  We discuss electromagnetic modes in cavities formed by metamaterials with
negative refraction and demonstrate that the straightforward approach to
substitute negative values of the electric permittivity and the magnetic
permeability leads to quasi-bound states with a negative quality factor. To
ensure positive quality factors and a consistent physical interpretation of the
quasi-bound states it is essential to include the frequency dispersion of the
permittivity and the permeability, as required by positive field energy and
causality. The basic mode equation and the boundary conditions including linear
frequency dispersion are derived. As an example we consider a disk-like cavity
with deformed cross sectional shape. The transition from the unphysical
nondispersive case with negative quality factors to the dispersive case with
positive quality factors is demonstrated numerically and in an analytical
perturbative treatment.
