Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Mitrokhin S. I. Multipoint Differential Operators: „Splitting“ of the Multiple in Main Eigenvalues. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2017, vol. 17, iss. 1, pp. 5-18. DOI: 10.18500/1816-9791-2017-17-1-5-18, EDN: YNBXZX

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
22.02.2017
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Russian
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UDC: 
517.926
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YNBXZX

Multipoint Differential Operators: „Splitting“ of the Multiple in Main Eigenvalues

Autors: 
Mitrokhin Sergei Ivanovich, Research Computing Center of Moscow State University named after M.V. Lomonosov
Abstract: 

We study the boundary value problem for the differential operator of the eighth order with a summable potential. The boundary conditions of the boundary value problem are multipoint. We derived the integral equation for solutions of differential equation which define the studied differential operator. The asymptotic formulas and estimates for the solutions of the corresponding differential equation for large values of the spectral parameter are obtained. By studying the boundary conditions, the equation for the eigenvalues as the determinant of the fourth order is derived. By using the properties of determinants and asymptotic formulas for solutions of differential equation we study the asymptotic behavior of the roots of the equation on eigenvalues of the operator. The coefficients of the boundary conditions of the studied boundary value problem are chosen in such a way that the main approach of the equation for the eigenvalues of the operator has two roots multiplicity three. The indicator diagram of the equation for the eigenvalues is studied in the detail. Studying one of the sectors of the indicator diagram, we derived the asymptotics of the eigenvalues of the studied operator. It is shown that the eigenvalues which are multiple in the main approximation „are split“ into three single series of eigenvalues. Similar properties of eigenvalues are observed in other sectors of the indicator diagram.

References: 
  1. Birkhoff G. D. On the asymptotic character of the solutions of the certain linear differential equations containing parameter. Trans. Amer. Math. Soc., 1908, vol. 9, pp. 219–231.
  2. Tamarkin J. D. On some general problems in the theory of ordinary linear differential equations and on the expansion in series of arbitrary functions. Petrograd, typography M. P. Frolova, 1917. 308 p. (in Russian). 
  3. Fedorjuk M. V. The asymptotics of solutions to ordinary linear differential equations of the n-th order. Differ. Uravn., 1966, vol. 2, no. 4, pp. 492–507 (in Russian).
  4. Gel’fand I. M., Levitan B. M. Ob odnom prostom tozhdestve dlia sobstvennykh znachenii differentsial’nogo operatora vtorogo poriadka [On a simple identity for eigenvalues of a differential operator of the second order]. Dokl. USSR Academy of Sciences, 1953, vol. 88, pp. 593–596 (in Russian).
  5. Levitan B. M., Gasymov M. G. Determination of a differential equation by two of its spectra. Russian Math. Surveys, 1964, vol. 19, iss. 2, pp. 1–63. DOI: https://doi.org/10.1070/RM1964v019n02ABEH001145.
  6. Lidskii V. B., Sadovnichii V. A. Asymptotic formulas for the zeros of a class of entire functions. Math. USSR-Sb., 1968, vol. 4, iss. 4, pp. 519–527. DOI: https://doi.org/10.1070/SM1968v004n04ABEH002812.
  7. Il’in V. A. Convergence of eigenfunction expansions at points of discontinuity of the coefficients of a differential operator. Math. Notes, 1977, vol. 22, iss. 5, pp. 870–882. DOI: https://doi.org/10.1007/BF01098352.
  8. Mitrokhin S.I. About formulas of regularized traces for differential operators of the second order with discontinuous coefficients. Vestnik Moskovskogo universiteta. Ser.: matematika, mehanika [Vestnik MGU. Ser.: Mathematics, mechanics], 1986, iss. 6, pp. 3–6 (in Russian).
  9. Hald O. H. Discontinuous inverse eigenvalue problems. Commun. Pure Appl. Math., 1984, vol. 37, iss. 5, pp. 539–577. DOI: https://doi.org/10.1002/cpa.3160370502.
  10. Mitrokhin S. I. About some spectral properties of differential operators of the second order with discontinuous weight function. Doklady Akademii nauk [Doklady Math.], 1997, vol. 356, iss. 1, pp. 13-15 (in Russian).
  11. Gottlieb H. P. W. Iso-spectral operators : some model examples with discontinuous coefficients. J. Math. Anal. and Appl., 1988, vol. 132, pp. 123–137.
  12. Vinokurov V. A., Sadovnichii V. A. Arbitrary-order asymptotics of the eigenvalues and eigenfunctions of the Sturm – Liouville boundary value problem on an interval with integrable potential. Differ. Equ., 1999, vol. 34, no. 10, pp. 1425–1429.
  13. Vinokurov V. A., Sadovnichii V. A. Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm – Liouville boundary-value problem on a segment with a summable potential. Izv. Math., 2000, vol. 64, iss. 4, pp. 695–754. DOI: https://doi.org/10.1070/im2000v064n04ABEH000295.
  14. Mitrokhin S.I. Asymptotics of the eigenvalues of the differential operator of the fourth order with summable coefficients. Vestnik Moskovskogo universiteta. Ser.: matematika, mehanika [Vestnik MGU. Ser.: Mathematics, mechanics], 2009, iss. 3, pp. 14–17 (in Russian).
  15. Mitrokhin S. I. On spectral properties of a differential operator with summable coefficients with a retarded argument. Ufimsk. Mat. Zh., 2011, vol. 3, no. 4, pp. 95–115 (in Russian).
  16. Mitrokhin S. I. The spectral properties of a differential operator with summable potential and smooth weight function. Vestnik of Samara State University. Natural Science Ser., 2008, no. 8/1(67), pp. 172–187 (in Russian).
  17. Naimark M. A. Lineynye differencial’nye operatory [Linear differential operators]. Moscow, Nauka, 1969. 528 p. (in Russian).
  18. Marchenko V. A. Operatory Shturma – Liuvillya i ikh prilozheniya [Sturm – Liouville operators and their applications]. Kiev, Naukova Dumka, 1977. 329 p. (in Russian).
  19. Lundina D. Sh. Exact relationship between the asymptotic expansions of the eigenvalues of boundary value problems of the Sturm-Liouville problem and the smoothness of the potential. Teoriya funktsiy, funktsional’nyy analiz i ikh prilozheniya [The theory of functions, functional analysis and their applications], 1982, no. 37, pp. 74–101 (in Russian).
  20. Bellman R., Cooke K. L. Differentsial’no-raznostnye uravneniya [Differential-difference equations]. Moscow, Mir, 1967. 548 p. (in Russian).
  21. Sadovnichii V. A., Lyubishkin V. A. Some new results of the theory of regularized traces of differential operators. Differ. Uravn., 1982, vol. 18, no. 1, pp. 109–116 (in Russian).
  22. Mitrokhin S. I. On the "splitting” in the main approximation of multiple eigenvalues of multipoint boundary value problems. Russian Math. [Iz. VUZ], 1997, no. 3, pp. 37–42
Received: 
11.09.2016
Accepted: 
25.01.2017
Published: 
28.02.2017