Exponential decay of Laplacian eigenfunctions in domains with branches of variable cross-sectional profiles

Delitsyn, A. L.; Nguyen, Binh-Thanh

doi:10.1140/epjb/e2012-30286-8

Cited by 15 publications

(20 citation statements)

References 23 publications

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“…The remaining analysis would be similar although statements about localization would be more subtle. In particular, the localization of the first eigenfunction does not necessarily occur in the subdomain with larger Lebesgue measure [44].…”

Section: On One Hand An Upper Bound Readsmentioning

confidence: 99%

Mode matching methods for spectral and scattering problems

Delitsyn

2018

The Quarterly Journal of Mechanics and Applied Mathematics

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We present several applications of mode matching methods in spectral and scattering problems. First, we consider the eigenvalue problem for the Dirichlet Laplacian in a finite cylindrical domain that is split into two subdomains by a "perforated" barrier. We prove that the first eigenfunction is localized in the larger subdomain, i.e., its L 2 norm in the smaller subdomain can be made arbitrarily small by setting the diameter of the "holes" in the barrier small enough. This result extends the well known localization of Laplacian eigenfunctions in dumbbell domains. We also discuss an extension to noncylindrical domains with radial symmetry. Second, we study a scattering problem in an infinite cylindrical domain with two identical perforated barriers. If the holes are small, there exists a low frequency at which an incident wave is fully transmitted through both barriers. This result is counter-intuitive as a single barrier with the same holes would fully reflect incident waves with low frequences.This domain can be naturally decomposed into two rectangular subdomains Ω 1 = (−a 1 , 0) × (0, h 1 ) and Ω 2 = (0, a 2 ) × (0, h 2 ). Without loss of generality, we assume h 1 ≥ h 2 . For each of these subdomains, one can explicitly write a general solution of the equation in (1.1) due to a separation of variables in perpendicular directions x

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Section: On One Hand An Upper Bound Readsmentioning

confidence: 99%

Mode matching methods for spectral and scattering problems

Delitsyn

2018

The Quarterly Journal of Mechanics and Applied Mathematics

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“…for a large class of domains Ω in R d (d = 2, 3, ...) which can be decomposed in a "basic" bounded domain V and a branch Q of a variable cross-sectional profile [10]. We proved that if the eigenvalue λ is smaller than the smallest eigenvalue µ among all cross-sections of the branch, then the associated eigenfunction u exponentially decays along that branch:…”

Section: Introductionmentioning

confidence: 95%

“…Proof. The proof relies on Maslov's differential inequality and follows the scheme that we used in [10] for Dirichlet boundary condition. We consider the squared L 2 -norm of the eigenfunction u in the "subbranch" Ω(x 0 ):…”

Section: Extension For Robin Boundary Conditionmentioning

confidence: 99%

“…Qualitatively, an eigenfunction cannot "squeeze" through a narrow connection when its typical wavelength is larger than the connection width. This qualitative picture has found many rigorous formulations for dumbbell shapes and classical and quantum waveguides [2][3][4][5][6][7][8][9][10][11][12]. Numerical and experimental evidence for localization in irregularly-shaped domains was also reported [13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

“…In turn, the Dirichlet boundary condition on the boundary of the branch Ω 2 was relevant. Many numerical illustrations for this theorem were given in [10].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Exponential Decay of Laplacian Eigenfunctions in Planar Domains with Branches

Nguyen¹,

L.²

2014

Geometric and Spectral Analysis

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We consider the eigenvalue problem for the Laplace operator in a planar domain which can be decomposed into a bounded domain of arbitrary shape and elongated "branches" of variable cross-sectional profiles. When the eigenvalue is smaller than a prescribed threshold, the corresponding eigenfunction decays exponentially along each branch. We prove this behavior for Robin boundary condition and illustrate some related results by numerically computed eigenfunctions.

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A Spectral Clustering Algorithm Based on Eigenvector Localization

Lucińska

2014

Artificial Intelligence and Soft Computing

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Exponential decay of Laplacian eigenfunctions in domains with branches of variable cross-sectional profiles

Cited by 15 publications

References 23 publications

Mode matching methods for spectral and scattering problems

Mode matching methods for spectral and scattering problems

On the Exponential Decay of Laplacian Eigenfunctions in Planar Domains with Branches

A Spectral Clustering Algorithm Based on Eigenvector Localization

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