  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.
