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Reductional Method in Perturbation Theory of Generalized Spectral E. Schmidt Problem

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In this a paper perturbations of multiple eigenvalues of Schmidt spectral problems is considered. At the usage of the reductional method suggested in the articles [11, 12] the investigation of the multiple Schmidt perturbation eigenvalues is reduced to the investigation of perturbation of simple ones. At the end, as application of the obtained results the problem about the boundary perturbation for the system of two Sturm–Liouville problems with Schmidt spectral parameter is considered.

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References

  1. I. S. Arzhanykh, “Generalization of the Hamilton–Cayley theorem,” Rep. Acad. Sci. Uzbek SSR, No. 7, 3–5 (1951).

  2. I. S. Arzhanykh and V. I. Gugnina, “Extension of the Krylov, LeVerrier, and Faddeev methods to polynomial matrices,” Proc. Romanovskii Math. Univ., 24, 33–67 (1962).

    Google Scholar 

  3. I. S. Arzhanykh and V. I. Gugnina, “On expansion of the characteristic equation,”Proc. Romanovskii Math. Univ., 26, 3–12 (1962).

    MathSciNet  Google Scholar 

  4. E. Goursat, Course d’Analyse Mathématique, Gautier–Villars, Paris (1933).

    Google Scholar 

  5. A. S. Il’inskiy and G. Ya. Slepyan, Oscillations and Waves in Electrodynamic Systems with Dissipation [in Russian], MGU, Moscow (1983).

    Google Scholar 

  6. A. N. Kuvshinova and B. V. Loginov, “Some concequences of the generalized Hamilton–Cayley theorem for matrices polynomially dependent on E. Schmidt spectral parameter,” ROMAI J., 10, No. 1, 81–92 (2014).

    MathSciNet  MATH  Google Scholar 

  7. B. V. Loginov and V. E. Pospeev, “On eigenvalues and eigenvectors of a perturbed operator,” Bull. Acad. Sci. Uzbek SSR. Ser. Phys.-Math. Sci., No. 6, 29–35 (1967).

  8. B. V. Loginov and D. G. Rakhimov, “On spectral problem for Laplace operator in domain with perturbed boundaries,” ROMAI J., 9, No. 2, 129–141 (2013).

    MathSciNet  MATH  Google Scholar 

  9. B. V. Loginov and Yu. B. Rusak, “Generalized Jordan structure in the branching theory,” In: Direct and Inverse Problems for Partial Differential Equations, Fan, Tashkent, pp. 133–148 (1978).

  10. Sh. I. Mogilevskiy, “On representation of a completely continuous operator in the abstract Hilbert separable space,” Bull. Higher Edu. Inst. Ser. Math., No. 3, 183–186 (1958).

  11. D. G. Rakhimov, “On perturbation of Fredholm eigenvalues of linear operators,” SVMO, 17, No. 3, 37–43 (2015).

    MATH  Google Scholar 

  12. D. G. Rakhimov, “On perturbation of Fredholm eigenvalues of linear operators,” Differ. Equ., 53, No. 5, 607–616 (2017).

    Article  MathSciNet  Google Scholar 

  13. F. Rellich, “Storungstheorie der Spektralzerlegung, I,” Math. Ann., 113, 600–619 (1936).

    Article  MathSciNet  Google Scholar 

  14. Yu. B. Rusak, “Generalized Jordan structure of an analytic operator-function and its conjugate one,” Bull. Acad. Sci. Uzbek SSR. Ser. Phys.-Math. Sci., No. 2, 15–19 (1978).

  15. E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. I,” Math. Ann., 63, 433–476 (1907).

    Article  MathSciNet  Google Scholar 

  16. E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. II,” Math. Ann., 64, 161–174 (1907).

    Article  MathSciNet  Google Scholar 

  17. E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. III,” Math. Ann., 65, 370–399 (1908).

    Article  MathSciNet  Google Scholar 

  18. N. V. Sidorov, A. V. Sinitsyn, and M. V. Falaleev, Lyapunov–Schmidt Methods in Nonlinear Analysis and Applications, Kluwer, Dordrecht (2002).

    Book  Google Scholar 

  19. M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Nonlinear Equations [in Russian], Nauka, Moscow (1969) .

    MATH  Google Scholar 

  20. D. V. Valovik and Yu. G. Smirnov, Propagation of Electromagnetic Waves in Nonlinear Foliated Media [in Russian], PGU, Penza (2010).

    Google Scholar 

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Authors and Affiliations

  1. Tashkent Branch of the Lomonosov Moscow State University, Tashkent, Uzbekistan

    D. G. Rakhimov

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  1. D. G. Rakhimov
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Correspondence to D. G. Rakhimov.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 65, No. 1, Contemporary Problems in Mathematics and Physics, 2019.

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Rakhimov, D.G. Reductional Method in Perturbation Theory of Generalized Spectral E. Schmidt Problem. J Math Sci 265, 69–78 (2022). https://doi.org/10.1007/s10958-022-06045-x

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  • DOI: https://doi.org/10.1007/s10958-022-06045-x

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