New L2 approach to quantum scattering: Theory

By exploiting the soluble infinite tridiagonal (Jacobi)-matrix problem generated by evaluating a zeroth-order scattering Hamiltonian ${H}_{0}$ in a certain ${L}^{2}$ basis set, we obtain phase shifts, wave functions, etc., which are exact for a full Hamiltonian $H$ in which only the potential $V$ is approximated. Only bound-bound (${L}^{2}$) matrix elements of the Hamiltonian and finite matrix manipulations are needed. The method is worked out here for $s$-wave scattering using Laguerre basis functions. Kato improvement of the results and necessary generalizations to many channels are treated.