Geodesic
Commutative properties of singularly perturbate operators
Teoretičeskaâ i matematičeskaâ fizika, Tome 102 (1995) no. 2, pp. 183-197

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Let a selfadjoint operator $A$ in Hilbert space $\mathcal H$ commutes with bounded operator $S$ and let $\widetilde A$ be singularly perturbate with respect to $A$, i.e. $\widetilde A$ coincides with $A$ on a dense domain in $\mathcal H$. The conditions under wich $\widetilde A$ commutes with $S$ are studied. The cases when $S$ is unbounded and when $S$ is replaced for singularly perturbate $\widetilde S$ are also investigated. As an example the Laplace operator in $L_2(\mathbf R^q)$ singularly perturbate by the set of $\delta$-functions and commuting with symmetrization in $\mathbf R^q$, $q=2,3$ or with regular representations of arbitrary isometric transformations in $\mathbf R^q$, $q\leqslant 3$ is considered. 
@article{TMF_1995_102_2_a1,
     author = {N. E. Dudkin and V. D. Koshmanenko},
     title = {Commutative properties of singularly perturbate operators},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {183--197},
     publisher = {mathdoc},
     volume = {102},
     number = {2},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1995_102_2_a1/}
}
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N. E. Dudkin; V. D. Koshmanenko. Commutative properties of singularly perturbate operators. Teoretičeskaâ i matematičeskaâ fizika, Tome 102 (1995) no. 2, pp. 183-197. http://geodesic.mathdoc.fr/item/TMF_1995_102_2_a1/