Surface Concentration of Transmission Eigenfunctions

Yat Tin CHOW; Youjun DENG; Hongyu LIU; Mahesh SUNKUlA

doi:10.1007/s00205-023-01887-4

Surface Concentration of Transmission Eigenfunctions

Yat Tin CHOW, Youjun DENG, Hongyu LIU^*, Mahesh SUNKUlA

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

7 Citations (Scopus)

Abstract

The transmission eigenvalue problem is a type of non-elliptic and non-selfadjoint spectral problem that arises in the wave scattering theory when invisibility/transparency occurs. The transmission eigenfunctions are the interior resonant modes inside the scattering medium. We are concerned with the geometric rigidity of the transmission eigenfunctions and show that they concentrate on the boundary surface of the underlying domain in two senses. This substantiates the recent numerical discovery in Chow et al. (SIAM J Imaging Sci, 14(3):946–975, 2021) on such an intriguing spectral phenomenon of the transmission resonance. Our argument is based a generalized Weyl’s law and certain novel ergodic properties of the coupled boundary layer-potential operators which are employed to analyze the generalized transmission eigenfunctions. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature.

Original language	English
Article number	54
Journal	Archive for Rational Mechanics and Analysis
Volume	247
Issue number	3
Online published	21 May 2023
DOIs	https://doi.org/10.1007/s00205-023-01887-4
Publication status	Published - Jun 2023

Funding

The work of Y. Deng was supported by NSF grant of China No. 11971487 and NSF grant of Hunan No. 2020JJ2038. The work of H Liu was supported by Hong Kong RGC General Research Funds (project numbers, 11300821, 12301420 and 12302919) and the NSFC/RGC Joint Research Fund, N_CityU101/21. The authors are grateful to the very helpful discussion with K.L. Lee, W.T. Leung and the two anonymous referees for their tremendously helpful suggestions.

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title = "Surface Concentration of Transmission Eigenfunctions",

abstract = "The transmission eigenvalue problem is a type of non-elliptic and non-selfadjoint spectral problem that arises in the wave scattering theory when invisibility/transparency occurs. The transmission eigenfunctions are the interior resonant modes inside the scattering medium. We are concerned with the geometric rigidity of the transmission eigenfunctions and show that they concentrate on the boundary surface of the underlying domain in two senses. This substantiates the recent numerical discovery in Chow et al. (SIAM J Imaging Sci, 14(3):946–975, 2021) on such an intriguing spectral phenomenon of the transmission resonance. Our argument is based a generalized Weyl{\textquoteright}s law and certain novel ergodic properties of the coupled boundary layer-potential operators which are employed to analyze the generalized transmission eigenfunctions. {\textcopyright} 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature.",

author = "CHOW, {Yat Tin} and Youjun DENG and Hongyu LIU and Mahesh SUNKUlA",

year = "2023",

month = jun,

doi = "10.1007/s00205-023-01887-4",

language = "English",

volume = "247",

journal = "Archive for Rational Mechanics and Analysis",

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T1 - Surface Concentration of Transmission Eigenfunctions

AU - CHOW, Yat Tin

AU - DENG, Youjun

AU - LIU, Hongyu

AU - SUNKUlA, Mahesh

PY - 2023/6

Y1 - 2023/6

N2 - The transmission eigenvalue problem is a type of non-elliptic and non-selfadjoint spectral problem that arises in the wave scattering theory when invisibility/transparency occurs. The transmission eigenfunctions are the interior resonant modes inside the scattering medium. We are concerned with the geometric rigidity of the transmission eigenfunctions and show that they concentrate on the boundary surface of the underlying domain in two senses. This substantiates the recent numerical discovery in Chow et al. (SIAM J Imaging Sci, 14(3):946–975, 2021) on such an intriguing spectral phenomenon of the transmission resonance. Our argument is based a generalized Weyl’s law and certain novel ergodic properties of the coupled boundary layer-potential operators which are employed to analyze the generalized transmission eigenfunctions. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature.

AB - The transmission eigenvalue problem is a type of non-elliptic and non-selfadjoint spectral problem that arises in the wave scattering theory when invisibility/transparency occurs. The transmission eigenfunctions are the interior resonant modes inside the scattering medium. We are concerned with the geometric rigidity of the transmission eigenfunctions and show that they concentrate on the boundary surface of the underlying domain in two senses. This substantiates the recent numerical discovery in Chow et al. (SIAM J Imaging Sci, 14(3):946–975, 2021) on such an intriguing spectral phenomenon of the transmission resonance. Our argument is based a generalized Weyl’s law and certain novel ergodic properties of the coupled boundary layer-potential operators which are employed to analyze the generalized transmission eigenfunctions. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature.

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U2 - 10.1007/s00205-023-01887-4

DO - 10.1007/s00205-023-01887-4

M3 - RGC 21 - Publication in refereed journal

SN - 0003-9527

VL - 247

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

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M1 - 54

ER -

Surface Concentration of Transmission Eigenfunctions

Abstract

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GRF: Mathematical Studies of Surface-localized Transmission Eigenstates and Applications

NSFC: A Mathematical Theory of Subwavelength Resonances in Elasticity with Applications and Beyond

GRF: Geometric Properties of Three Classes of Spectral Problems with Applications to Inverse Problems and Material Sciences

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Surface Concentration of Transmission Eigenfunctions

Abstract

Funding

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GRF: Mathematical Studies of Surface-localized Transmission Eigenstates and Applications

NSFC: A Mathematical Theory of Subwavelength Resonances in Elasticity with Applications and Beyond

GRF: Geometric Properties of Three Classes of Spectral Problems with Applications to Inverse Problems and Material Sciences

Cite this