  We consider quadrupole excitations of quasi-two dimensional interacting
quantum gas in an anisotropic harmonic oscillator potential at zero
temperature. Using the time-dependent variational approach, we calculate a few
low-lying collective excitation frequencies of a two dimensional anisotropic
Bose gas. Within the energy weighted sum-rule approach, we derive a general
dispersion relation of two quadrupole excitations of a two dimensional deformed
trapped quantum gas. This dispersion relation is valid for both statistics. We
show that the quadrupole excitation frequencies obtained from both methods are
exactly the same. Using this general dispersion relation, we also calculate the
quadrupole frequencies of a two dimensional unpolarized Fermi gas in an
anisotropic trap. For both cases, we obtain analytic expressions for the
quadrupole frequencies and the splitting between them for arbitrary value of
trap deformation. This splitting decreases with increasing interaction strength
for both statistics. For two dimensional anisotropic Fermi gas, the two
quadrupole frequencies and the splitting between them become independent of the
particle number within the Thomas-Fermi approach.
