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 \(d\ge 1\). 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.