Surface-Localized Transmission Eigenstates, Super-resolution Imaging, and Pseudo Surface Plasmon Modes

On new surface-localized transmission eigenmodes

“…In the current article, we show that there exists a new type of surface-localised transmission eigenmodes which are different from those found in [16]. In fact, we shall construct a sequence of transmission eigenfunctions (w m , v m ) m∈N associated with k m → ∞ such that both {w m } m∈N and {v m } m∈N are surface-localized, no matter n > 1 or 0 < n < 1.…”

“…The results mentioned above are all of a local feature which are localized around certain peculiar geometrical points on ∂Ω. In [16], a global geometric rigidity property was discovered for the transmission eigenfunctions. In fact, it is shown that there exists a sequence of eigenfunctions (w m , v m ) m∈N associated with k m → ∞ such that either {w m } m∈N or {v m } m∈N are surface-localized, depending on n > 1 or 0 < n < 1.…”

“…

Consider the transmission eigenvalue problemIt is shown in [16] that there exists a sequence of eigenfunctions (wm, vm) m∈N associated with km → ∞ such that either {wm} m∈N or {vm} m∈N are surfacelocalized, depending on n > 1 or 0 < n < 1. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions (wm, vm) m∈N associated with km → ∞ such that both {wm} m∈N and {vm} m∈N are surface-localized, no matter n > 1 or 0 < n < 1.

…”

“…Here, by surface-localization, we mean the L 2 -energy of the transmission eigenmode is concentrated in a sufficiently small neighbourhood of ∂Ω. Moreover, two interesting applications of practical importance were generated by using the surface-localization of the transmission eigenmodes in [16], including a super-resolution wave scheme and a possible pseudo plasmon sensing scheme.…”

“…From the inverse problem point of view, it seems more practical for one to characterize the range of F, namely Range(F) associated with all possible u i , which contains all the "visible" information. However, it is proposed in [16] that one can actually achieve super-resolution reconstruction for the inverse problem (4) if instead using Ker(F). In fact, it is shown in [16] that K ϵ (F) can be obtained by knowledge of Range(F), where ϵ ∈ R + is sufficiently small.…”