Hamiltonian formulation of systems with balanced loss–gain and exactly solvable models

Abstract

A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occur in a pairwise fashion. It is also shown that with the choice of a suitable co-ordinate, the Hamiltonian can always be reformulated in the background of a pseudo-Euclidean metric. If the equations of motion of some of the well-known many-body systems like Calogero models are generalized to include balanced loss and gain, it appears that the same may not be amenable to a Hamiltonian formulation. A few exactly solvable systems with balanced loss and gain, along with a set of integrals of motion is constructed. The examples 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. For the case of nonlinear oscillators, stable solution exists even if the loss and gain parameter has unbounded upper range. Further, the range of the parameter for which the stable solutions are obtained is independent of the total number of the oscillators. The set of coupled nonlinear equations are solved exactly for the case when the values of all the constants of motions except the Hamiltonian are equal to zero. Exact, analytical classical solutions are presented for all the examples considered.

Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction: general results and a case study with Landau Hamiltonian

“…There is no definite answer for the most general case with arbitrary Γ i and η ik . Even for a simpler case of η ik = (−1) i+1 δ ik , Γ i = ∂Γ ∂xi with even N and Γ being the potential of rational Calogero model, no definite answer is known for N > 2 [8]. The Hamiltonian formulation of systems with balanced loss and gain [8,10], which is summarized below, constitute a special case of Eq.…”

“…Moreover, all the Hamiltonian systems with balanced loss and gain considered in the literature [1,2,3,4,5,6,7], prior to the general formulation of such systems in Ref. [8,9,10], may be identified as defined in the background of a pseudo-Euclidean metric. The generalized momenta Π is defined by,…”

“…(14). Investigations [8,10,9] on Hamiltonian systems with balanced loss and gain have been so far restricted to a diagonal D which is a symmetric matrix. This is primarily because only the diagonal elements of D are relevant for determining whether a system is dissipative or nondissipative.…”

Integrable coupled Liénard-type systems with balanced loss and gain

“…A Hamiltonian formulation of many-particle systems with balanced loss and gain is presented in Ref. [16]. The loss/gain coefficient is constant in this approach and the Hamiltonian is written as,…”

“…A suggestion is made in Ref. [16] that this Hamiltonian formulation can also be generalized to include space-dependent balanced loss and gain co-efficients by redefining the generalized momenta Π as,…”