A Laplace transform approach to linear equations with infinitely many derivatives and zeta-nonlocal field equations

Abstract

We study existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives. We develop a natural framework based on Laplace transform as a correspondence between appropriate $L^p$ and Hardy spaces: this point of view allows us to interpret rigorously operators of the form $f(\partial_t)$ where $f$ is an analytic function such as (the analytic continuation of) the Riemann zeta function. We find the most general solution to the equation \begin{equation*} f(\partial_t) \phi = J(t) \; , \; \; \; t \geq 0 \; , \end{equation*} in a convenient class of functions, we define and solve its corresponding initial value problem, and we state conditions under which the solution is of class $C^k,\, k \geq 0$. More specifically, we prove that if some a priori information is specified, then the initial value problem is well-posed and it can be solved using only a {\em finite number} of local initial data. Also, motivated by some intriguing work by Dragovich and Aref'eva-Volovich on cosmology, we solve explicitly field equations of the form \begin{equation*} \zeta(\partial_t + h) \phi = J(t) \; , \; \; \; t \geq 0 \; , \end{equation*} in which $\zeta$ is the Riemann zeta function and $h > 1$. Finally, we remark that the $L^2$ case of our general theory allows us to give a precise meaning to the often-used interpretation of $f(\partial_t)$ as an operator defined by a power series in the differential operator $\partial_t$.