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

We study linear problems Sd 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∈N of the operator W1 = 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 classical unrestricted problem. In particular, for symmetric problems with λ1 ≤ 1 we give characterizations for polynomial tractability and strong polynomial tractability in terms of λ and the amount of the assumed symmetry. Finally, we apply our results to the approximation problem of solutions of the electronic Schrödinger equation.