Degenerate boundary conditions on a geometric graph

Cover Page


The boundary conditions of the Sturm-Liouville problem defined on a star-shaped geometric graph of three edges are studied. It is shown that if the lengths of the edges are different, then the Sturm-Liouville problem does not have degenerate boundary conditions. If the lengths of the edges and the potentials are the same, then the characteristic determinant of the Sturm-Liouville problem can not be equal to a constant different from zero. But the set of Sturm-Liouville problems for which the characteristic determinant is identically equal to zero is an infinite (continuum). In this way, in contrast to the Sturm-Liouville problem defined on an interval, the set of boundary-value problems on a star-shaped graph whose spectrum completely fills the entire plane is much richer. In the particular case when the minor A124 for matrix of coefficients is nonzero, it does not consist of two problems, as in the case of the Sturm-Liouville problem given on an interval, but of 18 classes, each containing two to four arbitrary constants.

About the authors

V. A. Sadovnichy

Lomonosov Moscow State University

Author for correspondence.

Russian Federation, 1, Leninskie gory, Moscow, 119991

Academician of the RAS

Ya. T. Sultanaev

Ufa Branch of the Russian Academy of Sciences; Bashkir State Pedagogical University


Russian Federation, 71, Prospect Oktyabrya, Ufa, 450054; 3A, October Revolution street, Republic of Bashkortostan, Ufa, 450000

A. M. Akhtyamov

Ufa Branch of the Russian Academy of Sciences; Bashkiria State University, Ufa


Russian Federation, 71, Prospect Oktyabrya, Ufa, 450054; 32, Validy Str., Ufa, 450076


  1. Марченко В. А. Операторы Штурма-Лиувилля и их приложения. Киев: Наук. думка, 1977. 332 c.
  2. Ширяев Е. А., Шкаликов А. А. // Мат. заметки. 2007. Т. 81. № 4. C. 636-640.
  3. Sadovnichii V. A., Sultanaev Ya.T., Akhtyamov A. M. // Azerbaijan J. Math. 2015. V. 5. № 2. Р. 96-108.
  4. Stone M. H. // Trans. Amer. Math. Soc. 1927. V. 29. № 1. P. 23-53.
  5. Садовничий В. А., Кангужин Б. Е. // ДАН. 1982. Т. 267. № 2. С. 310-313.
  6. Locker J. Eigenvalues and Сompleteness for Regular and Simply Irregular Two-Point Differential Operators. Providence: Amer. Math. Soc. 2008. 177 p. // Mem. Amer. Math. 2008. Soc. V. 195. № 911.
  7. Makin A. // Electronic J. Different. Equat. 2018. № 95. P. 1-7.
  8. Ахтямов А. М. // Мат. заметки. 2017. Т. 101. № 5. C. 643-646.
  9. Джумабаев С. А., Кангужин Б. Е. // Изв. АН КазССР. Сер. физ.-мат. 1988. № 1. C. 14-18.
  10. Макин А. С. // Дифференц. уравнения. 2014. Т. 50. № 10. C. 1408-1411.
  11. Маламуд М. М. // Функцион. анализ. 2008. Т. 42. № 3. C. 45-52.
  12. Ахтямов А. М. // Дифференц. уравнения. 2016. Т. 52. № 8. C. 1121-1123.
  13. Наймарк М. А. Линейные дифференциальные опе¬раторы. М.: Наука, 1969. 526 c.
  14. Akhtyamov A., Amram M., Mouftakhov A. // Int. J. Math. Education in Sci. and Technol. 2018. V. 49. № 2. Р. 268-321



Abstract - 261

PDF (Russian) - 187


Copyright (c) 2019 Russian academy of sciences

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies