Far-infrared edge modes in quantum dots

We have investigated edge modes of different multipolarity sustained by quantum dots submitted to external magnetic fields. We present a microscopic description based on a variational solution of the equation of motion for any axially symmetric confining potential and multipole mode. Numerical results for dots with different numbers of electrons, whose ground state is described within a local current density-functional theory, are discussed. Two sum rules, which are exact within this theory, are derived. In the limit of a large neutral dot at $B=0\mathrm{}$, we show that the classical hydrodynamic dispersion law for edge waves $\omega \mathrm{}\left(q\right)\mathrm{}$ $\sim \sqrt{q\ln {(q}_0/q)}$ holds when quantum and finite-size effects are taken into account.