Geodesic
Commutative properties of singularly perturbate operators
Teoretičeskaâ i matematičeskaâ fizika, Tome 102 (1995) no. 2, pp. 183-197 
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/
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     author = {N. E. Dudkin and V. D. Koshmanenko},
     title = {Commutative properties of singularly perturbate operators},
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     volume = {102},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1995_102_2_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Albeverio S., Gestezi F., Kheeg-Kron R., Kholden Kh., Reshaemye modeli v kvantovoi mekhanike, Per. s R47 angl., Mir, M., 1991 | MR

[2] Albeverio S., Karwowski W., Koshmanenko V. D., Square power of singularly perturbed operators, SFB 237-Preprint, No 176, Ruhr-Univ.-Bochum, 1992 ; Math. Nachr. (to appear) | MR

[3] Alonso A., Simon B., J. Oper. Theory, 1980, no. 4, 251–270 | MR | Zbl

[4] Akhiezer N. I., Glazman I. M., Teoriya lineinykh operatorov v gilbertovykh prostranstvakh, Nauka, M., 1966 | MR | Zbl

[5] Berezanskii Yu. M., Samosopryazhennye operatory v prostranstvakh funktsii beskonechnogo chisla peremennykh, Naukova dumka, Kiev, 1978 | MR

[6] Bokhonov Yu. E., Ukr. mat. zhurn., 42:5 (1990), 695–697 | MR | Zbl

[7] Jørgensen P. E. T., Amer. J. Math., 103:2 (1980), 273–287 | DOI | MR

[8] Koshmanenko V., Singulyarnye bilineinye formy v teorii vozmuschenii samosopryazhennykh operatorov, Naukova dumka, Kiev, 1993 | MR

[9] Koshmanenko V., Dence subspace in $A$-scale of Hilbert spaces, Preprint No 835, ITP UWr, 1993 | Zbl

[10] Kochubei A. N., Funkts. analiz i ego prilozh., 13:4 (1979), 77–78 | MR

[11] Ôta S., Proc. Amer. Math. Soc., 117:4 (1993), 8 | DOI | MR

[12] Ôta S., Proc. Amer. Math. Soc., 118:2 (1993), 5 | DOI | MR

[13] Schmüdgen K., Unbounded operator algebras and representations theory, Akademie-Verlag, Berlin, 1988