Hamiltonian formulation of linear non-Hermitian systems

Abstract In the case of a linear non-Hermitian system, I prove that it's possible to construct a Hamiltonian in such a way that the equations governing the non-Hermitian system can be exactly expressed using Hamilton's canonical equations. Initially, I demonstrate this within the discrete representation framework and subsequently extend it to continuous representation. Through this formulation employing the Hamiltonian, I can pinpoint a conserved charge using Noether's theorem and identify adiabatic invariants. When this approach is applied to Hermitian systems, all the obtained results converge to the well-known outcomes associated with the Schrödinger equation.