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

Using general local conservation laws we derive dispersion relations for edge modes in a slab of electron liquid confined by a symmetric 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. For a harmonic external potential the dispersion relations are expressed in terms of the exact static pressure and dynamic shear modulus of a homogeneous liquid with the density taken at the slab core. We also derive a simple expression for the frequency shift of the dipole (Kohn) modes in nearly parabolic quantum dots in a magnetic field.