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Time development of exponentially small nonadiabatic transitions
 Commun. Math. Phys
, 2004
"... Optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors. In this paper, we rigorously determine the leading order non–adiabatic corrections to these approximations for a particular family of two ..."
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Cited by 12 (6 self)
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Optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors. In this paper, we rigorously determine the leading order non–adiabatic corrections to these approximations for a particular family of two–level analytic Hamiltonian functions. Our results capture the time development of the exponentially small transition that takes place between optimal states by means of a particular switching function. Our results confirm the physics predictions of Sir Michael Berry in the sense that the switching function for this family of Hamiltonians has the form that he argues is universal.
Correlation between pole location and asymptotic behavior for Painlevé I solutions
 COMM. PURE AND APPL. MATH. VOL. LII
, 1997
"... We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show that the extension provides a method of finding singularities of solutions of nonlinear differential equations, using asymptotic information. This transasymptotic matching method is app ..."
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Cited by 10 (4 self)
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We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show that the extension provides a method of finding singularities of solutions of nonlinear differential equations, using asymptotic information. This transasymptotic matching method is applied to Painlev'e's first equation. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions, obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole xp(C) on R + of a solution is monotonic in a parameter C describing its asymptotics on antistokes lines, and obtain rigorous bounds for xp(C). We also derive the behavior of xp(C) for large C 2 C . The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles.
Analyticity of the free energy of a closed 3manifold
 AND ASYMPTOTICS OF GRAPH COUNTING PROBLEMS IN UNORIENTED SURFACES 23
"... Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative ChernSimons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold ..."
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Cited by 3 (1 self)
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Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative ChernSimons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold is uniformly Gevrey1. As a corollary, it follows that the genus g part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of BenderGaoRichmond on the asymptotics of the number of rooted maps for arbitrary genus. We illustrate our results with an explicit formula for the free energy of a Lens space. In addition, using the Painlevé differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of BenderGaoRichmond.
Movable singularities of solutions of difference equations in relation to solvability, and study of a superstable fixed point Theoretical and
 Mathematical Physics
, 2002
"... Abstract. We overview applications exponential asymptotics and analyzable function theory to difference equations, in defining an analog of the Painlevé property for them and we sketch the conclusions with respect to the solvability properties of first order autonomous ones. It turns out that if the ..."
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Cited by 2 (1 self)
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Abstract. We overview applications exponential asymptotics and analyzable function theory to difference equations, in defining an analog of the Painlevé property for them and we sketch the conclusions with respect to the solvability properties of first order autonomous ones. It turns out that if the Painlevé property is present the equations are explicitly solvable and in the contrary case, under further assumptions, the integrals of motion develop singularity barriers. We apply the method to the logistic map xn+1 = axn(1 − xn) where it turns out that the only cases with the Painlevé property are a = −2, 0, 2 and 4 for which explicit solutions indeed exist; in the opposite case an associated conjugation map develops singularity barriers. 1.
Rigorous bounds of Stokes constants for some nonlinear ordinary differential equations at rankone irregular singularities
"... Abstract. A rigorous way to obtain sharp bounds for Stokes constants is introduced and illustrated on a concrete problem arising in applications. ..."
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Cited by 1 (0 self)
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Abstract. A rigorous way to obtain sharp bounds for Stokes constants is introduced and illustrated on a concrete problem arising in applications.
RESURGENCE OF THE KONTSEVICHZAGIER POWER SERIES OVIDIU COSTIN AND STAVROS GAROUFALIDIS
, 2008
"... Abstract. The paper is concerned with the KontsevichZagier formal power series ..."
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Abstract. The paper is concerned with the KontsevichZagier formal power series
SINGULARITY BARRIERS AND BOREL PLANE ANALYTIC PROPERTIES OF 1 + DIFFERENCE EQUATIONS
, 2006
"... Abstract. The paper addresses generalized Borel summability of “1 + ” difference equations in “critical time”. We show that the Borel transform Y of a prototypical such equation is analytic and exponentially bounded for ℜ(p) < 1 but there is no analytic continuation from 0 toward +∞: the vertical li ..."
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Abstract. The paper addresses generalized Borel summability of “1 + ” difference equations in “critical time”. We show that the Borel transform Y of a prototypical such equation is analytic and exponentially bounded for ℜ(p) < 1 but there is no analytic continuation from 0 toward +∞: the vertical line ℓ: = {p: ℜ(p) = 1} is a singularity barrier of Y. There is a unique natural continuation through the barrier, based on the Borel equation dual to the difference equation, and the functions thus obtained are analytic and decaying on the other side of the barrier. In this sense, the Borel transforms are analytic and well behaved in C \ ℓ. The continuation provided allows for generalized Borel summation of the formal solutions. It differs from standard “pseudocontinuation ” [9]. This stresses the importance of the notion of cohesivity, a comprehensive extension of analyticity introduced and thoroughly analyzed by Écalle. We also discuss how, in some cases, Écalle acceleration can provide a procedure of natural continuation beyond a singularity barrier. 1.