Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 7

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

Pages: 1771 – 1804

DOI: https://dx.doi.org/10.4310/ATMP.2019.v23.n7.a2

Authors

A. Chávez (Departamento de Matemáticas, Universidad Nacional de Trujillo, Perú)

H. Prado (Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Santiago, Chile)

E. G. Reyes (Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Santiago, Chile)

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\[f(\partial_t) \phi = J(t) \quad , \quad t \geq 0 \: ,\]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 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\[\zeta(\partial_t + h) \phi = J(t) \quad , \quad t \geq 0 \; ,\]in which $\zeta$ is the Riemann zeta function and $h \gt 1$. 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 $\partial_t$.

Published 15 May 2020