Hamiltonian form of PT symmetric dimer

What is it about?

The standard $\mathcal{PT}$-symmetric dimer is a linearly-coupled two-site discrete nonlinear Schr\"odinger equation with one site losing and the other one gaining energy at the same rate. We show that despite gain and loss, the standard $\mathcal{PT}$-dimer is a Hamiltonian system. We also produce a Lagrangian formulation for this dimer.

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Why is it important?

PT-symmetric systems are commonly thought to occupy a niche between dissipative and conservative systems. I show that the most important PT-symmetric dimer (the "standard" dimer) is a conservative, Hamiltonian, system. Therefore this model belongs to the conservative world rather than lying halfway between dissipative and conservative domains as it was previously thought.