| Abstract: | Suppose the self-adjoint operatorA in the Hilbert spaceH commutes with the bounded operatorS. Suppose another self-adjoint operatorā is singularly perturbed with respect toA, i.e., it is identical toA on a certain dense set inH. We study the following question: Under what conditions doesā also commute withS? In addition, we consider the case whenS is unbounded and also the case whenS is replaced by a singularly perturbed operator S. As application, we consider the Laplacian inL 2(R q ) that is singularly perturbed by a set of δ functions and commutes with the symmetrization operator inR q ,q=2, 3, or with regular representations of arbitrary isometric transformations inR q ,q≤3. |
