Surface-localized transmission eigenstates, super-resolution imaging and pseudo surface plasmon modes

Chow, Yat Tin; Deng, Youjun; He, Youzi; Liu, Hongyu; Wang, Xianchao

doi:10.48550/arxiv.2003.02406

Cited by 3 publications

(8 citation statements)

References 45 publications

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“…That is, all the results hold for the partial-data transmission eigenvalue problem, namely in (1.2) the transmission boundary conditions on ∂Ω is required to hold only in a small neighbourhood of the corner point. It is mentioned that a global rigidity result of the geometric structure of the transmission eigenfunctions was presented in [14].…”

Section: Lemma 11 ( [23])mentioning

confidence: 99%

On vanishing near corners of conductive transmission eigenfunctions

Deng¹,

Duan²,

Liu³

2020

Preprint

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In this paper, we consider the transmission eigenvalue problem associated with a general conductive transmission condition and study the geometric structures of the transmission eigenfunctions. We prove that under a mild regularity condition in terms of the Herglotz approximations of one of the pair of the transmission eigenfunctions, the eigenfunctions must be vanishing around a corner on the boundary. The Herglotz approximation can be regarded as the Fourier transform of the transmission eigenfunction in terms of the plane waves, and the growth rate of the transformed function can be used to characterize the regularity of the underlying wave function. The geometric structures derived in this paper include the related results in [5,19] as special cases and verify that the vanishing around corners is a generic local geometric property of the transmission eigenfunctions.

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Section: Lemma 11 ( [23])mentioning

confidence: 99%

On vanishing near corners of conductive transmission eigenfunctions

Deng¹,

Duan²,

Liu³

2020

Preprint

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“…Moreover, it can be shown (cf. [7,26]) that invisibility occurs for an incident field u i if and only if (u| Ω , u i | Ω ) is a pair of transmission eigenfunctions associated with the transmission eigenvalue k 2 and the target scatterer (Ω, V ). Hence, for the good sake of invisibility, one should expect the transmission eigenvalues and transmission eigenfunctions of (3.20) should be as "dense" as possible, and this is in sharp difference to that for the LSM which requires the "sparse" the better; see our discussion in Section 2.2.…”

Section: More Connections To Inverse Scattering Problems and Invisibi...mentioning

confidence: 99%

“…Let 0 < k 2 1 ≤ k 2 2 ≤ • • • ≤ k 2 j → +∞ denote the real transmission eigenvalues, and (u k j , v k j ) be the corresponding pair of transmission eigenfunctions associated with the eigenvalue k 2 j . It is shown in [26,27] that there exists a subsequence (k jn ) ⊂ (k j ) with k 2 jn → +∞ as n → +∞ such that either (u jn ) is a sequence of surface-localized eigenstates or (v jn ) is a sequence of surface-localized eigenstates. The existence of surface-localized (u jn ) or (v jn ) depends on V .…”

Section: Lemma 42 ( [7]mentioning

confidence: 99%

On local and global structures of transmission eigenfunctions and beyond

Liu

2020

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The (interior) transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many aspects in a delicate way. The properties of the transmission eigenvalues have been extensively and intensively studied over the years, whereas the intrinsic properties of the transmission eigenfunctions are much less studied. Recently, in a series of papers, several intriguing local and global geometric structures of the transmission eigenfunctions are discovered. Moreover, those longly unveiled geometric properties produce some interesting applications of both theoretical and practical importance to direct and inverse scattering problems. This paper reviews those developments in the literature by summarizing the results obtained so far and discussing the rationales behind them. There are some side results of this paper including the general formulations of several types of transmission eigenvalue problems, some interesting observations on the connection between the transmission eigenvalue problems and several challenging inverse scattering problems, and several conjectures on the spectral properties of transmission eigenvalues and eigenfunctions, with most of them are new to the literature.

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“…There are several further studies on the locally vanishing property of the transmission eigenfunctions in different physical scenarios [6,8,12,14,17]. In [19], a global rigidity property is discovered, showing that the transmission eigenfunctions tend to localize on ∂Ω. Both the local and global geometric structures of transmission eigenfunctions can produce interesting and significant applications.…”

mentioning

confidence: 99%

“…Second, they have been used to establish novel unique identifiability results for the inverse scattering problems by a single far-field measurement [6, 8, 10-12, 17, 18, 26, 36, 37], which constitutes a longstanding problem in the inverse scattering theory [20,39]. Furthermore, in [19], a super-resolution wave imaging scheme was developed by making use the geometric properties of the transmission eigenfunctions.…”

mentioning

confidence: 99%

On a local geometric property of the generalized elastic transmission eigenfunctions and application

Diao,

Liu,

Sun

2021

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Consider the nonlinear and completely continuous scattering map S (Ω; λ, µ, V ), u i = u ∞ t (x), x ∈ S n−1 , which sends an inhomogeneous elastic scatterer (Ω; λ, µ, V ) to its far-field pattern u ∞ t due to an incident wave field u i via the Lamé system. Here, (λ, µ, V ) signifies the medium configuration of an elastic scatterer that is compactly supported in Ω. In this paper, we are concerned with the intrinsic geometric structure of the kernel space of S, which is of fundamental importance to the theory of inverse scattering and invisibility cloaking for elastic waves and has received considerable attention recently. It turns out that the study is contained in analysing the geometric properties of a certain nonselfadjoint and non-elliptic transmission eigenvalue problem. We propose a generalized elastic transmission eigenvalue problem and prove that the transmission eigenfunctions vanish locally around a corner of ∂Ω under generic regularity criteria. The regularity criteria are characerized by the Hölder continuity or a certain Fourier extension property of the transmission eigenfunctions. As an interesting and significant application, we apply the local geometric property to derive several novel unique identifiability results for a longstanding inverse elastic problem by a single far-field measurement.

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Surface-localized transmission eigenstates, super-resolution imaging and pseudo surface plasmon modes

Cited by 3 publications

References 45 publications

On vanishing near corners of conductive transmission eigenfunctions

On vanishing near corners of conductive transmission eigenfunctions

On local and global structures of transmission eigenfunctions and beyond

On a local geometric property of the generalized elastic transmission eigenfunctions and application

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