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Akhtyamov A.M. Survey of studies on degenerate boundary conditions and finite spectrum. Multiphase Systems. 14 (2019) 3. 184–201 (in Russian).
2019. Vol. 14. Issue 3, Pp. 184–201
URL: http://mfs.uimech.org/mfs2019.3.025,en
DOI: 10.21662/mfs2019.3.025
Survey of studies on degenerate boundary conditions and finite spectrum
Akhtyamov A.M.
Mavlutov Institute of Mechanics, UFRC RAS, Ufa, Russia
Bashkir State University, Ufa, Russia

Abstract

It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric diffusion operator, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false–periodic boundary conditions and is identically equal to a constant other than zero if and only if its boundary conditions are generalized Cauchy conditions. All degenerate boundary conditions for a spectral problem with a third–order differential equation y'''(x) = λy(x) are described. The general form of degenerate boundary conditions for the fourth–order differentiation operator D4 is found. 12 classes of boundary value eigenvalue problems are described for the operator D4, the spectrum of which fills the entire complex plane. It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem (eleventh problem): are there similar problems for odd–order differential equations? A positive answer is given to this question. It is proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved. John Locker posed a problem (tenth problem): can a spectral boundary–value problem have a finite spectrum? Boundary value problems with a polynomial occurrence of a spectral parameter in a differential equation are considered. It is shown that the corresponding boundary–value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. Thus, John Locker’s tenth problem is resolved.

Keywords

degenerate boundary conditions,
finite spectrum,
tenth and eleventh John Locker problems

Article outline

The work is devoted to the description of degenerate two-point boundary conditions of a homogeneous spectral problem for the diffusion operator. All degenerate boundary conditions of the spectral problem are described for the diffusion operator, for the third-order differentiation operator, for the fourth-order differentiation operator, and for differential equations of any odd order. One of the problems of John Locker (the tenth problem) related to the question: can a spectral boundary-value problem have a finite spectrum be also solved?

It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric operator of diffusion, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false-periodic boundary conditions, and is identically equal to a constant other than zero, if and only if its boundary conditions are generalized Cauchy conditions.

All degenerate boundary conditions for a spectral problem with a third-order differential equation y'''(x) = λy(x) are described. It is proved that the boundary conditions for this spectral problem are degenerate if and only if the coefficient matrix of the boundary conditions of size 2 by 6 consists of two square diagonal matrices of order three, on one diagonal of which there are units, and on the other, the roots of some numbers. consists of two diagonal matrices, on one diagonal of which there are units, and on the other - roots of minus one. It is shown that the third-order differential equation y'''(x) = λy(x) with general boundary conditions (not containing a spectral parameter) cannot have a finite spectrum.

The general form of degenerate boundary conditions for the fourth-order differentiation operator D4 is found. Shown, that the coefficient matrix of degenerate boundary conditions of size 4×8 consists of two fourth-order square diagonal matrices. One of the diagonal matrices is unit, and the diagonal of the second diagonal matrix consists of some numbers. We study the operators D4, whose spectrum fills the entire complex plane. A well-known example is a boundary value eigenvalue problem for an even-order differential operator whose spectrum fills the entire complex plane. These boundary conditions have the form Uj(y) = y(j−1)(0) + (−1)j−1 y(j−1)(1) = 0, j = 1, 2, 3, 4. However, in connection with this, another question arises whether there are other examples of such operators. This section shows that such examples exist. In addition, all 12 classes of boundary value eigenvalue problems are described for the operator D4, whose spectrum fills the entire complex plane. Each of the classes contains an arbitrary constant. Therefore, there is a continuum of degenerate boundary conditions for the fourth-order differentiation operator D4.

It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem: are there similar problems for odd-order differential equations? This section gives a positive answer to this question. It is2 proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved.

John Locker posed a problem (tenth problem): can a spectral boundary-value problem have a finite spectrum? Boundary value problems are considered. This paragraph shows that the corresponding boundary-value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. So John Locker’s tenth problem is solved.

References

  1. Marchenko V.A. [Sturm–Liouville operators and their applications] Operatory Shturma–Liuvillya i ix prilozheniya. Kiev: Naukova dumka, 1977. P. 329 (In Russian).
  2. Stone M.H. Irregular differential systems of order two and the related expansion problems. Trans. Amer. Math. Soc.. 1927. V. 29. Pp. 23–53.
    DOI: 10.2307/1989277
  3. Naimark M.A. [Linear Differential Operators] Linejnye differencial‘nye operatory. M.: Nauka, 1969. P. 526 (In Russian).
  4. Shiryaev E.A., Shkalikov A.A. Regular and completely regular differential operators. Mathematical Notes. 2007. V. 81, No. 4. Pp. 566–570.
    DOI: 10.1134/S0001434607030352
  5. Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M. General Inverse Sturm–Liouville Problem with Symmetric Potential. Azerbaijan Journal of Mathematics. 2015. V. 5, No. 2. Pp. 96–108.
    https://www.azjm.org/volumes/0502/0502-8.pdf
  6. Danford N., Shvarts Dzh.T. [Linear operators. Spectral Operators] Linejnye operatory. Spektral‘nye operatory. M.: Mir, 1974. P. 664 (In Russian).
  7. Dezin A.A. Spectral characteristics of general boundary–value problems for operator D2. Mathematical notes of the Academy of Sciences of the USSR. 1985. V. 37, No. 2. Pp. 142–146.
    DOI: 10.1007/BF01156759
  8. Biyarov B.N., Dzhumabaev S.A. A criterion for the Volterra property of boundary value problems for Sturm–Liouville equations. Mathematical Notes. 1994. V. 56, No. 1. Pp. 751–753.
    DOI: 10.1007/BF02110567
  9. Lang P., Locker J. Spectral theory of two-point differential operators determined by D2. I. Spectral properties. Journal of Mathematical Analysis and Applications. 1989. V. 141, No. 2. Pp. 538–558.
    DOI: 10.1016/0022-247X(89)90196-0
  10. Akhtyamov A.M. On degenerate boundary conditions in the Sturm–Liouville problem. Differential Equations. 2016. V. 52, No. 8. C. 1085–1087.
    DOI: 10.1134/S0012266116080140
  11. Akhtyamov A.M. Degenerate boundary conditions for the diffusion operator. Differential Equations. 2017. V. 53, No. 11. Pp. 1515–1518.
    DOI: 10.1134/S0012266117110143
  12. Sadovnichii V.A.; Kanguzhin B.E. A connection between the spectrum of a differential operator with symmetric coefficients and the boundary conditions. Sov. Math.,Dokl.. 1982. V. 26. Pp. 614–618.
    https://zbmath.org/?q=an:0521.34031
  13. Locker J. Eigenvalues and completeness for regular and simply irregular two–point differential operators. American Mathematical Society, 2006. P. 315.
  14. Locker J. Eigenvalues and completeness for regular and simply irregular two–point differential operators. American Mathematical Society, 2008. P. 177.
  15. Makin A.S. Two–point boundary–value problems with nonclassical asymptotics on the spectrum. Electronic Journal of Differential Equations. 2018. No. 95. P. 1–7.
    http://ejde.math.unt.edu/Volumes/2018/95/makin.pdf
  16. Akhtyamov A.M. On Degenerate Boundary Conditions for Operator D4. Springer Proceedings in Mathematics and Statistics. 2017. V. 216. Pp. 195–203.
    DOI: 10.1007/978-3-319-67053-9_18
  17. Akhtyamov A.M. On the spectrum of an odd–order differential operator. Mathematical Notes. 2017. V. 101, No. 5. Pp. 755–758.
    DOI: 10.1134/S0001434617050017
  18. Akhtyamov A.M. Degenerate Boundary Conditions for a ThirdOrder Differential Equation. Differential Equations. 2018. V. 54, No. 4. Pp. 419–426.
    DOI: 10.1134/S0012266118040018
  19. Dzhumabaev S.A., Kanguzhin B.E. [On an irregular problem on a finite interval]. Izvestiya AN KazSSR. Ser. fiz.–mat. [Bulletin of the Academy of Sciences of the KazSSR. Ser. Fiz.–Mat.]. No. 1. Pp. 14–18 (In Russian).
  20. Malamud M.M. On the completeness of the system of root vectors of the Sturm–Liouville operator with general boundary conditions. Functional Analysis and Its Applications. 2008. V. 42, No. 3. Pp. 198–204.
    DOI: 10.1007/s10688-008-0028-0
  21. Makin A.S. On an inverse problem for the Sturm–Liouville operator with degenerate boundary conditions. Differential Equations. 2014. V. 50, No. 10. Pp. 1402–1406.
    DOI: 10.1134/S0012266114100176
  22. Yurko V.A. The inverse problem for differential operators of second order with regular boundary conditions. Mathematical notes of the Academy of Sciences of the USSR. 1975. V. 18, No. 4. Pp. 928–932.
    DOI: 10.1007/BF01153046
  23. Akhtyamov A., Amram M., Mouftakhov A. On reconstruction of a matrix by its minors. International Journal of Mathematical Education in Science and Technology. 2018. V. 49, No. 2. Pp. 268–321.
    DOI: 10.1080/0020739X.2017.1383526
  24. Guseinov I.M. [Inverse spectral problems for a quadratic pencil of Sturm–Liouville operators on a finite interval]. Spektral‘naya teoriya operatorov i ee prilozheniya. Baku [Spectral theory of operators and its applications. Baku]. 1986. No. 7. Pp. 51–101 (In Russian).
  25. Guseinov I.M., Nabiev I.M. The inverse spectral problem for pencils of differential operators. Sb. Math. 2007. V. 198, No. 11. Pp. 1579–1598.
    DOI: 10.1070/SM2007v198n11ABEH003897
  26. Nabiev I.M., Shukurov A.Sh. Solution of inverse problem for the diffusion operator in a symmetric case. Izv. Saratov Univ. (N.S.) Ser. Math. Mech. Inform. 2009. V. 9, No. 4. Pp. 36–40.
    http://mi.mathnet.ru/eng/isu73
  27. Kamke E. [Handbook of Ordinary Differential Equations] Spravochnik po obyknovennym differencial‘nym uravneniyam. M.: Nauka, 1976. P. 576 (in Russian).
    http://elibrary.bsu.az/kitablar/1019.pdf
  28. Lankaster P. Theory of matrices. New York–London: Academic Press, 1969. P. 316.
  29. Lidskii V.B., Sadovnichii V.A. Regularized sums of zeros of a class of entire functions. Functional Analysis and Its Applications. 1967. V. 1, No. 2. Pp. 133–139.
    DOI: 10.1007/BF01076085
  30. Lidskii V.B., Sadovnichii V.A. Asymptotic formulas for the zeros of a class of entire functions. Mathematics of the USSR–Sbornik. 1968. V. 4, No. 4. Pp. 519–530.
    DOI: 10.1070/SM1968v004n04ABEH002812
  31. Kodington E.A., Levinson N. [Theory of ordinary differential equations] Teoriya obyknovennyx differencial‘nyx uravnenij. M.: Izd.–vo inostr. literatury, 1958. P. 474 (In Russian).
  32. Kalmenov T.Sh., Suragan D. [Determining the structure of regular boundary value problems for differential equations by the method of non–a priori estimates V.A. Ilyina]. Doklady Akademii Nauk [Doklady Mathematics]. 2008. No. 6. Pp. 730–732 (In Russian).
    https://elibrary.ru/item.asp?id=11634292
  33. Akhtyamov A.M. Finiteness of the Spectrum of Boundary Value Problems. Differential Equations. 2019. V. 55, No. 1. Pp. 142–142.
    DOI: 10.1134/S0012266119010154