282 research outputs found

    Effective resummation methods for an implicit resurgent function

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    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

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    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

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    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 PmP_m is a monic polynomial function of order mm 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 PmP_m. In particular, we obtain conditions for isomonodromic deformations when m=3m=3.Comment: 54 pages, 2 figures. To appear in Pac. Math.

    Bound States of Non-Hermitian Quantum Field Theories

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    The spectrum of the Hermitian Hamiltonian 12p2+12m2x2+gx4{1\over2}p^2+{1\over2}m^2x^2+gx^4 (g>0g>0), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian H=12p2+12m2x2−gx4H={1\over2}p^2+{1 \over2}m^2x^2-gx^4, where the coupling constant gg is real and positive, is PT{\cal PT}-symmetric. As a consequence, the spectrum of HH is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When gg 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 PT{\cal PT}-symmetric −gϕ4-g\phi^4 quantum field theory for all dimensions 0≀D<30\leq D<3 but is not present in the conventional Hermitian gϕ4g\phi^4 field theory.Comment: 14 pages, 3figure

    Introduction to the Écalle theory

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

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    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 PainlevĂ© 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 PainlevĂ© 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&#039;opérateur de Schrödinger stationnaire unidimensionnel à potentiel polynÎme trigonométrique

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    On Ă©tudie le spectre de l\u27opĂ©rateur de Schrödinger H= -x-2d2/dq2+V(q) pour un potentiel V (q) polynĂŽme trigonomĂ©trique rĂ©el de pĂ©riode 2 π,(1/h)Ă©tant considĂ©rĂ© comme un grand paramĂštre rĂ©el positif. On dĂ©crit la structure rĂ©surgente en x du problĂšme puis on applique la mĂ©thode semi-classique exacte au cas oĂč V (q)=1+cos (q). On dĂ©montre ainsi une conjecture de Zinn-Justi

    Exact WKB analysis near a simple turning point

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    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

    Hyperasymptotics for multidimensional Laplace integrals with boundaries

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    Singular integrals and the stationary phase methods

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    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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