256 research outputs found

    Integral representations for Padé-type operators

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    The main purpose of this paper is to consider an explicit form of the Padé-type operators. To do so, we consider the representation of Padé-type approximants to the Fourier series of the harmonic functions in the open disk and of the L p-functions on the circle by means of integral formulas, and, then we define the corresponding Padé-type operators. We are also oncerned with the properties of these integral operators and, in this connection, we prove some convergence results

    Convergence and Asymptotic of Multi-Level Hermite-Padé Polynomials

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    Mención Internacional en el título de doctorPrograma de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Francisco José Marcellán Español.- Secretario: Bernardo de la Calle Ysern.- Vocal: Arnoldus Bernardus Jacobus Kuijla

    Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them

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    We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree--Fock approximation and Rayleigh--Schr\"odinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within M{\o}ller--Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Pad\'e and quadratic approximants) that can improve the overall accuracy of the M{\o}ller--Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.Comment: 22 page, 14 figures, 4 table

    Uniformization and Constructive Analytic Continuation of Taylor Series

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    We analyze the general mathematical problem of global reconstruction of a function with least possible errors, based on partial information such as n terms of a Taylor series at a point, possibly also with coefficients of finite precision. We refer to this as the "inverse approximation theory problem, because we seek to reconstruct a function from a given approximation, rather than constructing an approximation for a given function. Within the class of functions analytic on a common Riemann surface Omega, and a common Maclaurin series, we prove an optimality result on their reconstruction at other points on Omega, and provide a method to attain it. The procedure uses the uniformization theorem, and the optimal reconstruction errors depend only on the distance to the origin. We provide explicit uniformization maps for some Riemann surfaces Omega of interest in applications. One such map is the covering of the Borel plane of the tritronquee solutions to the Painleve equations PI-PV. As an application we show that this uniformization map leads to dramatic improvement in the extrapolation of the PI tritronquee solution throughout its domain of analyticity and also into the pole sector. Given further information about the function, such as is available for the ubiquitous class of resurgent functions, significantly better approximations are possible and we construct them. In particular, any one of their singularities can be eliminated by specific linear operators, and the local structure at the chosen singularity can be obtained in fine detail. More generally, for functions of reasonable complexity, based on the nth order truncates alone we propose new efficient tools which are convergent as n to infty, which provide near-optimal approximations of functions globally, as well as in their most interesting regions, near singularities or natural boundaries.Comment: 39 pages, 9 figures; v2 some clarifications adde
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