Scale calculus and the Schrödinger equation

This paper is twofold. In a first part, we extend the classical differential calculus to continuous non differentiable functions by developping the notion of scale calculus. The scale calculus is based on a new approach of continuous non differentiable functions by constructing a one parameter family of differentiable functions f(t, ǫ) such that f(t, ǫ) → f(t) when ǫ goes to zero. This lead to several new notions as representation, fractal functions and ǫ-differentiability. The basic objets of the scale calculus are left and right quantum operators and the scale operator which generalize the classical derivative. We then discuss some algebraic properties of these operators. We define a natural bialgebra, called quantum bialgebra, associated to them. Finally, we discuss a convenient geometric object associated to our study. In a second part, we define a first quantization procedure of classical mechanics following the scale relativity theory developped by Nottale. We obtain a non linear Schrödinger equation via the classical Newton’s equation of dynamics using the scale operator. Under special assumptions we recover the classical Schrödinger equation and we discuss the relevance of these assumptions.