Lie algebras for systems with mixed spectra. I. The scattering Pöschl–Teller potential (original) (raw)

Starting from an N-body quantum space, we consider the Lie-algebraic framework where the Poschl-Teller Hamiltonian,-! a; + c sech 2 X + s csch 2 X' 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 .fj!+ X.fj!-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.