283 research outputs found
Effective resummation methods for an implicit resurgent function
Our main aim in this self-contained article is at the same time to detail the
relationships between the resurgence and the hyperasymptotic theories, and to
demonstrate how these theories can be used for an implicit resurgent function.
For this purpose we consider after Stokes the question of the effective
Borel-resummation of an exact Bohr-Sommerfeld-like implicit resurgent function
whose values on an explicit semi-lattice provide the zeros of the Airy
function. The resurgent structure encountered resembles what one usually gets
in nonlinear problems, so that the method described here is quite general
Exact solution for Morse oscillator in PT-symmetric quantum mechanics
The recently proposed PT-symmetric quantum mechanics works with complex
potentials which possess, roughly speaking, a symmetric real part and an
anti-symmetric imaginary part. We propose and describe a new exactly solvable
model of this type. It is defined as a specific analytic continuation of the
shape-invariant potential of Morse. In contrast to the latter well-known
example, all the new spectrum proves real, discrete and bounded below. All its
three separate subsequences are quadratic in n.Comment: 8 pages, submitted to Phys. Lett.
Resurgent Deformations for an Ordinary Differential Equation of Order 2
We consider in the complex field the differential equation \displaystyle
\frac{d^2}{d x^2} \Phi(x) = \frac{P_m(x,\a)}{x^2}\Phi(x), where is a
monic polynomial function of order with coefficients \a=(a_1, ..., a_m).
We investigate the asymptotic, resurgent, properties of the solutions at
infinity, focusing in particular on the analytic dependence on \a of the
Stokes-Sibuya multipliers. Taking into account the non trivial monodromy at the
origin, we derive a set of functional equations for the Stokes-Sibuya
multipliers. We show how these functional relations can be used to compute the
Stokes multipliers for a class of polynomials . In particular, we obtain
conditions for isomonodromic deformations when .Comment: 54 pages, 2 figures. To appear in Pac. Math.
Bound States of Non-Hermitian Quantum Field Theories
The spectrum of the Hermitian Hamiltonian
(), which describes the quantum anharmonic oscillator, is real and
positive. The non-Hermitian quantum-mechanical Hamiltonian , where the coupling constant is real and positive, is
-symmetric. As a consequence, the spectrum of is known to be
real and positive as well. Here, it is shown that there is a significant
difference between these two theories: When is sufficiently small, the
latter Hamiltonian exhibits a two-particle bound state while the former does
not. The bound state persists in the corresponding non-Hermitian -symmetric quantum field theory for all dimensions
but is not present in the conventional Hermitian field theory.Comment: 14 pages, 3figure
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.
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