On polynomial Trefftz spaces for the linear time-dependent Schrödinger equation
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We study the approximation properties of complex-valued polynomial Trefftz spaces for the (d+1)-dimensional linear time-dependent Schrödinger equation. More precisely, we prove that for the space-time Trefftz discontinuous Galerkin variational formulation proposed by Gómez, 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.
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