Coulomb interactions on planar structures: Inverting the square root of the Laplacian (Q2719254)

The pseudodifferential equation \((-\Delta)^{1/2}\psi=\omega\) is solved using a fast, adaptive, numerical method in the plane. The function \(\omega\in L^2(\mathbb{R}^2)\) and the solution of the above equation can be written in the form NEWLINE\[NEWLINE\psi(x)= \int_{\mathbb{R}^2} \bigl(\widehat \omega(k)/2 \pi(k)\bigr) \exp(2\pi ik.x) dk.NEWLINE\]NEWLINE Using a Green's function, solution of \((-\Delta)^{1/2} G(x)= \delta(x)\), the authors express \(\psi(x)\) by an inversion formula. The authors observe that the pseudodifferential equation and the inversion formula describe a three-dimensional Poisson equation for which \(\omega(x)\) is a singular density lying on the plane \(z=0\). For the computation of the above integral transform, the authors describe two special fast multipole methods, based on some results obtained previously by themselves.NEWLINENEWLINENEWLINEThe numerical-theoretical results obtained in the paper are applied to four particular cases of the following concrete problems: planar circuits in electrical engineering, problems in thin films, problems in tomography, quasi-geostrophic fluid dynamics.