On Unitarity of the Scattering Operator in Non-Hermitian Quantum Mechanics

We consider the Schrödinger operator with regular short range complex-valued potential in dimension d≥1. We show that, for d≥2, the unitarity of scattering operator for this Hamiltonian at high energies implies the reality of the potential (that is Hermiticity of Hamiltonian). In contrast, for d=1, we present complex-valued exponentially localized soliton potentials with unitary scattering operator for all positive energies and with unbroken PT symmetry. We also present examples of complex-valued regular short range potentials with real spectrum for d=3. Some directions for further research are formulated.