The initial value problem for ordinary differential equations with infinitely many derivatives

TL;DR

This paper develops a Lorentzian functional calculus to analyze the initial value problem for differential equations with infinitely many derivatives, establishing conditions for existence, uniqueness, and regularity of solutions.

Contribution

It introduces a rigorous operator interpretation for equations with infinitely many derivatives and characterizes well-posed initial value problems with finite initial conditions.

Findings

Established a Lorentzian calculus via Laplace transform for such equations

Derived the most general solution in exponentially bounded function space

Proved conditions for well-posed initial value problems with finite initial data

Abstract

We study existence, uniqueness and regularity of solutions for ordinary differential equations with infinitely many derivatives such as (linearized versions of) nonlocal field equations of motion appearing in particle physics, nonlocal cosmology and string theory. We develop an appropriate Lorentzian functional calculus via Laplace transform which allows us to interpret rigorously an operator of the form $f(\partial_t)$ on the half line, in which $f$ is an analytic function. We find the most general solution to the equation $f(\partial_t) \phi = J(t)$ (t greater or equal to 0) in the space of exponentially bounded functions, and we also analyze in full detail the delicate issue of the initial value problem. In particular, we state conditions under which the solution $\phi$ admits a finite number of derivatives, and we prove rigorously that if an a priori data directly connected with our Lorentzian calculus is specified, then the initial value problem is well-posed and it requires only a finite number of initial conditions.