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We present the discovery of a novel and intriguing global geometric structure of the (interior) transmission eigenfunctions associated with the Helmholtz system. It is shown in generic scenarios that there always exists a sequence of transmission eigenfunctions with the corresponding eigenvalues going to infinity such that those eigenfunctions are localized around the boundary of the domain. We provide a comprehensive and rigorous justification in the case within the radial geometry, whereas for the non-radial case, we conduct extensive numerical experiments to quantitatively verify the localizing behaviours. The discovery provides a new perspective on wave localization. As significant applications, we develop a novel inverse scattering scheme that can produce super-resolution imaging effects and propose a method of generating the so-called pseudo surface plasmon resonant (PSPR) modes with a potential sensing application.
Consider the transmission eigenvalue problem [ (Delta+k^2mathbf{n}^2) w=0, (Delta+k^2)v=0 mbox{in} Omega;quad w=v, partial_ u w=partial_ u v=0 mbox{on} partialOmega. ] It is shown in [12] that there exists a sequence of eigenfunctions $(w_m, v_
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