[Seminar] "On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations" by Gergő Nemes

Speaker: Gergő Nemes

Title: On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations

Abstract: We will consider a class of $n$th-order linear ordinary differential equations with a large parameter $u$. Analytic solutions of these equations can be described by (divergent) formal series in descending powers of $u$. We shall demonstrate that, given mild conditions on the potential functions of the equation, the formal solutions are Borel summable with respect to the parameter $u$ in large, unbounded domains of the independent variable. We will establish that the formal series expansions serve as asymptotic expansions, uniform with respect to the independent variable, for the Borel re-summed exact solutions. Additionally, the exact solutions can be expressed using factorial series in the parameter, and these expansions converge in half-planes, uniformly with respect to the independent variable. To illustrate our theory, we apply it to a third-order Airy-type equation.

Profile: Gergő Nemes is a specially appointed associate professor at Tokyo Metropolitan University. His research interests include exponential asymptotics, hyperasymptotics, exact WKB analysis and special functions. He completed his PhD under the supervision of Árpád Tóth at the Central European University in Budapest. He subsequently held positions at the University of Edinburgh, at Kindai University and at the Alfréd Rényi Institute of Mathematics. Since 2022, he is a contributing developer of the NIST Digital Library of Mathematical Functions. https://users.renyi.hu/~gergonemes/

This seminar is a part of the first TSVP Thematic Program "Exact Asymptotics: From Fluid Dynamics to Quantum Geometry" (https://groups.oist.jp/tsvp/exact-asymptotics). OIST students and researchers are welcome to participate in all scientific sessions of the program without registration.

List and profiles of program participants can be found here: https://groups.oist.jp/tsvp/23ea-program-participants