Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 10

The Borel transform and linear nonlocal equations: applications to zeta-nonlocal field models

Pages: 3487 – 3535

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n10.a3

Authors

Alan Chavez (Instituto de Investigación en Matemáticas & Departamento de Matemáticas, Universidad Nacional de Trujillo, Perú)

Humberto Prado (Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile (USACH), Santiago, Chile)

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

Abstract

We define rigorously operators of the form $f(\partial_t)$, in which $f$ is an analytic function on a simply connected domain. Our formalism is based on the Borel transform on entire functions of exponential type. We study existence and regularity of real-valued solutions for the nonlocal in time equation\[f(\partial_t)\phi = J(t) \quad , t \quad \in \mathbb{R} \quad ,\]and we find its more general solution as a restriction to $\mathbb{R}$ of an entire function of exponential type. As an important special case, we solve explicitly the linear nonlocal zeta field equation[\zeta (\partial^2_t + h) \phi = J(t) \quad ,\]in which $h$ is a real parameter, $\zeta$ is the Riemann zeta function, and $J$ is an entire function of exponential type. We also analyse the case in which $J$ is a more general analytic function (subject to some weak technical assumptions). This latter case turns out to be rather delicate: we need to re-interpret the symbol $\zeta (\partial^2_t + h)$. We prove that in this case the zeta-nonlocal equation above admits an analytic solution on a simply connected domain determined by $J$.

<p>The linear zeta field equation is a linear version of a field model depending on the Riemann zeta function arising in $p$-adic string theory [B. Dragovich, Zeta-nonlocal scalar fields, Theoret. Math.Phys., 157 (2008), 1671–1677].

Published 25 March 2024