Nonstationary SQM/IST Correspondence and ${\cal CPT/PT}$-Invariant Paired Hamiltonians on the Line

Abstract

Abstract We fill some of the existing gaps in the correspondence between supersymmetric quantum mechanics and the inverse scattering transform by extending the consideration to the case of paired stationary and nonstationary Hamiltonians. We formulate the Goursat problem corresponding to the case and explicitly construct the kernel of the nonlocal inverse scattering transform, which solves it. As a result, we find a way of constructing non-Hermitian Hamiltonians from the initially Hermitian ones that leads, in the case of real-valued spectra of both potentials, to pairing of ${\cal CPT/PT}$-invariant Hamiltonians. The relevance of our proposal to quantum optics and optical waveguide technology, as well as to nonlinear dynamics and black hole physics, is briefly discussed.