Exact solution of $D_N$ type quantum Calogero model through a mapping to free harmonic oscillators

TL;DR

This paper presents an exact solution to the $D_N$ type quantum Calogero model by transforming it into free harmonic oscillators, revealing its eigenfunctions and orthogonality properties.

Contribution

The authors provide a novel mapping of the $D_N$ Calogero model to decoupled harmonic oscillators, enabling explicit eigenfunction construction and analysis of orthogonality.

Findings

Eigenfunctions constructed from bosonic harmonic oscillators with specific parity

Eigenfunctions are orthogonal under a nontrivial inner product

The model's eigenvalues and eigenfunctions are explicitly obtained

Abstract

We solve the eigenvalue problem of the $D_N$ type of Calogero model by mapping it to a set of decoupled quantum harmonic oscillators through a similarity transformation. In particular, we construct the eigenfunctions of this Calogero model from those of bosonic harmonic oscillators having either all even parity or all odd parity. It turns out that the eigenfunctions of this model are orthogonal with respect to a nontrivial inner product, which can be derived from the quasi-Hermiticity property of the corresponding conserved quantities.