A multiplicative product of distributions and a global formulation of the confined Schrodinger equation

TL;DR

This paper introduces a new associative multiplicative product for Schwartz distributions, enabling a global formulation of the confined Schrödinger equation in quantum systems.

Contribution

It presents a novel multiplicative product for distributions and applies it to derive a global formulation of the confined Schrödinger equation.

Findings

New associative product for Schwartz distributions introduced.

Provides a framework for global quantum confined systems.

Enables derivation of a global Schrödinger equation formulation.

Abstract

A concise derivation of a new multiplicative product of Schwartz distributions is presented. The new product $\star$ is defined in the vector space ${\cal A}$ of piecewise smooth functions on $\bkR$ and all their (distributional) derivatives; it is associative, satisfies the Leibnitz rule and reproduces the usual product of functions for regular distributions. The algebra $({\cal A},+,\star)$ yields a sufficiently general setting to address some interesting problems. As an application we consider the problem of deriving a global formulation for quantum confined systems.