation with variable potential. The proposed methodis well-posed and quasi-optimal in mesh-dependent norms for very generaldiscrete spaces. Optimal $h$-convergence error estimates are derived for themethod when test and trial spaces are chosen either as piecewise polynomials,or as a novel quasi-Trefftz polynomial space. The latter allows for asubstantial reduction of the number of degrees of freedom and admitspiecewise-smooth potentials. Several numerical experiments validate theaccuracy and advantages of the proposed method.

A space-time DG method for the Schr\"odinger equation with variable potential