The complexity of linear tensor product problems in (anti)symmetric Hilbert spaces

We study linear problems SdSd defined on tensor products of Hilbert spaces with an additional (anti)symmetry property. We construct a linear algorithm that uses finitely many continuous linear functionals and show an explicit formula for its worst case error in terms of the eigenvalues λ=(λm)m∈Nλ=(λm)m∈N of the operator W1=S1†S1W1=S1†S1 of the univariate problem. Moreover, we show that this algorithm is optimal with respect to a wide class of algorithms and investigate its complexity. We clarify the influence of different (anti)symmetry conditions on the complexity, compared to the case for the classical unrestricted problem. In particular, for symmetric problems with λ1≤1λ1≤1 we give characterizations for polynomial tractability and strong polynomial tractability in terms of λλ and the amount of the assumed symmetry. Finally, we apply our results to the approximation problem of solutions of the electronic Schrödinger equation.