256 research outputs found
Integral representations for Padé-type operators
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
MenciĂłn Internacional en el tĂtulo de doctorPrograma de Doctorado en IngenierĂa Matemática por la Universidad Carlos III de MadridPresidente: Francisco JosĂ© 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
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
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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