We study the approximation properties of complex-valued polynomial Trefftz spaces for the $(d+1)$-dimensional linear time-dependent Schr\"odinger equation. More precisely, we prove that for the space-time Trefftz discontinuous Galerkin variational formulation proposed by G\'omez, Moiola (SIAM. J. Num. Anal. 60(2): 688-714, 2022), the same $h$-convergence rates as for polynomials of degree $p$ in $(d + 1)$ variables can be obtained in a mesh-dependent norm by using a space of Trefftz polynomials of anisotropic degree. For such a space, the dimension is equal to that of the space of polynomials of degree $2p$ in $d$ variables, and bases are easily constructed.

翻译：我们研究了(d+1)维线性时变薛定谔方程的复值多项式Trefftz空间的逼近性质。具体而言，针对Gómez和Moiola (SIAM. J. Num. Anal. 60(2): 688-714, 2022)提出的时空Trefftz间断伽辽金变分公式，我们证明：通过使用各向异性次数的Trefftz多项式空间，在网格依赖范数下可以获得与(d+1)变量中p次多项式相同的h-收敛速率。对于此类空间，其维数等于d变量中2p次多项式空间的维数，且基函数的构造十分简便。