Integrable and Exactly Solvable models with Balanced Loss and Gain

A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients will be presented. It will be shown that using a suitable choice of coordinates, the Hamiltonian can always be reformulated as a many-particle system in the background of a pseudo-Euclidean metric and subjected to an analogous inhomogeneous magnetic field with a functional form that is identical with space-dependent loss/gain co-efficient. Partially integrable systems will be presented for two distinct cases, namely, systems with (i) translational symmetry or (ii) rotational invariance in a pseudo-Euclidean space. Quantization of the Hamiltonian will also be discussed with the construction of the integrals of motion for specific choices of the potential and gain-loss coefficients. A few exact solutions for both the cases will be presented for specific choices of the potential and space-dependent gain/loss coefficients. This will include a coupled chain of nonlinear oscillators and a many particle Calogero type model with four-body inverse square plus two-body pair-wise harmonic interactions. A few quasi-exactly solvable models admitting bound states in appropriate Stoke wedges will also be discussed.