We consider the Schrödinger operator with regular short range complex-valued potential in dimension \(d\ge 1\) . We show that, for \(d\ge 2\) , 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.
