The characteristics of a one-dimensional subwavelength cavity resonator ~SWCR! formed by a left-handed material layer and a right-handed material layer are studied. Some previously published conclusions for some fundamental issues are discussed and rectified. Our analysis for the stability and the quality factor of the SWCR shows that the two materials should be mismatched in the impedance in order to obtain a useful SWCR. It is shown that the resonant frequency of the cavity is closely related to the total thickness of the cavity. The total thickness can still be far less than the resonant wavelength if the thickness ratio of the two layers is chosen appropriately. The compactness of the SWCR is restricted by its stability and quality factor.