  Systems that involve N identical interacting particles under quantum
confinement appear throughout many areas of physics, including chemical,
condensed matter, and atomic physics. In this paper, we present the methods of
dimensional perturbation theory, a powerful set of tools that uses symmetry to
yield simple results for studying such many-body systems. We present a detailed
discussion of the dimensional continuation of the N-particle Schrodinger
equation, the spatial dimension D -> infinity equilibrium (D^0) structure, and
the normal-mode (D^{-1}) structure. We use the FG matrix method to derive
general, analytical expressions for the many-body normal-mode vibrational
frequencies, and we give specific analytical results for three confined N-body
quantum systems: the N-electron atom, N-electron quantum dot, and N-atom
inhomogeneous Bose-Einstein condensate with a repulsive hardcore potential.
