Abstract
For left-definite Sturm-Liouville problems with h-periodic coefficients and each integer \(k>2\), it is well known that the eigenvalues of some self-adjoint complex boundary conditions on the interval \([a,a+h]\) are the same as the periodic eigenvalues on the interval \([a,a+kh]\). For each k, we identify explicitly which of the uncountable number of complex conditions generates these periodic eigenvalues. Based on this condition, we give the inequality relation of periodic eigenvalues on the interval \([a,a+kh]\). Moreover, an analogous result for semi-periodic eigenvalues is also obtained.
Similar content being viewed by others
Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
References
Blumenson, L.E.: On the eigenvalues of Hill’s equation. Comm. Pure Appl. Math. 16, 261–266 (1963)
Eastham, M.S.P.: The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, Edinburgh/London (1973)
Eastham, M.S.P.: The first stability interval of the periodic Schroedinger equation. J. London Math. Soc. 4(2), 587–592 (1972)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Zettl, A.: Sturm-Liouville Theory, Mathematical Surveys and Monographs, vol. 121, Amer. Math. Soc., (2005)
Freiling, G., Yurko, V.: Inverse Sturm-Liouville problems and their applications. Nova Science Publishers Inc, Huntington, NY (2001)
Bennewitz, C., Brown, M., Weikard, R.: Spectral and scattering theory for ordinary differential equations. Vol. I: Sturm-Liouville equations. Universitext. Springer, Cham, (2020)
Constantin, A., McKean, H.P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999)
Kong, Q., Wu, H., Zettl, A.: Left-definite Sturm-Liouville problems. J. Diff. Eq. 177, 1–26 (2001)
Binding, P., Browne, P.: Asymptotics of eigencurves for second order ordinary differential equations, I. J. Diff. Eq. 88, 30–45 (1990)
Binding, P., Volkmer, H.: Eigenvalues for two-parameter Sturm-Liouville equations. SIAM Rev. 38, 27–48 (1996)
Constantin, A.: A general-weighted Sturm-Liouville problem. Ann. Scuola Norm. Sup. Pisa 24, 767–782 (1997)
Ince, E.: Ordinary Differential Equations. Dover, New York (1956)
Mingarelli, A.: A survey of the regular weighted Sturm-Liouville problem -the nondefinite case, In: Xiao S., Pu F., (eds.) International Workshop on Applied Differential Equations, 109-137, World Scientific, Singaoore, (1986)
Yuan, Y., Sun, J., Zettl, A.: Eigenvalues of periodic Sturm-Liouville problems. Linear Algebra Appl. 517, 148–166 (2017)
Kong, Q., Wu, H., Zettl, A.: Dependence of the nth Sturm-Liouville eigenvalue on the problem. J. Diff. Eq. 156, 328–354 (1999)
Evertt, W.N., Race, D.: On necessary and sufficient conditions for the existence of Caratheodory solutions of ordinary differential equation. Quaest. Math. 3, 507–512 (1976)
Yuan, Y., Sun, J., Zettl, A.: Inequalities among eigenvalues of Sturm-Liouville equations with periodic coefficients. Electr. J. Differ. Equ. 264, 1–13 (2017)
Acknowledgements
The authors are grateful to the referees for a careful reading and very helpful suggestions which improved and strengthened the presentation of this manuscript. This project was supported by the Natural Science Foundation of Shandong Province (No. ZR2020QA009, ZR2021MA047), the National Nature Science Foundation of China (Nos. 11561050, 11801286, 12101356), Natural Science Foundation of Inner Mongolia (No. 2018MS01021), the China Postdoctoral Science Foundation (Grant 2019M662313)) and the Youth Creative Team Sci-Tech Program of Shandong Universities (No.2019KJI007).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Adrian Constantin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hao, X., Sun, L., Li, K. et al. Eigenvalues of left-definite Sturm-Liouville problems with periodic coefficients. Monatsh Math 200, 819–834 (2023). https://doi.org/10.1007/s00605-022-01797-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-022-01797-9