The convex structure of the cone of \(N\)-fermion positive-semidefinite 1-particle operators (Q1580728)

Let \(\mathcal{H}^{\wedge N}\) denote an \(N\)-particle antisymmetric Hilbert space consisting of functions depending on space-spin variables of \(N\) fermions. \(\mathcal{H}^{\wedge N}\) is a \(N\)-th antisymmetric product of \(\mathcal{H}^1\), the \(1\)-particle Hilbert space. A \(1\)-particle Hermitian operator \(h^1 : \mathcal{H}^1 \longrightarrow \mathcal{H}^1\) is \(N\)-fermion positive semidefinite if its \(N\)-fermion antisymmetric extention \(h^1\wedge I^{\wedge(N-1)} : \mathcal{H}^{\wedge N} \longrightarrow \mathcal{H}^{\wedge N}\) is positive semidefinite (\(h^1\wedge I^{\wedge(N-1)} \geq 0\)). A set of all \(h^1\) such that \(h^1\wedge I^{\wedge(N-1)} \geq 0\), denoted by \(\widetilde{\mathcal{P}}^1_N\), is a convex cone. The cone \(\widetilde{\mathcal{P}}^1_N\) being convex can be characterized by its extreme elements. In \textit{H. Grudziński} [Rep. Math. Phys. 37, No. 1-3, 233-259 (1996; Zbl 0885.47028)], the canonical convex decomposition of an element \(h^1 \in \widetilde{\mathcal{P}}^1_N\) into the extreme elements of \(\widetilde{\mathcal{P}}^1_N\) has been given. In the present paper, the dependence of the convex structure on the dimension of the underlying \(1\)-particle Hilbert space is studied. The three following cases are considered: \(\dim \mathcal{H}^1=n \geq N+2\), \(n=N+1\), \(n=N\), where \(N\) is a number of fermions.