Singular Sturmian separation theorems for nonoscillatory symplectic difference systems

Šepitka, Peter; Hilscher, Roman Šimon

doi:10.1080/10236198.2018.1544247

Cited by 8 publications

(2 citation statements)

References 32 publications

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“…from the oscillation theory of linear Hamiltonian systems (H). It can be further developed in the direction of the oscillation theory on discrete time domains [10,37] or in the direction of singular comparison theorems for the oscillation numbers, as we recently presented in [38,39] for conjoined bases of nonoscillatory linear Hamiltonian systems. Such results may have fundamental applications in the theory of Maslov index on unbounded intervals, or in the study of the rotation number of a family of linear Hamiltonian systems, such as in [28,29].…”

Section: Discussionmentioning

confidence: 99%

Oscillation numbers for continuous Lagrangian paths and Maslov index

Elyseeva¹,

Šepitka²,

Hilscher³

2021

Preprint

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In this paper we present the theory of oscillation numbers and dual oscillation numbers for continuous Lagrangian paths in R 2n . Our main results include a connection of the oscillation numbers of the given Lagrangian path with the Lidskii angles of a special symplectic orthogonal matrix. We also present Sturmian type comparison and separation theorems for the difference of the oscillation numbers of two continuous Lagrangian paths. These results, as well as the definition of the oscillation number itself, are based on the comparative index theory (Elyseeva, 2009). The applications of these results are directed to the theory of Maslov index of two continuous Lagrangian paths. We derive a formula for the Maslov index via the Lidskii angles of a special symplectic orthogonal matrix, and hence we express the Maslov index as the oscillation number of a certain transformed Lagrangian path. The results and methods are based on a generalization of the recently introduced oscillation numbers and dual oscillation numbers for conjoined bases of linear Hamiltonian systems (Elyseeva, 2019 and and on the connection between the comparative index and Lidskii angles of symplectic matrices ( Šepitka and Šimon Hilscher, 2020).

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Section: Discussionmentioning

confidence: 99%

Oscillation numbers for continuous Lagrangian paths and Maslov index

Elyseeva¹,

Šepitka²,

Hilscher³

2021

Preprint

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“…On the other hand, we will utilize concepts related to higher order equations that link Equation (1.1) to the Hamiltonian system. Such ideas were developed and investigated among others in [1,[3][4][5] (see also recent contributions to the topic concerning Symplectic systems in [47][48][49]56]).…”

Section: Introductionmentioning

confidence: 99%

Criticality of general two‐term even‐order linear difference equation via a chain of recessive solutions

Jekl

2024

Mathematische Nachrichten

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In this paper, the author investigates particular disconjugate even‐order linear difference equations with two terms and classify them based on the properties of their recessive solutions at plus and minus infinity. The main theorem described states that the studied equation is ‐critical whenever a specific second‐order linear difference equation is ‐critical. In the proof, the author derived closed‐form solutions for the studied equation wherein the solutions of the said second‐order equation appear. Furthermore, the solutions were organized, in order to determine recessive solutions, into a linear chain by sequence ordering that compares the solutions at .

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