The complex Dirac Delta, Plemelj formula, and integral representations

TL;DR

This paper extends the Dirac Delta distribution to complex fields using Gaussian regularization, providing a new interpretation and extension of the Sokhotski-Plemelj formula relevant for quantum scattering theory.

Contribution

It introduces a novel complex extension of the Dirac Delta and Plemelj formula using Gaussian regularization, with a new distribution interpretation based on path prescriptions.

Findings

Extended Dirac Delta to complex plane with Gaussian regularization

Derived an alternative Sokhotski-Plemelj formula extension

Provided a new distribution framework for complex-energy solutions

Abstract

The extension of the Dirac Delta distribution (DD) to the complex field is needed for dealing with the complex-energy solutions of the Schr\"odinger equation, typically when calculating their inner products. In quantum scattering theory the DD usually arises as an integral representation involving plane waves of real momenta. We deal with the complex extension of these representations by using a Gaussian regularization. Their interpretation as distributions requires prescribing the integration path and a corresponding space of test functions. An extension of the Sokhotski-Plemelj formula is obtained. This definition of distributions is alternative to the historic one referred to surface integrations on the complex plane.