Non-unitary transformation of quantum time-dependent non-hermitian systems

Anotace: We provide a new perspective on non-Hermitian evolution in quantum mechanics by emphasizing the same method as in the Hermitian quantum evolution. We first give a precise description of the non unitary transformation and the associated evolution, and collecting the basic results around it and postulating the norm preserving. This cautionary postulate imposing that the time evolution of a non Hermitian quantum system preserves the inner products between the associated states must not be read naively. We also give an example showing that the solutions of time-dependent non Hermitian Hamiltonian systems given by a linear combination of SU(1,1) and SU(2) are obtained thanks to time-dependent non-unitary transformation.