Lie algebras for systems with mixed spectra. I. The scattering Pöschl–Teller potential

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Abstract
Starting from an N-body quantum space, we consider the Lie-algebraic framework where the Pöschl–Teller Hamiltonian, − 1/2 ∂2χ +c sech2 χ+s csch2 χ, is the single sp(2,R) Casimir operator. The spectrum of this system is mixed: it contains a finite number of negative-energy bound states and a positive-energy continuum of free states; it is identified with the Clebsch–Gordan series of the 𝒟+×𝒟− representation coupling. The wave functions are the sp(2,R) Clebsch–Gordan coefficients of that coupling in the parabolic basis. Using only Lie-algebraic techniques, we find the asymptotic behavior of these wave functions; for the special pure-trough potential (s=0) we derive thus the transmission and reflection amplitudes of the scattering matrix.