Commutative properties of singularly perturbed operators
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 inL2(Rq) that is singularly perturbed by a set of δ functions and commutes with the symmetrization operator inRq,q=2, 3, or with regular representations of arbitrary isometric transformations inRq,q≤3.
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