  We investigate the quantum breathing mode (monopole oscillation) of trapped
fermionic particles with Coulomb and dipole interaction in one and two
dimensions. This collective oscillation has been shown to reveal detailed
information on the many-particle state of interacting trapped systems and is
thus a sensitive diagnostics for a variety of finite systems, including cold
atomic and molecular gases in traps and optical lattics, electrons in metal
clusters and in quantum confined semiconductor structures or nanoplasmas. An
improved sum rule formalism allows us to accurately determine the breathing
frequencies from the ground state of the system, avoiding complicated
time-dependent simulations. In combination with the Hartree-Fock and the
Thomas-Fermi approximations this enables us to extend the calculations to large
particle numbers $N$ on the order of several million. Tracing the breathing
frequency to large $N$ as a function of the coupling parameter of the system
reveals a surprising difference of the asymptotic behavior of one-dimensional
and two-dimensional harmonically trapped Coulomb systems.
