On the Eigenvalues of Finitely Perturbed Laplace Operators in Infinite Cylindrical Domains

Grushin, V. V.

doi:10.1023/b:matn.0000023312.41107.72

Cited by 26 publications

(15 citation statements)

References 8 publications

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“…The obtained results about the asymptotic expansion of the eigenvalue λ ε > π 2 are significantly different from the usual results, i.e., from (9), even in their statement. We introduce the necessary notation and requirements on the profile function H, which must be written as…”

Section: Statement Of the Problemcontrasting

confidence: 94%

Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide

Назаров

2011

Theor Math Phys

111

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We establish that by choosing a smooth local perturbation of the boundary of a planar quantum waveguide, we can create an eigenvalue near any given threshold of the continuous spectrum and the corresponding trapped wave exponentially decaying at infinity. Based on an analysis of an auxiliary object, a unitary augmented scattering matrix, we asymptotically interpret Wood's anomalies, the phenomenon of fast variations in the diffraction pattern due to variations in the near-threshold wave frequency.Keywords: quantum waveguide, regular perturbation of the boundary, asymptotic expansion of eigenvalue on the continuous spectrum, Wood's anomalies

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Section: Statement Of the Problemcontrasting

confidence: 94%

Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide

Назаров

2011

Theor Math Phys

111

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“…Spectral stability of the Laplace operator in bent three-dimensional tubes as a consequence of the Hardy inequality is discussed in [EKK08] and later in more detail in [Kr08]. A different approach to the repulsive effect of twisting, without appeal to a Hardytype inequality, was used in [Gr04,Gr05], see also [BMT07]. Other consequences of twisting on the behavior of the discrete spectrum, apart from the absence of bound states in mildly bent tubes, will be discussed in Chap.…”

Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

146

133

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More information about this series at http://www.springer.com/series/720The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals that the series is a suitable publication platform for both the mathematical and the theoretical physicist. The wider scope of the series is reflected by the composition of the editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research.

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“…The authors derived Lieb-Thirring-type inequalities for eigenvalue moments of order σ > 1/2 . Other spectral aspects of twisted tubes were treated in [14,15] (existence/non-existence of bound states), [2,8] (Hardy type inequalities), [4] (asymptotic behavior of the spectrum as the thickness of the tube cross section goes to zero), [3] (eigenvalue asymptotics in the case when the rotation velocity decays slowly at infinity).…”

Section: Introductionmentioning

confidence: 99%

Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity

Barseghyan

Khrabustovskyi²

2019

Operators and Matrices

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We study the spectrum of the Dirichlet Laplacian on an unbounded twisted tube with twisting velocity exploding to infinity. If the tube cross section does not intersect the axis of rotation, then its spectrum is purely discrete under some additional conditions on the twisting velocity (D. Krejčiřík, 2015). In the current work we prove a Berezin type upper bound for the eigenvalue moments.Mathematics subject classification (2010): 35P20, 35P15, 81Q10, 81Q37.

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On the Eigenvalues of Finitely Perturbed Laplace Operators in Infinite Cylindrical Domains

Cited by 26 publications

References 8 publications

Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide

Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide

Quantum Waveguides

Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity

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