Commutative properties of singularly perturbate operators

Let a selfadjoint operator A in Hilbert space Η commutes with bounded operator S and let A-tilde be singularly perturbate with respect to A, i.e. A-tilde coincides with A on a dense domain in Η. THe conditions under which A-tilde commutes with S are studied. The cases when S is unbounded and when S is replaced for singularly perturbate S-tilde are also investigated. As an example the Laplace operator in L2(Rq) singularly perturbate by the set of δ-functions and commuting with symmetrization in Rq,q=2,3 or with regular representations of arbitrary isomeric transformations in Rq,q≤3 is considered. 13 refs