Spectral-shift and scattering-equivalent Hamiltonians on a coarse momentum grid

TL;DR

This paper introduces a new spectral shift method for calculating phase shifts in scattering problems that maintains unitarity on a finite momentum grid, improving accuracy over previous methods.

Contribution

It presents a novel spectral shift approach based on Chebyshev-angle variables that preserves the unitary equivalence of the S-matrix on discretized momentum grids.

Findings

Spectral shift formula yields more accurate phase shifts.

Method is effective for non-relativistic NN scattering in specific channels.

Approach is competitive with standard phase shift calculation procedures.

Abstract

The solution of the scattering problem based on the Lippmann-Schwinger equation requires in many cases a discretization of the spectrum in the continuum which does not respect the unitary equivalence of the S-matrix on the finite grid. We present a new prescription for the calculation of phase shifts based on the shift that is produced in the spectrum of a Chebyshev-angle variable. This is analogous to the energy shift that is produced in the energy levels of a scattering process in a box, when an interaction is introduced. Our formulation holds for any momentum grid and preserves the unitary equivalence of the scattering problem on the finite momentum grid. We illustrate this procedure numerically considering the non-relativistic NN case for $^1S_0$ and $^3S_1$ channels. Our spectral shift formula provides much more accurate results than the previous ones and turns out to be at least as competitive as the standard procedures for calculating phase shifts.