Trace Formulas Applied to the Riemann ζ-Function

Ashbaugh, Mark S.; Gesztesy, Fritz; Hermi, Lotfi; Kirsten, Klaus; Littlejohn, Lance L.; Tossounian, Hagop

doi:10.1007/978-3-030-20087-9_8

Cited by 2 publications

(3 citation statements)

References 24 publications

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“…This is an expansion on old ideas due to Kato [31]. The spectral questions explored here connect in a fundamental way with the works [9,8,22,21].…”

Section: Introductionmentioning

confidence: 74%

“…While apparently similar in formulation to the nonlocal integral operator commuting with the Laplacian corresponding to the free space Green's function, the Kreȋn-von Neumann problem was shown to be the "farthest", while the anti-periodic problem proved to be the "closest" when seen as finite-rank perturbations. In [30], we will develop the spectral theory of the associate iterated Brownian bridge kernels corresponding to this GSARC framework; see also [13,18,8,9].…”

Section: Discussionmentioning

confidence: 99%

“…This problem received special attention in the literature, both in one dimension, and in higher dimensions, [3,8,9,7,5,6], partially in connection with work on self-adjoint extensions of the operator…”

Section: Kreȋn-von Neumann Boundary Conditionsmentioning

confidence: 99%

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On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

Hermi

Saito

2022

Preprint

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We reformulate all general real coupled self-adjoint boundary value problems as integral operators and show that they are all finite rank perturbations of the free space Green's function on the real line. This free space Green's function corresponds to the nonlocal boundary value problem proposed earlier by Saito [41]. These are polynomial perturbations of rank up to 4. They encapsulate in a fundamental way the corresponding boundary conditions.

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“…This is an expansion on old ideas due to Kato [31]. The spectral questions explored here connect in a fundamental way with the works [9,8,22,21].…”

Section: Introductionmentioning

confidence: 74%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

Hermi

Saito

2022

Preprint

Self Cite

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On a nonlocal integral operator commuting with the Laplacian and the Sturm–Liouville problem: Low rank perturbations of the operator

Hermi,

Saito

2024

Journal of Mathematical Physics

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We reformulate all general real coupled self-adjoint boundary value problems as integral operators and show that they are all finite rank perturbations of the free space Green’s function on the real line. This free space Green’s function corresponds to the nonlocal boundary value problem proposed earlier by Saito [Appl. Comput. Harmon. Anal. 25, 68–97 (2008)]. We prove these perturbations to be polynomials of rank up to 4. They encapsulate in a fundamental way the corresponding boundary conditions.

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Trace Formulas Applied to the Riemann ζ-Function

Abstract: We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.

Cited by 2 publications

References 24 publications

On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

On a nonlocal integral operator commuting with the Laplacian and the Sturm–Liouville problem: Low rank perturbations of the operator

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