Script PScript T-symmetric regularizations in supersymmetric quantum mechanics

Within supersymmetric quantum mechanics the necessary regularization of the poles of the superpotentials on the real line of coordinates x may be most easily mediated by a small constant shift of this axis into the complex plane. Detailed attention is paid here to the resulting recipe which works, in effect, with non-Hermitian (a.k.a. -symmetric or pseudo-Hermitian) Hamiltonians. Besides an exhaustive discussion of the role of the complex spike in harmonic oscillator, we mention also some applications concerning the regularized versions of the Smorodinsky–Winternitz and Calogero models and of the relativistic Klein–Gordon equation.