The classification of solutions for some dissipative systems by the information of the spectrum is established. Its generator is non self-adjoint Schrödinger operator with rank one singular perturbation. For the proof, a generalized Parseval formula is constructed.
"On the asymptotics of solutions for some Schrödinger equations with dissipative perturbations of rank one." Hiroshima Math. J. 34 (3) 345 - 369, November 2004. https://doi.org/10.32917/hmj/1150998510