Scattering and bound states: A Lorentzian function-based spectral filter approach

We present a Lorentzian function-based spectral filter method for computing the elements of the quantum scattering matrix (S-matrix) and molecular bound-state spectra. For computing bound-state eigenvalues in a predefined energy window, we use the Lorentzian function within the filter diagonalization framework. From the spectral filter point of view, we find that the formal theoretical structure for computing the S-matrix is the same as those of overlap and Hamiltonian matrix elements necessary for filter diagonalization, and hence, the same computational protocol can be utilized for scattering as well as bound-state studies. Furthermore, we argue that the appropriate scattering boundary conditions can be accurately built while preparing the initial wave packets. For numerical implementation, we have utilized the Lorentzian filter in two complementary series forms: (1) using Chebyshev polynomials with the Hamiltonian as its argument, which is useful for a fully quantum mechanical study; and (2) in terms of a discrete set of short-time quantum propagators, which can additionally be extended to approximate dynamical regimes. The computation of matrix elements for a filter diagonalization application and the scattering matrix requires a product of a series representation of two filter operators for which we have been able to perform a partial resummation of both the series analytically, giving thereby a very compact and rapidly convergent expression. The exponential damping term associated with the Lorentzian filter is very useful for controlling the convergence and removing unwanted features from the computed spectrum. As is true of previous discrete time expansion of the spectral density operator, the present formalism can also be utilized for inverting discrete time signals obtained from various experiments. We illustrate the validity of the present approach by test calculations on a model one-dimensional quantum scattering problem.