N identical particles under quantum confinement: A many-body dimensional perturbation theory approach

TL;DR

This paper applies dimensional perturbation theory to analyze N identical particles under quantum confinement, deriving analytical results for their vibrational modes across various physical systems.

Contribution

It introduces a detailed methodology for using dimensional perturbation theory to study many-body quantum systems with symmetry, providing analytical expressions for vibrational frequencies.

Findings

Derived general analytical expressions for normal-mode frequencies.

Applied methods to N-electron atom, quantum dot, and BEC systems.

Provided insights into the equilibrium structure at infinite dimension.

Abstract

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.