A note on eigenvalues of a class of singular continuous and discrete linear Hamiltonian systems

Zhu, Hao

doi:10.1016/j.laa.2019.02.004

Cited by 2 publications

(1 citation statement)

References 23 publications

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“…Discrete Hamiltonian systems are of growing interest in recent years because of their wide applications (see [3][4][5][6][7][8][9][10][11][12] and references therein). Although discrete Hamiltonian systems originate from the discretization of continuous Hamiltonian systems, there is an important difference between them.…”

Section: Introductionmentioning

confidence: 99%

Stability of deficiency indices for discrete Hamiltonian systems under bounded perturbations

Liu

2020

Adv Differ Equ

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This paper is concerned with stability of deficiency indices for discrete Hamiltonian systems under perturbations. By applying the perturbation theory of Hermitian linear relations we establish the invariance of deficiency indices for discrete Hamiltonian systems under bounded perturbations. As a consequence, we obtain the invariance of limit types for the systems under bounded perturbations. In particular, we build several criteria of the invariance of the limit circle and limit point cases for the systems. Some of these results improve and extend some previous results.

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Section: Introductionmentioning

confidence: 99%

Stability of deficiency indices for discrete Hamiltonian systems under bounded perturbations

Liu

2020

Adv Differ Equ

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Jump phenomena of the n-th eigenvalue of discrete Sturm–Liouville problems with application to the continuous case

Ren

Zhu

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we characterize jump phenomena of the $n$ -th eigenvalue of self-adjoint discrete Sturm–Liouville problems in any dimension. For a fixed Sturm–Liouville equation, we completely characterize jump phenomena of the $n$ -th eigenvalue. For a fixed boundary condition, unlike in the continuous case, the $n$ -th eigenvalue exhibits jump phenomena and we describe the singularity under a non-degenerate assumption. Compared with the continuous case in Hu et al. (2019, J. Differ. Equ.266, 4106–4136) and Kong et al. (1999, J. Differ. Equ.156, 328–354), the jump set here is involved with coefficients of the Sturm–Liouville equations. This, along with arbitrariness of the dimension, causes difficulty when dividing the jump areas. We study the singularity by partitioning and analysing the local coordinate systems, and provide a Hermitian matrix which can determine the areas’ division. To prove the asymptotic behaviour of the $n$ -th eigenvalue, we generalize the method developed in Zhu and Shi (2016, J. Differ. Equ.260, 5987–6016) to any dimension. As an application, by transforming the continuous Sturm–Liouville problem of Atkinson type to a discrete one, we determine the number of eigenvalues and obtain complete characterization of jump phenomena of the $n$ -th eigenvalue for the Atkinson type.

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A note on eigenvalues of a class of singular continuous and discrete linear Hamiltonian systems

Cited by 2 publications

References 23 publications

Stability of deficiency indices for discrete Hamiltonian systems under bounded perturbations

Stability of deficiency indices for discrete Hamiltonian systems under bounded perturbations

Jump phenomena of the n-th eigenvalue of discrete Sturm–Liouville problems with application to the continuous case

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