On the jth Eigenvalue of Sturm–Liouville Problem and the Maslov Index

In the previous papers (Hu et al. in J Differ Equ 266(7):4106–4136, 2019; Kong et al. in J Differ Equ 156(2):328–354, 1999), the jump phenomena of the j -th eigenvalue were completely characterized for Sturm–Liouville problems. In this paper, we show that the jump number of these eigenvalue branches...

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Bibliographic Details
Published inJournal of dynamics and differential equations Vol. 34; no. 3; pp. 1949 - 1967
Main Authors Hu, Xijun, Liu, Lei, Wu, Li, Zhu, Hao
Format Journal Article
LanguageEnglish
Published New York Springer US 01.09.2022
Springer Nature B.V
Subjects
Applications of Mathematics
Boundary conditions
Dirichlet problem
Eigenvalues
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Partial Differential Equations
34B24
53D12
Eigenvalue
34L15
Boundary condition
Sturm–Liouville problem
Maslov index
34B09
Lagrangian subspace
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Summary:In the previous papers (Hu et al. in J Differ Equ 266(7):4106–4136, 2019; Kong et al. in J Differ Equ 156(2):328–354, 1999), the jump phenomena of the j -th eigenvalue were completely characterized for Sturm–Liouville problems. In this paper, we show that the jump number of these eigenvalue branches is exactly the Maslov index for the path of corresponding boundary conditions. Then we determine the sharp range of the j th eigenvalue on each layer of the space of boundary conditions. Finally, we prove that the graph of monodromy matrix tends to the Dirichlet boundary condition as the spectral parameter goes to - ∞ .
ISSN:1040-7294
1572-9222
DOI:10.1007/s10884-021-10107-0