  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.
