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.
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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).
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Communicated by Adrian Constantin.
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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
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DOI: https://doi.org/10.1007/s00605-022-01797-9