On eigenfunctions corresponding to a small resurgent eigenvalue

Article type: Research Article

Authors: Getmanenko, Alexander

Affiliations: Department of Mathematics, Northwestern University, Evanston IL, USA; Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany; Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8568, Japan. Fax: +81 4 7136 4941; E-mail: Alexander.Getmanenko@ipmu.jp

Abstract: The paper is devoted to some foundational questions in resurgent analysis. As a main technical result, it is shown that under appropriate conditions the infinite sum of endlessly continuable majors commutes with the Laplace transform. A similar statement is proven for compatibility of a convolution and of an infinite sum of majors. We generalize the results of Candelpergher–Nosmas–Pham and prove a theorem about substitution of a small (extended) resurgent function into a holomorphic parameter of another resurgent function. Finally, we discuss an application of these results to the question of resurgence of eigenfunctions of a one-dimensional Schrödinger operator corresponding to a small resurgent eigenvalue.

Keywords: resurgence, Laplace transform, linear ODEs

DOI: 10.3233/ASY-2011-1068

Journal: Asymptotic Analysis, vol. 76, no. 2, pp. 87-114, 2012