Local density of states in parabolic quantum corrals

Abstract

Atomic manipulation and scanning tunnel microscope experiments on metal surfaces have shown that electronic states in a ``quantum corral'' can be locally monitored and used to analyze the nonlocal effects of perturbations. We study new corral geometries defined by families of confocal parabolas. General solutions of the Schr\"odinger equation for the interior problem with Dirichlet (hard wall) boundary conditions are found exactly in terms of zeroes of hypergeometric functions. We show that the Hilbert space of solutions is separated in subspaces with odd and even symmetry. We perform numerical evaluation of the zeroes and study the effects of the parabolic curvatures on the eigenvalues and eigenfunctions of the parabolic quantum corral. The evolution of the local density of states with energy as a function of parabolic corral geometry is also analyzed. We find that under suitable conditions, the distribution of state antinodes can be described as directed intensity beams, which could be used as ``quantum beacons'' in future generations of ``quantum mirage'' experiments or optical and acoustic analogs of quantum corrals for the state node distribution.