Lie algebras for systems with mixed spectra. I. The scattering Poeschl--Teller potential

Starting from an N-body quantum space, we consider the Lie-algebraic framework where the Poeschl--Teller Hamiltonian, - 1/2 partial2/sub chi/ +c sech2 chi+s csch2 chi, 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+ x 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