Longitudinal oscillations in a non-uniform spatially dispersive plasma

Longitudinal oscillations of the electron fluid in the hydrodynamic model of a metal are examined with pressure effects taken into account. It is well-known that this entails spatial dispersion. The equilibrium electron number density is taken to be non-uniform and a non-self-adjoint fourth order differential equation obeyed by the electric potential is derived. A velocity potential necessary for the description of sound waves is introduced in the standard fashion and the generalized version of Bloch orthogonality appropriate to a non-uniform background is deduced. We observe a duality between electric and velocity potentials in the sense that the respective differential operators are adjoint to each other. The spectrum is calculated in the special case of an exponential profile for the equilibrium electron number density. The surface plasmons are connected with the analytic properties of the scattering amplitude in the complex plane. The phase shift at threshold is expressed in terms of the number of surface plasmon modes via an expression reminiscent of Levinson's statement in quantum mechanics.