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Solutions with prescribed numbers of focal points of nonoscillatory linear Hamiltonian systems

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Abstract

In this paper we present an existence result for conjoined bases of nonoscillatory linear Hamiltonian systems on an unbounded interval, which have prescribed numbers of left and right proper focal points. The result is based on a singular Sturmian separation theorem on an unbounded interval by the authors (2019) and it extends a similar property, which was recently derived for linear Hamiltonian systems on compact interval (2021). At the same time it is new even for completely controllable linear Hamiltonian systems, including higher order Sturm–Liouville differential equations. As the main tools we use the comparative index and properties of the minimal principal solution at infinity, which serves as the reference solution for calculating the numbers of proper focal points. We also provide several examples illustrating the presented theory.

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Acknowledgements

Both authors contributed to the results obtained in this paper equally. Both authors read and approved the final manuscript. This research was supported by the Czech Science Foundation under grant GA19–01246S. The authors wish to thank an anonymous referee for valuable comments, which helped improve the overall presentation of the results.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ-61137, Brno, Czech Republic

    Peter Šepitka & Roman Šimon Hilscher

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  1. Peter Šepitka
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  2. Roman Šimon Hilscher
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Correspondence to Roman Šimon Hilscher.

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Communicated by Gerald Teschl.

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Šepitka, P., Šimon Hilscher, R. Solutions with prescribed numbers of focal points of nonoscillatory linear Hamiltonian systems. Monatsh Math 200, 359–387 (2023). https://doi.org/10.1007/s00605-022-01780-4

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  • DOI: https://doi.org/10.1007/s00605-022-01780-4

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