PT symmetry and the square well potential: Antilinear symmetry rather than Hermiticity in scattering processes

Abstract

A real potential Hamiltonian has real energy bound states below the scattering threshold and complex energy resonances above it. Scattering states are not square integrable, being instead delta function normalized. This lack of square integrability breaks the connection between Hermiticity and real eigenvalues, to thus allow for real bound state sector eigenvalues and complex scattering sector eigenvalues. When written as contour integrals delta functions take support in the complex plane, with the scattering amplitude being able to take support in the complex plane too. However, the scattering amplitude is CPT symmetric. For resonance scattering this antilinear symmetry requires the presence of a complex conjugate pair of energies, one to describe the excitation of the resonance and the other to describe its decay, with it being their interplay that enforces probability conservation. Each complex pair of energy eigenvalues corresponds to only one observable resonance not two, to thus modify the standard pure decaying complex energy pole discussion of resonances. We show that the non-relativistic real potential square-well Schr\"odinger equation possesses C and PT symmetry in both the bound and scattering sectors, with there being complex conjugate pairs of energy eigenvalue solutions in the scattering sector. The Hamiltonian thus acts as a Hermitian operator below the scattering threshold and as a non-Hermitian one above it. For those values of the potential for which bound states lie right at the top of the well the scattering amplitude threshold branch point is an exceptional point, a characteristic of systems with antilinear symmetry at which there are more independent solutions to the Schr\"odinger equation than there are eigenstates of a then non-Hermitian Hamiltonian. The square well provides an explicit realization of how antilinearity is more general than Hermiticity.