Singular Sturmian separation theorems on unbounded intervals for linear Hamiltonian systems |
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Authors: | Peter Šepitka Roman Šimon Hilscher |
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Institution: | Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlá?ská 2, CZ-61137 Brno, Czech Republic |
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Abstract: | In this paper we develop new fundamental results in the Sturmian theory for nonoscillatory linear Hamiltonian systems on an unbounded interval. We introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and, based on this notion, we prove singular Sturmian separation theorems on an unbounded interval. The main results are formulated in terms of the (minimal) principal solutions at both endpoints of the considered interval, and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases. As a natural tool we use the comparative index, which was recently implemented into the theory of linear Hamiltonian systems by the authors and independently by J. Elyseeva. Throughout the paper we do not assume any controllability condition on the system. Our results turn out to be new even in the completely controllable case. |
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Keywords: | 34C10 Sturmian separation theorem Linear Hamiltonian system Proper focal point Minimal principal solution Antiprincipal solution Comparative index |
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