Asymptotically exact dispersion relations for collective modes in a confined charged Fermi liquid

Using the general local conservations laws for densities of particles and momentum we derive dispersion relations for two lowest collective modes in a slab of electron liquid confined by a symmetric external potential. The dispersion relations are exact up to $\lambda^{2} q^{2}$, where $q$ is a wave vector and $\lambda$ is an effective screening length which is assumed to be smaller than the slab width. For a harmonic external potential the dispersion relations are expressed in terms of the {\em exact} static pressure and dynamic shear modulus of a homogeneous electron liquid with the density taken at the slab core.