Comparative index and Sturmian theory for linear Hamiltonian systems

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• 作者：Peter &Scaron ; epitka ^{sepitkap@math.muni.cz" class="auth_mail" title="E-mail the corresponding author} ; Roman &Scaron ; imon Hilscher ; ^{hilscher@math.muni.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：34C10
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：262
• 期：2
• 页码：914-944
• 全文大小：539 K

文摘

The comparative index was introduced by J. Elyseeva (2007) as an efficient tool in matrix analysis, which has fundamental applications in the discrete oscillation theory. In this paper we implement the comparative index into the theory of continuous time linear Hamiltonian systems, study its properties, and apply it to obtain new Sturmian separation theorems as well as new and optimal estimates for left and right proper focal points of conjoined bases of these systems on bounded intervals. We derive our results for general possibly abnormal (or uncontrollable) linear Hamiltonian systems. The results turn out to be new even in the case of completely controllable systems. We also provide several examples, which illustrate our new theory.