arXiv subjects

Guglielmo Fucci

Publications and source records attributed to Guglielmo Fucci.

At least 37 records · Page 2Linked to original sources

arXiv2014

Expansion of Infinite Series Containing Modified Bessel Functions of the Second Kind

The aim of this work is to analyze general infinite sums containing modified Bessel functions of the second kind. In particular we present a method for the construction of a proper asymptotic expansion for such series valid when one of the parameters in the argument of the modified Bessel function of the second kind is small compared to the others. We apply the results obtained for the asymptotic expansion to specific problems that arise in the ambit of quantum field theory.

math-ph↗

arXiv2014

Casimir Pistons with General Boundary Conditions

In this work we analyze the Casimir energy and force for a scalar field endowed with general self-adjoint boundary conditions propagating in a higher dimensional piston configuration. The piston is constructed as a direct product $I\times N$, with $I=[0,L]\subset\mathbb{R}$ and $N$ a smooth, compact Riemannian manifold with or without boundary. The study of the Casimir energy and force for this configuration is performed by employing the spectral zeta function regularization technique. The obtained analytic results depend explicitly on the spectral zeta function associated with the manifold $N$ and the parameters describing the general boundary conditions imposed. These results are then specialized to the case in which the manifold $N$ is a $d$-dimensional sphere.

hep-th↗

arXiv2014

Heat Kernel Asymptotic Expansion on Unbounded Domains with Polynomially Confining Potentials

In this paper we analyze the small-t asymptotic expansion of the trace of the heat kernel associated with a Laplace operator endowed with a spherically symmetric polynomially confining potential on the unbounded, d-dimensional Euclidean space. To conduct this study, the trace of the heat kernel is expressed in terms of its partially resummed form which is then represented as a Mellin-Barnes integral. A suitable contour deformation then provides, through the use of Cauchy's residue theorem, closed formulas for the coefficients of the asymptotic expansion. The general expression for the asymptotic expansion, valid for any dimension and any polynomially confining potential, is then specialized to two particular cases: the general quartic and sestic oscillator potentials.

math-ph↗

arXiv2013

Spectral Functions for Regular Sturm-Liouville Problems

In this paper we provide a detailed analysis of the analytic continuation of the spectral zeta function associated with one-dimensional regular Sturm-Liouville problems endowed with self-adjoint separated and coupled boundary conditions. The spectral zeta function is represented in terms of a complex integral and the analytic continuation in the entire complex plane is achieved by using the Liouville-Green (or WKB) asymptotic expansion of the eigenfunctions associated with the problem. The analytically continued expression of the spectral zeta function is then used to compute the functional determinant of the Sturm-Liouville operator and the coefficients of the asymptotic expansion of the associated heat kernel.

math-ph↗

arXiv2013

Analytic Continuation of the Doubly-periodic Barnes Zeta Function

The aim of this work is to study the analytic continuation of the doubly-periodic Barnes zeta function. By using a suitable complex integral representation as a starting point we find the meromorphic extension of the doubly periodic Barnes zeta function to the entire complex plane in terms of a real integral containing the Hurwitz zeta function and the first Jacobi theta function. These allow us to explicitly give expressions for the derivative at all non-positive integer points.

math-ph↗

arXiv2012

Casimir Effect in the Presence of External Fields

In this work the Casimir effect is studied for scalar fields in the presence of boundaries and under the influence of arbitrary smooth potentials of compact support. In this setting, piston configurations are analyzed in which the piston is modeled by a potential. For these configurations, analytic results for the Casimir energy and force are obtained by employing the zeta function regularization method. Also, explicit numerical results for the Casimir force are provided for pistons modeled by a class of compactly supported potentials that are realizable as delta-sequences. These results are then generalized to higher dimensional pistons by considering additional Kaluza-Klein dimensions.

hep-th↗

arXiv2012

The Casimir Effect for Generalized Piston Geometries

In this paper we study the Casimir energy and force for generalized pistons constructed from warped product manifolds of the type $I\times_{f}N$ where $I=[a,b]$ is an interval of the real line and $N$ is a smooth compact Riemannian manifold either with or without boundary. The piston geometry is obtained by dividing the warped product manifold into two regions separated by the cross section positioned at $R\in(a,b)$. By exploiting zeta function regularization techniques we provide formulas for the Casimir energy and force involving the arbitrary warping function $f$ and base manifold $N$.

hep-th↗

arXiv2012

Spectral methods in quantum field theory and quantum cosmology

We review the application of the spectral zeta-function to the 1- loop properties of quantum field theories on manifolds with boundary, with emphasis on Euclidean quantum gravity and quantum cosmology. As was shown in the literature some time ago, the only boundary conditions that are completely invariant under infinitesimal diffeomorphisms on metric perturbations suffer from a drawback, i.e. lack of strong ellipticity of the resulting boundary-value problem. Nevertheless, at least on the Euclidean 4-ball background, it remains possible to evaluate the zeta(0) value, which describes in this case a universe which, in the limit of small 3-geometry, has vanishing probability of approaching the cosmological singularity. An assessment of this result is here performed, discussing its physical and mathematical implications.

hep-th↗

arXiv2011

The Spectral Zeta Function for Laplace Operators on Warped Product Manifolds of the type $I\times_{f} N$

In this work we study the spectral zeta function associated with the Laplace operator acting on scalar functions defined on a warped product of manifolds of the type $I\times_{f} N$ where $I$ is an interval of the real line and $N$ is a compact, $d$-dimensional Riemannian manifold either with or without boundary. Starting from an integral representation of the spectral zeta function, we find its analytic continuation by exploiting the WKB asymptotic expansion of the eigenfunctions of the Laplace operator on $M$ for which a detailed analysis is presented. We apply the obtained results to the explicit computation of the zeta regularized functional determinant and the coefficients of the heat kernel asymptotic expansion.

math-ph↗

arXiv2011

Heat Kernel Coefficients for Laplace Operators on the Spherical Suspension

In this paper we compute the coefficients of the heat kernel asymptotic expansion for Laplace operators acting on scalar functions defined on the so called spherical suspension (or Riemann cap) subjected to Dirichlet boundary conditions. By utilizing a contour integral representation of the spectral zeta function for the Laplacian on the spherical suspension we find its analytic continuation in the complex plane and its associated meromorphic structure. Thanks to the well known relation between the zeta function and the heat kernel obtainable via Mellin transform we compute the coefficients of the asymptotic expansion in arbitrary dimensions. The particular case of a $d$-dimensional sphere as the base manifold is studied as well and the first few heat kernel coefficients are given explicitly.

math-ph↗

arXiv2011

Pistons modeled by potentials

In this article we consider a piston modelled by a potential in the presence of extra dimensions. We analyze the functional determinant and the Casimir effect for this configuration. In order to compute the determinant and Casimir force we employ the zeta function scheme. Essentially, the computation reduces to the analysis of the zeta function associated with a scalar field living on an interval $[0,L]$ in a background potential. Although, as a model for a piston, it seems reasonable to assume a potential having compact support within $[0,L]$, we provide a formalism that can be applied to any sufficiently smooth potential.

hep-th↗

arXiv2011

Bose-Einstein Condensation on Product Manifolds

We investigate the phenomenon of Bose-Einstein condensation on manifolds constructed as a product of a three-dimensional Euclidian space and a general smooth, compact $d$-dimensional manifold possibly with boundary. By using spectral $ζ$-function methods, we are able to explicitly provide thermodynamical quantities like the critical temperature and the specific heat when the gas of bosons is confined, in the three-dimensional manifold, by the experimentally relevant anisotropic harmonic oscillator potential.

hep-th↗

arXiv2011

On the Hurwitz Zeta Function of Imaginary Second Argument

In this work we exploit Jonquière's formula relating the Hurwitz zeta function to a linear combination of polylogarithmic functions in order to evaluate the real and imaginary part of $ζ_{H}(s,ia)$ and its first derivative with respect to the first argument $s$. In particular, we obtain expressions for the real and imaginary party of $ζ_{H}(s,i a)$ and its derivative for $s=m$ with $m\in\mathbb{Z}\backslash\{1\}$ involving simpler transcendental functions.

math-ph↗

arXiv2011

Conical Casimir Pistons with Hybrid Boundary Conditions

In this paper we compute the Casimir energy and force for massless scalar fields endowed with hybrid boundary conditions, in the setting of the bounded generalized cone. By using spectral zeta function regularization methods, we obtain explicit expressions for the Casimir energy and force in arbitrary dimensions in terms of the zeta function defined on the piston. Our general formulas are, subsequently, specialized to the case in which the piston is modelled by a $d$-dimensional sphere. In this particular situation, explicit results are given for $d=2,3,4,5$.

hep-th↗

arXiv2011

The Casimir Effect for Conical Pistons

In this paper we utilize $ζ$-function regularization techniques in order to compute the Casimir force for massless scalar fields subject to Dirichlet and Neumann boundary conditions in the setting of the conical piston. The piston geometry is obtained by dividing the bounded generalized cone into two regions separated by its cross section positioned at $a$ with $a\in(0,b)$ with $b>0$. We obtain expressions for the Casimir force that are valid in any dimension for both Dirichlet and Neumann boundary conditions in terms of the spectral $ζ$-function of the piston. As a particular case, we specify the piston to be a $d$-dimensional sphere and present explicit results for $d=2,3,4,5$.

hep-th↗

arXiv2010

Zeta Determinant for Laplace Operators on Riemann Caps

The goal of this paper is to compute the zeta function determinant for the massive Laplacian on Riemann caps (or spherical suspensions). These manifolds are defined as compact and boundaryless $D-$dimensional manifolds deformed by a singular Riemannian structure. The deformed spheres, considered previously in the literature, belong to this class. After presenting the geometry and discussing the spectrum of the Laplacian, we illustrate a method to compute its zeta regularized determinant. The special case of the deformed sphere is recovered as a limit of our general formulas.

math-ph↗

arXiv2010

Non-commutative Corrections in Spectral Matrix Gravity

We study a non-commutative deformation of general relativity based on spectral invariants of a partial differential operator acting on sections of a vector bundle over a smooth manifold. We compute the first non-commutative corrections to Einstein equations in the weak deformation limit and analyze the spectrum of the theory. Related topics are discussed as well.

gr-qc↗

arXiv2009

Small Mass Expansion of Functional Determinants on the Generalized Cone

In this paper we compute the small mass expansion for the functional determinant of a scalar Laplacian defined on the bounded, generalized cone. In the framework of zeta function regularization, we obtain an expression for the functional determinant valid in any dimension for both Dirichlet and Robin boundary conditions in terms of the spectral zeta function of the base manifold. Moreover, as a particular case, we specify the base to be a $d$-dimensional sphere and present explicit results for $d=2,3,4,5$.

hep-th↗