Доклады Академии наукДоклады Академии наук0869-5652The Russian Academy of Sciences1582010.31857/S0869-56524874365-369Research ArticleOn singular spectrum of finite dimensional perturbations (to the Aronszajn-Donoghue-Kac theory)MalamudM. M.malamud3m@gmail.comPeoples Friendship University of Russia2708201948743653692308201923082019Copyright © 2019, Russian academy of sciences2019<p>The main results of the Aronszajn-Donoghue-Kac theory are extended to the case of <em>n</em>-dimensional (in the resolvent sense) perturbations of an operator A<sub>0 = </sub>A<sub>0<sup>*</sup></sub> defined on a Hilbert space H. Applying technique of boundary triplets we describe singular continuous and point spectra of extensions <em>A<sub>B</sub></em> of a symmetric operator <em>A</em> acting in H in terms of the Weyl function <em>M</em>() of the pair {<em>A</em>, <em>A</em><sub>0</sub>} and boundary <em>n</em>-dimensional operator <em>B</em> = <em>B</em><sup>*</sup>. Assuming that the multiplicity of singular spectrum of <em>A</em><sub>0</sub> is maximal it is established orthogonality of singular parts E<sup>s</sup><sub>Aв</sub>and E<sup>s</sup><sub>Ao</sub> of the spectral measures E<sub>Aв</sub> and E<sub>Ao</sub> of the operators <em>A<sub>B</sub></em> and <em>A</em><sub>0</sub>, respectively. It is shown that the multiplicity of singular spectrum of special extensions of direct sums <em>A</em> = <em>A</em><sup>(1)</sup> <em>A</em><sup>(2)</sup> cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. In particular, it is obtained a generalization of the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line as well as its clarification. The multiplicity of singular spectrum of special extensions of direct sums <em>A</em> = <em>A</em><sup>(1)</sup> <em>A</em><sup>(2)</sup> are investigated. In particular, it is shown that it cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. This result generalizes the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line and clarifies it.</p>finite dimensional perturbationssingular spectrumpoint spectrummultiplicity of spectrumконечномерные возмущениясингулярный спектрточечный спектркратность спектра[Aronszajn N. // Amer. J. Math. 1957. V. 79. P. 597-610.][Brasche J., Malamud M., Neidhardt H. // Int. Eq. Oper. Theory. 2002. V. 43. № 3. P. 264-289.][Горбачук В. И., Горбачук М. Л. Граничные задачи для операторных дифференциальных уравнений. Киев: Наук. думка, 1984.][Derkach V. A., Malamud M. M. // J. Funct. Anal. 1991. V. 95. P. 1-95.][Derkach V. A., Hassi S., Malamud M. M., de Snoo H. // Meth. Funct. Anal. Topology. 2000. V. 6. № 3. P. 45-65.][Donoghue W. // Communs Pure Appl. Math. 1965. V. 18. P. 559-576.][Кац И. С. // Изв. АН СССР. 1963. Т. 27. № 5. С. 1081-1112.][Kato T. Perturbation Theory for Linear Operators. B.; Heidelberg; N.Y.: Springer Verlag, 1966.][Liaw C., Treil S. // Matrix Measures and Finite Rank Perturbations of Selfadjoint Operators. arXiv:1806. 08856v1, [Math.SP], 2018][Маламуд М. М., Маламуд С. М. // Алгебра и анализ. 2003. Т. 15. № 3. С. 1-77.][Malamud M. M., Neidhardt H. // J. Funct. Anal. 2011. V. 260. № 3. P. 613-638.][Reed M., Simon B. // Methods of Modern Mathematical Physics. II. Functional Analysis. 2nd ed. N.Y.: Acad. Press, 1980.][Рофе-Бекетов Ф. С. // Мат. сб. 1960. Т. 51. № 3. С. 293-342.][Simon B., Wolff T. // Communs Pure Appl. Math. 1986. V. 39. P. 75-90.][Simonov S., Worachek N. // Integr. Equat. Oper. Theory. 2014. V. 78. № 4. P. 523-575.]