The initial value problem for ordinary differential equations with infinitely many derivatives

Details The initial value problem for ordinary differential equations with infinitely many derivatives The initial value problem for ordinary differential equations with infinitely many derivatives

Mar 2012

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2012cqgra..29f5017g&link_type=abstract

Classical and Quantum Gravity, Volume 29, Issue 6, pp. 065017 (2012).

Physics

Scientific paper

We study existence, uniqueness and regularity of solutions for ordinary differential equations with infinitely many derivatives such as (linearized versions of) nonlocal field equations of motion appearing in particle physics, nonlocal cosmology and string theory. We develop a Lorentzian functional calculus via Laplace transform which allows us to interpret rigorously operators of the form f(∂t) on the half line, in which f is an analytic function. We find the most general solution to the equation \begin{equation*}
f(\partial _t) \phi = J(t), \quad t \ge 0,
\end{equation*}
in the space of exponentially bounded functions, and we also analyze in full detail the delicate issue of the initial value problem. In particular, we state conditions under which the solution ϕ admits a finite number of derivatives, and we prove rigorously that if a finite set of a priori data directly connected with our Lorentzian calculus is specified, then the initial value problem is well posed and it requires only a finite number of initial conditions.

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