A semiclassical theory of linear response in finite Fermi systems, based on the Vlasov equation, and its applications to the study of isoscalar vibrations in heavy nuclei are reviewed. It is argued that the Vlasov equation can be used to study the response of small quantum systems like (heavy) nuclei in regimes for which the finite size of the system is more important than the collisions between constituents. This requires solving the linearized Vlasov equation for finite systems, however, in this case the problem of choosing appropriate boundary conditions for the fluctuations of the phase-space density is nontrivial. Calculations of the isoscalar response functions performed by using different boundary conditions, corresponding to fixed and moving nuclear surface, are compared for different multipoles and it is found that, in a sharp-surface model, the moving-surface boundary conditions give better agreement with experiment. The semiclassical strength functions given by this theory are strikingly similar to the results of analogous quantum calculations, in spite of the fact that shell effects are not included in the theory. This happens because of a well-known close relation between classical trajectories and shell structure.

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