A novel way of calculating scattering integrals

TL;DR

This paper demonstrates how the NDIM technique, originally used for Feynman integrals, can be effectively applied to evaluate improper integrals in quantum mechanics, offering a more straightforward alternative to traditional methods.

Contribution

The work extends NDIM to improper integrals, providing a new, pole-free approach for their calculation in quantum mechanical scattering problems.

Findings

NDIM successfully evaluates improper integrals in quantum mechanics.

The method avoids complex residue calculations and contour integrations.

Results align with traditional approaches, confirming validity.

Abstract

The technique coined as NDIM - Negative Dimensional Integration Method by their discoverers, relies on a three-pronged basis: Gaussian integration, series expansion and analytic continuation. The technique has been successfully applied to the calculation of covariant and non covariant Feynman integrals in a generic dimensional regularization space, i.e., D-dimensional space-time for D including the negative domain values. Since the dimensionality is general, we can use specifically for one-dimensional integrals. In this work we show how this technique can be applied to tackle certain improper integrals and give an example of a particular improper integral that appears in quantum mechanical scattering process. Traditionally, improper integrals are ascribed certain values through the limiting approach or as is known, by the Cauchy principal value via residues concept technique. Here we use the NDIM approach to do the calculations and show it works fine for the improper integrals. This novel approach we believe is more straightforward and does not require to handle poles, residues, or difficult closed contours as in the traditional approach.