arXiv subjects

Guglielmo Fucci

Publications and source records attributed to Guglielmo Fucci.

At least 19 recordsLinked to original sources

arXiv2026

Complex analytic theory of Sturm-Liouville operators with Schatten $p$-class resolvents

We use the theory of entire functions of finite order to prove a universal spectral dependence of the blowup/decay rate of solutions of the Sturm-Liouville eigenvalue equation for problems with Schatten $p$-class resolvents. The general form of the asymptotics turns out to depend exclusively on the largest integer $\mathfrak{p}$ such that the underlying resolvents fail to be in the Schatten $\mathfrak{p}$-class. We then use the above result to construct a characteristic function of minimal order for Sturm-Liouville problems with Schatten $p$-class resolvents. This immediately yields contour integral representations of spectral $ζ$-functions that were previously only known for quasi-regular problems (except for a few examples). We also demonstrate how our methods lead to new results in connection to important classic topics of Liouville-Green (or WKB) asymptotics and the approximation of the spectrum of singular problems via underlying truncated regular problems. All our applications are accompanied by illustrative examples, including the Airy differential equation, harmonic oscillator (and general power potentials), and the Laguerre differential equation.

math.SP↗

arXiv2025

$ζ$-functions via contour integrals and universal sum rules

This work develops an analytic framework for the study of the $ζ$-function associated with general sequences of complex numbers. We show that a contour integral representation, commonly used when studying spectral $ζ$-functions associated with self-adjoint differential operators, can be extended far beyond its traditional setting. In contrast to representations utilizing integrals of $θ$-functions, our method applies to arbitrary sequences of complex numbers with minimal assumptions. This leads to a set of universal identities, including sum rules and meromorphic properties, that hold across a broad class of $ζ$-functions. Additionally, we discuss the connection to regularized (modified) Fredholm determinants of $p$-Schatten--von Neumann class operators. We illustrate the versatility of this representation by computing special values and residues of the $ζ$-function for a variety of sequences of complex numbers, in particular, the zeros of Airy functions, parabolic cylinder functions, and confluent hypergeometric functions. Furthermore, we employ the adaptive Antoulas--Anderson (AAA) algorithm for rational interpolation in the study of the Airy $ζ$-function.

math.CA↗

arXiv2024

The exotic structure of the spectral $ζ$-function for the Schrödinger operator with Pöschl--Teller potential

This work focuses on the analysis of the spectral $ζ$-function associated with a Schrödinger operator endowed with a Pöschl--Teller potential. We construct the spectral $ζ$-function using a contour integral representation and, for particular self-adjoint extensions, we perform its analytic continuation to a larger region of the complex plane. We show that the spectral $ζ$-function in these cases can possess a very unusual and remarkable structure consisting of a series of logarithmic branch points located at every nonpositive integer value of $s$ along with infinitely many additional branch points (and finitely many simple poles) whose locations depend on the parameters of the problem. By comparing the Pöschl--Teller potential to the classic Bessel potential, we further illustrate that perturbing a given potential by a smooth potential on a finite interval can greatly affect the meromorphic structure and branch points of the spectral $ζ$-function in surprising ways.

math-ph↗

arXiv2024

The spectral $ζ$-function for quasi-regular Sturm--Liouville operators

In this work we analyze the spectral $ζ$-function associated with the self-adjoint extensions, $T_{A,B}$, of quasi-regular Sturm--Liouville operators that are bounded from below. By utilizing the Green's function formalism, we find the characteristic function which implicitly provides the eigenvalues associated with a given self-adjoint extension $T_{A,B}$. The characteristic function is then employed to construct a contour integral representation for the spectral $ζ$-function of $T_{A,B}$. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the $ζ$-function to a larger region of the complex plane. We also present a method for computing the value of the spectral $ζ$-function of $T_{A,B}$ at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral $ζ$-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of $s$.

math-ph↗

arXiv2023

Vacuum energy of scalar fields on spherical shells with general matching conditions

In this work we analyze the spectral zeta function for massless scalar fields propagating in a $D$-dimensional flat space under the influence of a shell potential. The shell potential is defined in terms of the two-interval self-adjoint extensions of the Hamiltonian describing the dynamics of the scalar field. After performing the necessary analytic continuation, we utilize the spectral zeta function of the system to compute the vacuum energy of the field.

hep-th↗

arXiv2023

Casimir energy for spinor fields with $δ$-shell potentials

This work analyzes the Casimir energy of a massive spinor field propagating in flat space endowed with a spherically symmetric $δ$-function potential. By utilizing the spectral zeta function regularization method, the Casimir energy is evaluated after performing a suitable analytic continuation. Explicit numerical results are provided for specific cases in which the Casimir energy is unambiguously defined. The results described in this work represent a generalization of the MIT bag model for spinor fields.

hep-th↗

arXiv2022

The Krein-von Neumann extension revisited

We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.

math.FA↗

arXiv2022

Spectral zeta-Functions and zeta-Regularized Functional Determinants for Regular Sturm-Liouville Operators

The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $ζ$-functions to efficiently compute values of spectral $ζ$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $τ$. Depending on the underlying boundary conditions, we express the $ζ$-function values in terms of a fundamental system of solutions of $τy = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $ζ$-function through a Liouville transformation and provide an explicit expression for the $ζ$-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schrödinger operators with zero, piecewise constant, and a linear potential on a compact interval.

math.SP↗

arXiv2021

One-loop effective action of the ${\mathbb C}P^{N-1}$ model at large $μβ$

In this note we consider a non-linear, large-$N$ ${\mathbb C}P^{N-1}$ sigma model on a finite size interval with periodic boundary conditions, at finite temperature and chemical potential in the regime of $βμ$ large. Our goal is to extend previous calculations and obtain the coefficients of the derivative expansion of the one-loop effective action in the region of $βμ$ large by carrying out the appropriate analytical continuation. This calculation complements previous results and allows us to conclude that the ground state remains homogeneous in this regime as long as it is assumed to be a slowly varying function of the spatial coordinates. While this is reasonable at the two extremes of small or large chemical potential, for intermediate values of the chemical potential and small enough temperature, one might expect (by analogy with other models) that lower energy crystalline solutions may exist. In this case a simple derivative expansion, like the one discussed here, would need to be modified in order to capture these features.

hep-th↗

arXiv2021

Casimir pistons with generalized boundary conditions: a step forward

In this work we study the Casimir effect for massless scalar fields propagating in a piston geometry of the type $I\times N$ where $I$ is an interval of the real line and $N$ is a smooth compact Riemannian manifold. Our analysis represents a generalization of previous results obtained for pistons configurations as we consider all possible boundary conditions that are allowed to be imposed on the scalar fields. We employ the spectral zeta function formalism in the framework of scattering theory in order to obtain an expression for the Casimir energy and the corresponding Casimir force on the piston. We provide explicit results for the Casimir force when the manifold $N$ is a $d$-dimensional sphere and a disk.

math-ph↗

arXiv2019

Ground state modulations in the ${\mathbb C}P^{N-1}$ model

In this work we examine a system consisting of a confined one-dimensional arrangement of atoms that we describe by using the 2-dimensional ${\mathbb C}P^{N-1}$ model, restricted to an interval and at finite temperature. We develop a method to obtain the bulk and boundary parts of the one-loop effective action as a function of the effective mass of the fluctuations. The formalism has the advantage of allowing for a systematic analysis of a large class of boundary conditions and to model the (adiabatic) response of the ground state to changes in the boundary conditions. In the case of periodic boundary conditions, we find that inhomogeneous phases are disfavored for intervals of large size. Away from periodic boundary conditions, our numerical results show that the ground state has a generic crystal-like structure that can be modulated by variations of the boundary conditions. The results presented here could be relevant for experimental implementations of nonlinear sigma models and could be tested by lattice numerical simulations.

hep-th↗

arXiv2018

Asymptotic Expansion of the Heat Kernel Trace of Laplacians with Polynomial Potentials

It is well-known that the asymptotic expansion of the trace of the heat kernel for Laplace operators on smooth compact Riemmanian manifolds can be obtained through termwise integration of the asymptotic expansion of the on-diagonal heat kernel. It is the purpose of this work to show that, in certain circumstances, termwise integration can be used to obtain the asymptotic expansion of the heat kernel trace for Laplace operators endowed with a suitable polynomial potential on unbounded domains. This is achieved by utilizing a resummed form of the asymptotic expansion of the on-diagonal heat kernel.

math-ph↗

arXiv2017

The Casimir effect for pistons with transmittal boundary conditions

This work focuses on the analysis of the Casimir effect for pistons subject to transmittal boundary conditions. In particular we consider, as piston configuration, a direct product manifold of the type $I\times N$ where $I$ is a closed interval of the real line and $N$ is a smooth compact Riemannian manifold. By utilizing the spectral zeta function regularization technique, we compute the Casimir energy of the system and the Casimir force acting on the piston. Explicit results for the force are provided when the manifold $N$ is a $d$-dimensional ball.

hep-th↗

arXiv2017

Vacuum instability in Kaluza-Klein manifolds

The purpose of this work in to analyze particle creation in spaces with extra dimensions. We consider, in particular, a massive scalar field propagating in a Kaluza-Klein manifold subject to a constant electric field. We compute the rate of particle creation from vacuum by using techniques rooted in the spectral zeta function formalism. The results we obtain show explicitly how the presence of the extra-dimensions and their specific geometric characteristics, influence the rate at which pairs of particles and anti-particles are generated.

hep-th↗

arXiv2017

Some new results for the one-loop mass correction to the compactified $λϕ^{4}$ theory

In this work we consider the one-loop effective action of a self-interacting $λϕ^{4}$ field propagating in a $D$ dimensional Euclidean space endowed with $d\leq D$ compact dimensions. The main purpose of this paper is to compute the corrections to the mass of the field due to the presence of the compactified dimensions. Although results for the one-loop correction to the mass of a $λϕ^{4}$ field are very well known for compactified toroidal spaces, where the field obeys periodic boundary conditions, similar results do not appear to be readily available for cases in which the scalar field is subject to Dirichlet and Neumann boundary conditions. We apply the results for the one-loop mass correction to the study of the critical temperature in Ginzburg-Landau models.

hep-th↗

arXiv2016

Functional determinants and Casimir energy in higher dimensional spherically symmetric background potentials

In this paper we analyze the spectral zeta function associated with a Laplace operator acting on scalar functions on an N-dimensional Euclidean space in the presence of a spherically symmetric background potential. The obtained analytic continuation of the spectral zeta function is then used to derive very simple results for the functional determinant of the operator and the Casimir energy of the scalar field.

hep-th↗

arXiv2015

The Casimir Effect for Thick Pistons

In this work we analyze the Casimir energy and force for a {\it thick} piston configuration. This study is performed by utilizing the spectral zeta function regularization method. The results we obtain for the Casimir energy and force depend explicitly on the parameters that describe the general self-adjoint boundary conditions imposed. Numerical results for the Casimir force are provided for specific types of boundary conditions and are also compared to the corresponding force on an infinitely thin piston.

hep-th↗

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↗