Calogero-like Model without Rearrangement Symmetry

Abstract

Reinterpretation of mathematics behind the exactly solvable Calogero’s A-particle quantum model is used to propose its generalization. Firstly, it is argued that the strongly singular nature of Calogero’s particle–particle interactions makes the original permutation-invariant Hamiltonian tractable as a direct sum 𝐻=⨁𝐻𝑎 of isospectral components, which are mutually independent. Secondly, after the elimination of the center-of-mass motion, the system is reconsidered as existing in the reduced Euclidean space ℝ𝐴−1 of relative coordinates and decaying into a union of subsets 𝑊𝑎 called Weyl chambers. The mutual independence of the related reduced forms of operators 𝐻𝑎 enables us to makes them nonisospectral. This breaks the symmetry and unfolds the spectral degeneracy of H. A new multiparametric generalization of the conventional A-body Calogero model is obtained. Its detailed description is provided up to 𝐴=4.