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Paolo Amore

Publications and source records attributed to Paolo Amore.

At least 109 records · Page 6Linked to original sources
arXiv2007

Rational Approximation for Two-Point Boundary value problems

We propose a method for the treatment of two--point boundary value problems given by nonlinear ordinary differential equations. The approach leads to sequences of roots of Hankel determinants that converge rapidly towards the unknown parameter of the problem. We treat several problems of physical interest: the field equation determining the vortex profile in a Ginzburg--Landau effective theory, the fixed--point equation for Wilson's exact renormalization group, a suitably modified Wegner--Houghton's fixed point equation in the local potential approximation, a Riccati equation, and the Thomas--Fermi equation. We consider two models where the approach does not apply in order to show the limitations of our Padé--Hankel approach.

math-ph↗
arXiv2007

A new approach to resummation: Parametric Perturbation Theory

We present a {\sl non--perturbative} method, called {\sl Parametric Perturbation Theory} (PPT), which is alternative to the ordinary perturbation theory. The method relies on a principle of simplicity for the observable solutions, which are constrained to be linear in a certain (unphysical) parameter. The perturbative expansion is carried out in this parameter and not in the physical coupling (as in ordinary perturbation theory). We show that the method is capable to resum the divergent perturbative series, to {\sl extract} the leading asymptotic (strong coupling) behavior and {\sl predict} with high accuracy the coefficients of the perturbative series.

cond-mat.other↗
arXiv2007

Going beyond perturbation theory: Parametric Perturbation Theory

We devise a {\sl non--perturbative} method, called {\sl Parametric Perturbation Theory} (PPT), which is alternative to the ordinary perturbation theory. The method relies on a principle of simplicity for the observable solutions, which are constrained to be linear in a certain (unphysical) parameter. The perturbative expansion is carried out in this parameter and not in the physical coupling (as in ordinary perturbation theory). We provide a number of nontrivial examples, where our method is capable to resum the divergent perturbative series, {\sl extract} the leading asymptotic (strong coupling) behavior and {\sl predict} with high accuracy the coefficients of the perturbative series. In the case of a zero dimensional field theory we prove that PPT can be used to provide the imaginary part of the solution, when the problem is analytically continued to negative couplings. In the case of a $ϕ^4$ lattice model 1+1 and of elastic theory we have shown that the observables resummed with PPT display a branch point at a finite value of the coupling, signaling the transition from a stable to a metastable state. We have also applied the method to the prediction of the virial coefficients for a hard sphere gas in two and three dimensions; in this example we have also found that the solution resummed with PPT has a singularity at finite density. Predictions for the unknown virial coefficients are made.

hep-ph↗
arXiv2007

A new representation for non--local operators and path integrals

We derive an alternative representation for the relativistic non--local kinetic energy operator and we apply it to solve the relativistic Salpeter equation using the variational sinc collocation method. Our representation is analytical and does not depend on an expansion in terms of local operators. We have used the relativistic harmonic oscillator problem to test our formula and we have found that arbitrarily precise results are obtained, simply increasing the number of grid points. More difficult problems have also been considered, observing in all cases the convergence of the numerical results. Using these results we have also derived a new representation for the quantum mechanical Green's function and for the corresponding path integral. We have tested this representation for a free particle in a box, recovering the exact result after taking the proper limits, and we have also found that the application of the Feynman--Kac formula to our Green's function yields the correct ground state energy. Our path integral representation allows to treat hamiltonians containing non--local operators and it could provide to the community a new tool to deal with such class of problems.

hep-th↗
arXiv2006

Comment to ``Quantization of FRW spacetimes in the presence of a cosmological constant and radiation''

The quantization of the Friedmann-Robertson-Walker spacetime in the presence of a negative cosmological constant was used in a recent paper to conclude that there are solutions that avoid singularities (big bang--big crunch) at the quantum level. We show that a proper study of their model{\sl does not prevent} the occurrence of singularities at the quantum level, in fact the quantum probability of such event is larger than the classical one. Our numerical simulations based on the powerful variational sinc collocation method (VSCM) also show that the precision of the results of that paper is much lower than the 20 significant digits reported by the authors.

gr-qc↗
arXiv2006

The emergence of AdS(2) from quantum fluctuations

We have shown how the quantization of two-dimensional quantum gravity with an action which contains only a positive cosmological constant and boundary cosmological constants leads to the emergence of a spacetime which can be described as a constant negative curvature spacetime with superimposed quantum fluctuations.

gr-qc↗
arXiv2006

Iterative solution of differential equations

We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.

math-ph↗
arXiv2006

Variational collocation on finite intervals

In this paper we study a new family of sinc--like functions, defined on an interval of finite width. These functions, which we call ``little sinc'', are orthogonal and share many of the properties of the sinc functions. We show that the little sinc functions supplemented with a variational approach enable one to obtain accurate results for a variety of problems. We apply them to the interpolation of functions on finite domain and to the solution of the Schrödinger equation, and compare the performance of present approach with others.

quant-ph↗
arXiv2006

Wronskian perturbation theoryWronskian perturbation theory

We develop a perturbation method that generalizes an approach proposed recently to treat velocity--dependent quantum--mechanical models. In order to test present approach we apply it to some simple trivial and nontrivial examples.

quant-ph↗
arXiv2006

Development of accurate solutions for a classical oscillator

We present a method to obtain arbitrarily accurate solutions for conservative classical oscillators. The method that we propose here works both for small and large nonlinearities and provides simple analytical approximations. A comparison with the standard Lindstedt-Poincaré method is presented, from which the advantages of our method are clear.

math-ph↗
arXiv2006

Analytical formulas for gravitational lensing

In this paper we discuss a new method which can be used to obtain arbitrarily accurate analytical expressions for the deflection angle of light propagating in a given metric. Our method works by mapping the integral into a rapidly convergent series and provides extremely accurate approximations already to first order. We have derived a general first order formula for a generic spherically symmetric static metric tensor and we have tested it in four different cases.

gr-qc↗
arXiv2006

Analytical approximations to the spectra of quark-antiquark potentials

A method, recently devised to obtain analytical approximations to certain classes of integrals, is used in combination with the WKB expansion to derive accurate analytical expressions for the spectrum of quantum potentials. The accuracy of our results is verified by comparing them both with the literature on the subject and with the numerical results obtained with a Fortran code. As an application of the method that we propose, we consider the meson spectroscopy with various phenomenological potentials.

hep-ph↗
arXiv2006

A variational sinc collocation method for strong-coupling problems

We have devised a variational sinc collocation method (VSCM) which can be used to obtain accurate numerical solutions to many strong-coupling problems. Sinc functions with an optimal grid spacing are used to solve the linear and non-linear Schrödinger equations and a lattice $ϕ^4$ model in $(1+1)$. Our results indicate that errors decrease exponentially with the number of grid points and that a limited numerical effort is needed to reach high precision.

cond-mat.other↗
arXiv2005

Convergence acceleration of series through a variational approach

By means of a variational approach we find new series representations both for well known mathematical constants, such as $π$ and the Catalan constant, and for mathematical functions, such as the Riemann zeta function. The series that we have found are all exponentially convergent and provide quite useful analytical approximations. With limited effort our method can be applied to obtain similar exponentially convergent series for a large class of mathematical functions.

math-ph↗
arXiv2005

Non perturbative series for the calculation of one loop integrals at finite temperature

The calculation of one loop integrals at finite temperature requires the evaluation of certain series, which converge very slowly or can even be divergent. Here we review a new method, recently devised by the author, for obtaining accelerated analytical expressions for these series. The fundamental properties of the new series are studied and an application to a physical example is considered. The relevance of the method to other physical problems is also discussed.

hep-th↗