Numerical solution of the time-independent-wave-packet form of the Lippmann–Schwinger equation: Nystrom method vs Chebyshev expansion of the resolvent operator

This paper computationally explores some lesser-known results from the quantum scattering theory that gives the S-matrix in terms of the resolvent operator through the mediation of time-independent wave packets (TIWP). The S-matrix at a given energy is given by the ratio of the interacting and free TIWP’s evaluated at the corresponding on-shell momentum. This expression can also be expressed as the ratio of full and free resolvents at a given energy acting on the pre-collision wave packet. Another expression involves the ratio of the matrix elements of the interacting and free resolvents between post- and pre-collision wave packets. Computational implementations of these formal results involve the solution of the so-called TIWP form of the Lippmann–Schwinger equation (TIWP-LS). In the present contribution, the partial-wave TIWP-LS equation is solved using two schemes: (i) Direct numerical solution of the TIWP-LS equation via the quadrature discretization, and (ii) Construction of the resolvent operator by an expansion in Chebyshev polynomials of the scaled Hamiltonian. Computational performances of these procedures to compute the S-matrix are tested on a model potential for the p-wave case.