Summing the spectral representations of Pöschl–Teller and Rosen–Morse fixed-energy amplitudes

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Abstract
The spectral representations of the fixed-energy amplitude of the symmetric and the general Pöschl–Teller potentials are summed via a Sommerfeld–Watson transformation which leads to a simple closed-form expression. The result is used to write down a similar expression for the symmetric and general Rosen–Morse potentials, exploiting the close correspondence that exists between the two systems within the Schrödinger theory and the path integral formalism via a Duru–Kleinert transformation. From the singularities of the latter amplitude the bound and continuum states of the general Rosen–Morse potential are extracted.