Casimir energy of smooth compact surfaces

We discuss the formalism of Balian and Duplantier for the calculation of the Casimir energy for an arbitrary smooth compact surface, and use it to give some examples: a finite cylinder with hemispherical caps, the torus, ellipsoid of revolution, a "cube" with rounded corners and edges, and a "drum" made of disks and part of a torus. We propose a model function which approximately captures the shape dependence of the Casimir energy.Comment: 9 pages, 4 figures, minor changes, published versio

Kelvin-Mach wake in a two-dimensional Fermi sea

The dispersion law for plasma oscillations in a two-dimensional electron gas in the hydrodynamic approximation interpolates between $\Omega \propto \sqrt{q}$ and $\Omega \propto q$ dependences as the wave vector $q$ increases. As a result, downstream of a charged impurity in the presence of a uniform supersonic electric current flow, a wake pattern of induced charge density and potential is formed whose geometry is controlled by the Mach number $M$. For $1<M\leqslant \sqrt{2}$ the wake consists of transverse wavefronts confined within a sector whose angle is given by the classic Mach condition. An additional wake of larger angle resembling the Kelvin ship wake and consistsing of both transverse and diverging wavefronts is found outside the Mach sector for $M>\sqrt{2}$. These wakes also trail an external charge traveling supersonically a fixed distance away from the electron gas.Comment: 4+ pages, 3 figures, minor changes, version to be published in Phys. Rev. Let