To the Bicommutant Theorem for Algebras Generated by the Symmetries of Finite Point Sets in \({\mathbb{R}}^{3}\)

The problem of describing invariant extensions of the 3D Schrödinger operator H with finite number of point interactions leads to the need for studying matrices of a special type, the permutation matrices. A large class of such extensions considered in a certain boundary triplet is in one-to-one correspondence with a set of the so-called boundary operators (matrices). The extension of H with point interactions concentrated on X = {x₁, . . . , x_m} ⊂ \({\mathbb{R}}^{3}\) ${\mathbb{R}}^3$ is invariant under the symmetry group of X (or its subgroup) if and only if the corresponding boundary matrix commutes with the set of m-by-m permutation matrices induced by the symmetry group, i.e., belongs to its commutant.

The bicommutant theorem for such a set of matrices is proven for an arbitrary finite point set. For the special cases of regular polygon, tetrahedron, and cube, the basis of the bicommutant, considered as a vector space, is specified explicitly.