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Matrix representations of Sturm-Liouville problems with transmission conditions. (English) Zbl 1247.34039

Summary: We identify a class of Sturm-Liouville equations with transmission conditions such that any Sturm-Liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self-adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem. Conversely, given any matrix eigenvalue problem of certain type and an arbitrary separated or real coupled self-adjoint boundary condition and transmission condition, we construct a class of Sturm-Liouville problems with this specified boundary condition and transmission condition, each of which is equivalent to the given matrix eigenvalue problem.

MSC:

34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
15A18 Eigenvalues, singular values, and eigenvectors
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[1] Kong, Q.; Volkmer, H.; Zettl, A., Matrix representations of Sturm-Liouville problems with finite spectrum, Results. Math., 54, 103-116 (2009) · Zbl 1185.34032
[2] Kong, Q.; Zettl, A., The study of Jacobi and cyclic Jacobi matrix eigenvalue problems using Sturm-Liouville theory, Linear Algebra Appl., 434, 1648-1655 (2011) · Zbl 1210.15011
[3] Kong, Q.; Zettl, A., Inverse Sturm-Liouville problems with finite spectrum, J. Math. Anal. Appl., 386, 1-9 (2012) · Zbl 1232.34023
[4] Ao, J. J.; Sun, J.; Zhang, M. Z., The finite spectrum of Sturm-Liouville problems with transmission conditions, Appl. Math. Comput., 218, 1166-1173 (2011) · Zbl 1242.34042
[5] Atkinson, F. V., Discrete and Continuous Boundary Value Problems (1964), Academic Press: Academic Press New York, London · Zbl 0117.05806
[6] Everitt, W. N.; Race, D., On necessary and sufficient conditions for the existence of caratheodory solutions of ordinary differential equations, Quaest. Math., 3, 507-512 (1976) · Zbl 0392.34002
[7] Kong, Q.; Wu, H.; Zettl, A., Sturm-Liouville problems with finite spectrum, J. Math. Anal. Appl., 263, 748-762 (2001) · Zbl 1001.34019
[8] Kong, Q.; Wu, H.; Zettl, A., Dependence of the \(n\) th Sturm-Liouville eigenvalue on the problem, J. Differential Equations, 156, 328-354 (1999) · Zbl 0932.34081
[9] Chanane, B., Sturm-Liouville problems with impulse effects, Appl. Math. Comput., 190, 610-626 (2007) · Zbl 1122.65378
[10] Sun, J.; Wang, A., Sturm-Liouville operators with interface conditions, (The Progress of Research for Math., Mech., Phys. and High New Tech., vol. 12 (2008), Science Press: Science Press Beijing), 513-516
[11] Zettl, A., Sturm-Liouville Theory, Amer. Math. Soc., Math. Surveys Monogr., 121 (2005) · Zbl 1074.34030
[12] Kadakal, M.; Mukhtarov, O. Sh., Sturm-Liouville problems with discontinuities at two points, Comput. Math. Appl., 54, 1367-1379 (2007) · Zbl 1140.34012
[13] Volkmer, H., Eigenvalue problems of Atkinson, Feller and Krein and their mutual relationship, Electron. J. Differential Equations, 48, 15 (2005), (electronic) · Zbl 1075.34027
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