A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators (Q2018342)

The Green's function for the Helmholtz equation is analysed in the upper half space \(\mathbb{R}^{2}\times]0,\infty[\) with a number of translated, bounded, cylinder-shaped extensions \(\left(B\left(0,1\right)\times]-h,0[\:\cup\epsilon\Lambda\right)+\left(z_{0,j},z_{1,j},0\right)\), \(j=1,\ldots,M\), \(\epsilon\in]0,1[\:\), into the lower half space -- the Helmholtz resonators -- connected via the small opening \(\epsilon\Lambda\), where \(\left(0,0,0\right)\in\Lambda\subseteq B\left(0,1\right)\times\left\{ 0\right\} \). Homogeneous Neumann boundary conditions are assumed. The asymptotic behaviour of the resonances induced by the eigenvalue \(0\) of the Neumann problem in the cavities and of the Green's function are derived for small \(\epsilon\). The analysis of the leading order terms of the asymptotic expansion of the resolution function is offered as an explanation for the super-resolution phenomenon.