On the bound states and correlation functions of a class of Calogero-type quantum many-body problems with balanced loss and gain

TL;DR

This paper explores the quantization, bound states, and correlation functions of Calogero-type quantum many-body systems with balanced loss and gain, revealing spectral properties and stability conditions through analytical methods and mappings to random matrix theory.

Contribution

It introduces new Calogero-type models with balanced loss and gain, constructs integrals of motion, and analyzes their spectra and correlation functions, extending understanding of non-Hermitian quantum systems.

Findings

Models exhibit both quantized and continuous spectra.

Bound states can be obtained via box-normalization.

Correlation functions are derived using random matrix theory mappings.

Abstract

The quantization of many-body systems with balanced loss and gain is investigated. Two types of models characterized by either translational invariance or rotational symmetry under rotation in a pseudo-Euclidean space are considered. A partial set of integrals of motion are constructed for each type of model. Specific examples for the translationally invariant systems include Calogero-type many-body systems with balanced loss and gain, where each particle is interacting with other particles via four-body inverse-square potential plus pair-wise two-body harmonic terms. A many-body system interacting via short range four-body plus six-body inverse square potential with pair-wise two-body harmonic terms in presence of balanced loss and gain is also considered. In general, the eigen values of these two models contain quantized as well as continuous spectra. A completely quantized spectra and bound states involving all the particles may be obtained by employing box-normalization on the particles having continuous spectra. The normalization of the ground state wave functions in appropriate Stoke wedges is discussed. The exact n-particle correlation functions of these two models are obtained through a mapping of the relevant integrals to known results in random matrix theory. It is shown that a rotationally symmetric system with generic many-body potential does not have entirely real spectra, leading to unstable quantum modes. The eigenvalue problem of a Hamiltonian system with balanced loss and gain and admitting dynamical O(2, 1) symmetry is also considered.