2019/07/04 by Chávez, Alan, Prado, Humberto, Reyes, Enríque G.
#30D99 #44A05 #44A10 #47G30 #58J40 #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1907.02617
We define rigorously operators of the form f(∂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(∂t) ϕ= J(t) , t∈ ℝ ,. and we find its more general solution as a restriction to ℝ of an entire function of exponential type. As an important special case, we solve explicitly the linear nonlocal zeta field equation ζ(∂t2+h)ϕ= J(t) , in which h is a real parameter, ζ is the Riemann zeta function, and J is an entire function of exponential type. We also analyze the case in which J is a more general analytic function (subject to some weak technical assumptions). This case turns out to be rather delicate: we need to re-interpret the symbol ζ(∂t2+h) and to leave the class of functions of exponential type. We prove that in this case the zeta-nonlocal equation above admits an analytic solution on a Runge domain determined by J. The linear zeta field equation is a linear version of a field model depending on the Riemann zeta function arising from p-adic string theory.