Finite spectrum of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:math>th order boundary value problems

Ao, Ji-jun; Sun, Jian‐Qiao; Zettl, Anton

doi:10.1016/j.aml.2014.10.003

Cited by 11 publications

(9 citation statements)

References 14 publications

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“…Since P −1 0 (σ σ σ t 2 ) − P −1 0 (σ σ σ t 1 ) is a positive semi-definite matrix for t 1 < t 2 < 0, we infer from Corollary 3.10 and Lemma 3.13 that λ n (σ σ σ t 1 ) ≥ λ n (σ σ σ t 2 ) for each 1 ≤ n ≤ N d − r 0 . Hence, by Lemma 3.16 (1),…”

Section: Singularity Of the N-th Eigenvalue On The Sturm-liouville Eq...mentioning

confidence: 83%

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Discrete Sturm-Liouville problems: singularity of the $n$-th eigenvalue with application to Atkinson type

Ren,

Zhu

2019

Preprint

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In this paper, we characterize singularity of the n-th eigenvalue of self-adjoint discrete Sturm-Liouville problems in any dimension. For a fixed Sturm-Liouville equation, we completely characterize singularity 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 [8,12], the singular set here is involved heavily with coefficients of the Sturm-Liouville equations. This, along with arbitrariness of the dimension, causes difficulty when dividing areas in layers of the considered space such that the n-th eigenvalue has the same singularity in any given area. We study the singularity by partitioning and analyzing the local coordinate systems, and provide a Hermitian matrix which can determine the areas' division. To prove the asymptotic behavior of the n-th eigenvalue, we generalize the method developed in [21] to any dimension. Finally, by transforming the Sturm-Liouville problem of Atkinson type in any dimension to a discrete one, we can not only determine the number of eigenvalues, but also apply our approach above to obtain the complete characterization of singularity of the n-th eigenvalue for the Atkinson type.

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Section: Singularity Of the N-th Eigenvalue On The Sturm-liouville Eq...mentioning

confidence: 83%

“…A d-dimensional Sturm-Liouville problem is said to be of Atkinson type if it consists of (5.1) of Atkinson type and a self-adjoint boundary condition. 1-dimensional case has been studied in [1,4,11,14]. In this section, we always assume that (5.1)-(5.2) is of Atkinson type.…”

Section: Applications To D-dimensionalmentioning

confidence: 99%

Discrete Sturm-Liouville problems: singularity of the $n$-th eigenvalue with application to Atkinson type

Ren,

Zhu

2019

Preprint

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“…, i = 0, 1, … , n. By (7), w i > 0,ŵ i > 0,w i > 0, i = 0, 1, … , n. Since k = n + 1, by Theorem 4.1, there exists a block Jacobi matrix M ∈ M 2n+2 in the form of (14) such that…”

Section: Main Results and Proofmentioning

confidence: 98%

“…[4][5][6][7][8][9][10][11][12] On the other hand, boundary value problems of Atkinson type which have finite spectrum have been considered in recent years. [13][14][15][16][17][18] These problems are connected with some physical problems such as frequencies of vibrating strings and diffusion operators. 19 Kong and Zettl 20 considered the inverse Sturm-Liouville problems with finite spectrum of Atkinson type by using the so-called matrix representations of these problems given in Kong et al 17 and the inverse matrix eigenvalue problems from Xu.…”

Section: Introductionmentioning

confidence: 99%

Inverse spectral problem of fourth‐order boundary value problems with finite spectrum

Zhang

2019

Math Methods in App Sciences

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In the present paper, we consider a class of inverse spectral problem of fourth-order boundary value problems. Under the so-called "Atkinson type" conditions, the problem has finite spectrum and corresponding matrix representations. By using the method of inverse matrix eigenvalue problems of two-banded matrix, the leading coefficient and potential functions are reconstructed from the given three sets of interlacing real numbers satisfying certain conditions. KEYWORDS block Jacobi matrix, boundary eigenvalue problems, finite spectrum, fourth-order problems, inverse problems 4472

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“…Therefore, there exists t 0 > 0 such that B t0 ∈ U (r 0 ,r + ,r − ) ε and λ n (B t0 ) < c 1 , 1 n r + − r + 1 , which yields that λn (B t0 ) = λ n (B t0 ). According to lemma 3.15 (1), λn (U…”

Section: Jump Phenomena Of the N-th Eigenvalue Of Discrete Sturm-liou...mentioning

confidence: 96%

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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Finite spectrum of2nth order boundary value problems

Cited by 11 publications

References 14 publications

Discrete Sturm-Liouville problems: singularity of the $n$-th eigenvalue with application to Atkinson type

Discrete Sturm-Liouville problems: singularity of the $n$-th eigenvalue with application to Atkinson type

Inverse spectral problem of fourth‐order boundary value problems with finite spectrum

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

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