Divergent Series, Summability and Resurgence III. Resurgent Methods and the First Painlevé Equation

The aim of this volume is two-fold. First, to show how the resurgent methods can be applied efficiently in a non-linear setting; to this end further properties of the resurgence theory are developed. Second, to analyze the fundamental example of the First Painlevé equation. The resurgent analysis of singularities is pushed all the way up to the so-called “bridge equation”, which concentrates all information about the non-linear Stokes phenomenon at infinity of the First Painlevé equation.  The third in a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists who are interested in divergent power series and related problems, such as the Stokes phenomenon.

Spectre de l'opérateur de Schrödinger stationnaire unidimensionnel à potentiel polynôme trigonométrique

On étudie le spectre de l\u27opérateur de Schrödinger H= -x-2d2/dq2+V(q) pour un potentiel V (q) polynôme trigonométrique réel de période 2 π,(1/h)étant considéré comme un grand paramètre réel positif. On décrit la structure résurgente en x du problème puis on applique la méthode semi-classique exacte au cas où V (q)=1+cos (q). On démontre ainsi une conjecture de Zinn-Justi

Exact WKB analysis near a simple turning point

We extend and propose a new proof for a reduction theorem near a simple turning point due to Aoki et al., in the framework of the exact WKB analysis. Our scheme of proof is based on a Laplace-integral representation derived from an existence theorem of holomorphic solutions for a singular linear partial differential equation

Singular integrals and the stationary phase methods

The paper is based on a course given in 2007 at an ICTP school in Alexandria, Egypt. It aims at introducing young scientists to methods to calculate asymptotic developments of singular integrals.

Spectral analysis of the complex cubic oscillator

Using the `exact semiclassical analysis\u27, we study the spectrum of a one-parameter family of complex cubic oscillators. The PT-invariance property of the complex Hamiltonians and the reality property of the spectrum are discussed. Analytic continuations of the spectrum in the complex parameter and their connections with the resonance problem for the real cubic oscillator are investigated. The global analytic structure of the spectrum yields a branch point structure similar to the multivalued analytic structure discovered by Bender and Wu for the quartic oscillator

Sommation effective d’une somme de Borel par séries de factorielles / Effective Borel-resummation by factorial series

In this article, we consider the effective resummation of a Borel sum by its associated factorial series expansion. Our approach provides concrete estimates for the remainder term when truncating this factorial series. We then generalize a theorem of Nevanlinna which gives us the natural framework to extend the factorial series method for Borel-resummable fractional power series expansions