Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-30T18:01:59.727Z Has data issue: false hasContentIssue false
  • English
  • Français

The lower bounds of non-real eigenvalues for singular indefinite Sturm–Liouville problems

Part of: Ordinary differential operators Special classes of linear operators Boundary value problems

Published online by Cambridge University Press:  22 December 2023

Fu Sun
Fu Sun*
Affiliation:
School of Statistics and Data Science, Qufu Normal University, Qufu 273165, People's Republic of China (sfmath@163.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

The present paper deals with the non-real eigenvalues for singular indefinite Sturm–Liouville problems. The lower bounds on non-real eigenvalues for this singular problem associated with a special separated boundary condition are obtained.

Keywords

Sturm–Liouville problemnon-real eigenvalueindefinitelower bounds

MSC classification

Primary: 34B24: Sturm-Liouville theory
Secondary: 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 47B50: Operators on spaces with an indefinite metric

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 155 , Issue 3 , June 2025 , pp. 1051 - 1060
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Atkinson, F. and Jabon, D.. Indefinite Sturm-Liouville problems. In ‘Proceedings of the Focused Research Program on Spectral Theory and Boundary Value Problems Vol.I’, (ed. H. Kaper, M. K. Kwong, and A. Zettle) (Argonne National Laboratory, Argonne, IL, 1984), pp. 31–45.Google Scholar
Behrndt, J., Chen, S., Philipp, F. and Qi, J.. Estimates on the non-real eigenvalues of regular indefinite Sturm-Liouvilleproblems. Proc. Roy. Soc. Edinburgh. A. 144 (2014), 1113–1126.CrossRefGoogle Scholar
Behrndt, J., Katatbeh, Q. and Trunk, C.. Non-real eigenvalues of singular indefinite Sturm-Liouville operators. Proc. Amer. Math. Soc. 137 (2009), 3797–3806.CrossRefGoogle Scholar
Behrndt, J., Philipp, F. and Trunk, C.. Bounds on the non-real spectrum of differential operators with indefinite weights. Math. Ann. 357 (2013), 185–213.CrossRefGoogle Scholar
Behrndt, J., Schmitz, P. and Trunk, C.. Bounds on the non-real spectrum of a singular indefinite Sturm-Liouville operator on $\mathbb {R}$. Proc. Appl. Math. Mech. 16 (2016), 881–882.CrossRefGoogle Scholar
Behrndt, J., Schmitz, P. and Trunk, C.. Spectral bounds for singular indefinite Sturm-Liouville operators with $L^1\text {-potentials},$ arXiv:1709.04994v2 [math.SP] 18 Dec 2017.CrossRefGoogle Scholar
Behrndt, J. and Trunk, C.. On the negative squares of indefinite Sturm-Liouville operators. J. Differ. Equ. 238 (2007), 491–519.CrossRefGoogle Scholar
Binding, P. and Volkmer, H.. Eigencurves for two-parameter Sturm-Liouville equations. SIAM Rev. 38 (1996), 27–48.CrossRefGoogle Scholar
Čurgus, B. and Langer, H.. A Krein space approach to symmetric ordinary differential operators with an indefinite weight functions. J. Differ. Equ. 79 (1989), 31–61.CrossRefGoogle Scholar
Fleckinger, J. and Mingarelli, A. B., On the eigenfunctions of non-definite elliptic operators. In Differential Equations Vol. 92, (ed. I. W. Knowles and R. T. Lewis) (North-Holland, The Netherlands, 1984), pp. 219–227.Google Scholar
Haupt, O.. Über eine Methode zum Beweis von Oszillationstheoremen. Math. Ann. 76 (1915), 67–104.CrossRefGoogle Scholar
Kong, L., Kong, Q., Wu, H. and Zettl, A.. Regular approximations of singular Sturm-Liouville problems with limit-circle endpoints. Result. Math. 45 (2004), 274–292.CrossRefGoogle Scholar
Kong, Q., Möller, M., Wu, H. and Zettl, A.. Indefinite Sturm-Liouville problems. Proc. Roy. Soc. Edinburgh. Sect. A. 133 (2003), 639–652.CrossRefGoogle Scholar
Mingarelli, A. B., Indefinite Sturm-Liouville problems, Lecture Notes in Math., Vol. 964, (Springer, Berlin, New York, 1982, pp. 519–528).CrossRefGoogle Scholar
Mingarelli, A. B., A survey of the regular weighted Sturm-Liouville problem – the non-definite case, e-print arXiv:1106.6013v1 [math.CA], 2011.Google Scholar
Niessen, H. D. and Zettl, A.. Singular Sturm-Liouville problem: the Friedrichs extension and comparison of eigenvalues. Proc. London Math. Soc. 64 (1992), 545–578.CrossRefGoogle Scholar
Qi, J. and Chen, S.. A priori bounds and existence of non-real eigenvalues of indefinite Sturm-Liouville problems. J. Spectr. Theory. 4 (2014), 53–63.CrossRefGoogle Scholar
Qi, J., Xie, B. and Chen, S.. The upper and lower bounds on non-real eigenvalues of indefinite Strum-Liouville problems. Pro. Amer. Math. Soc. 144 (2016), 547–559.CrossRefGoogle Scholar
Richardson, R. G. D.. Theorems of oscillation for two linear differential equations of second order with two parameters. Trans. Amer. Math. Soc. 13 (1912), 22–34.CrossRefGoogle Scholar
Sun, F. and Qi, J.. A priori bounds and existence of non-real eigenvalues for singular indefinite Sturm-Liouville problems with limit-circle type endpoints. Proc. Roy. Soc. Edinburgh A. 150 (2020), 2607–2619.CrossRefGoogle Scholar
Xie, B. and Qi, J.. Non-real eigenvalues of indefinite Sturm-Liouville problems. J. Differ. Equ. 8 (2013), 2291–2301.CrossRefGoogle Scholar
Zettl, A., Sturm-Liouville Theory, Math. Surveys Monogr., Vol. 121, (Amer. Math. Soc., Providence, RI, 2005).Google Scholar