Noether's Theorem with Momentum and Energy Terms for Cresson's Quantum Variational Problems

TL;DR

This paper extends Noether's theorem to Cresson's quantum variational calculus, establishing symmetry and conservation laws for nondifferentiable functionals, with applications to the Schrödinger equation.

Contribution

It introduces a Noether symmetry theorem and DuBois-Reymond condition within Cresson's quantum calculus, addressing nondifferentiable functions.

Findings

Derived a constant of motion for the Schrödinger equation

Established a DuBois-Reymond optimality condition

Proved a Noether symmetry theorem in quantum calculus

Abstract

We prove a DuBois-Reymond necessary optimality condition and a Noether symmetry theorem to the recent quantum variational calculus of Cresson. The results are valid for problems of the calculus of variations with functionals defined on sets of nondifferentiable functions. As an application, we obtain a constant of motion for a linear Schrodinger equation.