Abstract
Differential equations with infinitely many derivatives, sometimes also referred to as “nonlocal” differential equations, appear frequently in branches of modern physics such as string theory, gravitation and cosmology. We properly interpret and solve linear equations in this class with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous studies of nonlocal equations via Laplace and Fourier transforms, see for instance (Barnaby and Kamran, J High Energy Phys 02:40, 2008; Górka et al., Class Quantum Gravity 29:065017, 2012; Górka et al., Ann Henri Poincaré 14:947–966, 2013). We reconsider “generalized” initial value problems within the present approach and we disprove various conjectures found in modern physics literature. We illustrate various analytic phenomena that can occur with concrete examples, and we also treat efficient implementations of the theory.
Article PDF
Similar content being viewed by others
References
Aref’eva, I.Y., Volovich, I.V.: Cosmological daemon. J. High Energy Phys. 08, 102 (2011). arXiv:1103.0273
Barnaby N.: A new formulation of the initial value problem for nonlocal theories. Nucl. Phys. B 845, 1–29 (2011)
Barnaby, N., Biswas, T., Cline, J.M.: p-Adic inflation. J. High Energy Phys. 04, 35 (2007) (Paper 056)
Barnaby, N., Kamran, N.: Dynamics with infinitely many derivatives: the initial value problem. J. High Energy Phys. 02, 40 (2008) (Paper 008)
Barnaby, N., Kamran, N.: Dynamics with infinitely many derivatives: variable coefficient equations. J. High Energy Phys. 12, 27 (2008) (Paper 022)
Boas R.P.: Entire Functions. Academic Press, New York (1954)
Calcagni G., Montobbio M., Nardelli G.: Route to nonlocal cosmology. Phys. Rev. D 76, 126001 (2007)
Calcagni G., Montobbio M., Nardelli G.: Localization of nonlocal theories. Phys. Lett. B 662, 285–289 (2008)
Carleson L.: On infinite differential equations with constant coefficients. I. Math. Scand. 1, 31–38 (1953)
Carmichael R.D.: On non-homogeneous linear differential equations of infinite order with constant coefficients. Am. J. Math. 58(3), 473–486 (1936)
Carmichael R.D.: Linear differential equations of infinite order. Bull. AMS 42, 193–218 (1936)
Cooley J.W., Tukey J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)
Dubinskii Y.A.: The algebra of pseudodifferential operators with analytic symbols and its applications to mathematical physics. Russ. Math. Surv. 37, 109–153 (1982)
Eliezer D.A., Woodard R.P.: The problem of nonlocality in string theory. Nucl. Phys. B 325, 389–469 (1989)
Górka P., Prado H., Reyes E.G.: Nonlinear equations with infinitely many derivatives. Complex Anal. Oper. Theory 5, 313–323 (2011)
Górka P., Prado H., Reyes E.G.: Functional calculus via Laplace transform and equations with infinitely many derivatives. J. Math. Phys. 51, 103512 (2010)
Górka P., Prado H., Reyes E.G.: The initial value problem for ordinary equations with infinitely many derivatives. Class. Quantum Gravity 29, 065017 (2012)
Górka P., Prado H., Reyes E.G.: On a general class of nonlocal equations. Ann. Henri Poincaré 14, 947–966 (2013)
Hörmander L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (1990)
Hörmander L.: The Analysis of Linear Partial Differential Operators II. Springer, Berlin (1983)
Hörmander L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlin (1994)
Kawai T., Struppa D.C.: On the existence of holomorphic solutions of systems of linear differential equations of infinite order and with constant coefficients. Int. J. Math. 1, 63–82 (1990)
Kawai T., Struppa D.C.: Overconvergence phenomena and grouping in exponential representation of solutions of linear differential equations of infinite order. Adv. Math. 161, 131–140 (2001)
Koosis P.: Introduction to H p-Spaces, 2nd edn. Cambrige University Press, Cambrige (1998)
Levin, B.J.: Distribution of zeros of entire functions. In: Translations of Mathematical Monographs, revised edn, vol. 5. American Mathematical Society, Providence (1980)
Moeller, N., Zwiebach, B.: Dynamics with infinitely many time derivatives and rolling tachyons. J. High Energy Phys. 10, 38 (2002) (Paper 34)
Pólya G.: Analytische Fortsetzung una konvexe Kurven. Math. Ann. 89, 179–191 (1923)
Ritt J.F.: On a general class of linear homogeneous equations of infinite order with constant coefficients. Trans. Am. Math. Soc. 3, 27–49 (1917)
Sato M., Kashiwara M., Kawai T.: Linear differential equations of infinite order and theta functions. Adv. Math. 47, 300–325 (1983)
Schwartz L.: Théorie générale des fonctions moyenne-périodiques. Ann. Math. 48, 857–929 (1947)
Teixeira E.: On infinite order and fully nonlinear partial differential evolution equations. J. Differ. Equ. 238, 43–63 (2007)
Van Tran Duc, Hào Dinh Nho: Differential Operators of Infinite Order with Real Arguments and Their Applications. World Scientific, Singapore (1994)
Vladimirov V.S.: The equation of the p-adic open string for the scalar tachyon field. Izv. Math. 69, 487–512 (2005)
Vladimirov, V.S., Volovich, Ya.I.: Nonlinear dynamics equation in p-adic string theory. Teoret. Mat. Fiz. 138, 355–368 (2004) [English transl., Theor. Math. Phys. 138, 297–309 (2004)]
Witten E.: Noncommutative Geometry and String Field Theory. Nucl. Phys. B 268, 253–294 (1986)
Young R.M.: An Introduction to Non-Harmonic Fourier Series. Academic Press, New York (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Nader Masmoudi.
M. Carlsson was supported by DICYT-USACH and STINT; H. Prado’s research was partially supported by FONDECYT # 1130554; E. G. Reyes’ research was partially supported by FONDECYT Grant # 1111042 and the USACH DICYT Grant Código 041533RG.
Rights and permissions
About this article
Cite this article
Carlsson, M., Prado, H. & Reyes, E.G. Differential Equations with Infinitely Many Derivatives and the Borel Transform. Ann. Henri Poincaré 17, 2049–2074 (2016). https://doi.org/10.1007/s00023-015-0447-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-015-0447-4