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One of the fundamental eigenmodes of finite interacting systems is the modeof {\em uniform radial expansion and contraction} -- the ``breathing'' mode(BM). Here we show in a general way that this mode exists only under specialconditions: i) for harmonically trapped systems with interaction potentials ofthe form $1/r^\gamma$ $(\gamma\in\mathbb{R}_{\neq0})$ or $\log(r)$, or ii) forsome systems with special symmetry such as single shell systems formingplatonic bodies. Deviations from the BM are demonstrated for two examples:clusters interacting with a Lennard-Jones potential and parabolically trappedsystems with Yukawa repulsion. We also show that vanishing of the BM leads tothe occurence of multiple monopole oscillations which is of importance forexperiments.

Existence and Vanishing of the Breathing Mode in Strongly Correlated Finite Systems