  We study the interaction between low-lying transverse collective oscillations
and thermal excitations of an elongated Bose-Einstein condensate by means of
perturbation theory. We consider a cylindrically trapped condensate and
calculate the transverse elementary excitations at zero temperature by solving
the linearized Gross-Pitaevskii equations in two dimensions. We use them to
calculate the matrix elements between thermal excited states coupled with the
quasi-2D collective modes. The Landau damping of transverse collective modes is
investigated as a function of temperature. At low temperatures, the damping
rate due to the Landau decay mechanism is in agreement with the experimental
data for the decay of the transverse quadrupole mode, but it is too small to
explain the slow experimental decay of the transverse breathing mode. The
reason for this discrepancy is discussed.
