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

TL;DR

The work argues that antilinearity, encapsulated by CPT (or PT) symmetry, is a more general guiding principle than Hermiticity for quantum theories, especially in scattering where plane waves are delta-normalized and not square-integrable. By analyzing a real square-well potential, the authors show PT symmetry across bound and scattering sectors, yielding complex-conjugate energy pairs $E_0 \pm i\Gamma$ while preserving probability through a pseudo-Hermitian framework with a metric $\hat{V}$. A key finding is that two complex poles in the scattering amplitude correspond to a single observable resonance, with an exceptional point at the scattering threshold where the Hamiltonian becomes non-diagonalizable. This work strengthens the view that antilinearity can govern quantum dynamics beyond Hermiticity and provides explicit realizations via contour representations and square-well scattering.

Abstract

While a Hamiltonian with a real potential acts as a Hermitian operator when it operates on bound states, it produces resonances with complex energies in a scattering experiment. The 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 taking support in the complex energy plane too. However, the scattering amplitude is $CPT$ symmetric (or $PT$ symmetric if $C$ is conserved), regardless of whether or not states are square integrable. 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 square-well problem with a real potential possesses $PT$ symmetry in both the bound and scattering sectors, with there being complex conjugate pairs of energy eigenvalues in the scattering sector. Despite this there is just the same number of observable resonances as in the pure decaying sector. We show that the square-well scattering threshold branch point is an exceptional point (a characteristic of antilinear symmetry) at which the Hamiltonian becomes of non-diagonalizable, and thus non-Hermitian, Jordan-block form. The square-well potential thus provides an explicit realization of how antlinearity is more general than Hermiticity.