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

摘要

We study existence, uniqueness and regularity of solutions for linearequations in infinitely many derivatives. We develop a natural framework basedon Laplace transform as a correspondence between appropriate $L^p$ and Hardyspaces: this point of view allows us to interpret rigorously operators of theform $f(partial_t)$ where $f$ is an analytic function such as (the analyticcontinuation of) the Riemann zeta function. We find the most general solutionto the equation egin{equation*} f(partial_t) phi = J(t) ; , ; ; ; tgeq 0 ; , end{equation*} in a convenient class of functions, we define andsolve its corresponding initial value problem, and we state conditions underwhich the solution is of class $C^k,, k geq 0$. More specifically, we provethat if some a priori information is specified, then the initial value problemis well-posed and it can be solved using only a {em finite number} of localinitial data. Also, motivated by some intriguing work by Dragovich andAref'eva-Volovich on cosmology, we solve explicitly field equations of the formegin{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 givea precise meaning to the often-used interpretation of $f(partial_t)$ as anoperator defined by a power series in the differential operator $partial_t$.