arXiv subjects

Fabio Pizzichillo

Publications and source records attributed to Fabio Pizzichillo.

13 recordsLinked to original sources

arXiv2026

Relativistic Virial Operators

When studying Dirac operators, it is well known that the phenomenon of Zitterbewegung leads to a lack of convexity of the variance, which creates difficulties in the analysis of dispersive properties. In particular, standard virial methods are harder to implement in the Dirac setting. In this paper, we introduce a new approach based on the center-of-energy operator, leading to a family of relativistic virial identities. As an application, we establish spectral stability results for perturbed Dirac operators and prove local smoothing estimates for the associated evolution equation.

math.AP↗

arXiv2024

A relativistic Hardy-type inequality with minimisers

In this paper, we prove a sharp, weighted Hardy-type inequality for the Dirac operator. A key feature of our result is that the inequality is not only sharp but also attained, and we construct explicit minimizers that satisfy the equality case. This extends previous work on the spectral properties of Dirac operators, especially in the context of relativistic quantum mechanics and Coulomb-like potentials.

math.AP↗

arXiv2023

Dirac operators with infinite mass boundary conditions on unbounded domains with infinite corners

We investigate the self-adjointness of the two dimensional Dirac operator with infinite mass boundary conditions on an unbounded domain with an infinite number of corners. We prove that if the domain has no concave corners, then the operator is self-adjoint. On the other hand, when concave corners are present, the operator is no longer self-adjoint and self-adjoint extensions can be constructed. Among these, we characterize the distinguished extension as the unique one whose domain is included in the Sobolev space $H^{s}$, where $s>1/2$ depends on the amplitude of the corners. Lastly, we study the spectrum of this distinguished self-adjoint extension.

math.AP↗

arXiv2022

Keller and Lieb-Thirring estimates of the eigenvalues in the gap of Dirac operators

We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schr\"odinger operator, which are equivalent to Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. The Keller estimate is then extended to a Lieb-Thirring inequality for the eigenvalues in the gap. Most of our result are established in the Birman-Schwinger reformulation.

math.AP↗

arXiv2022

Dirac-Coulomb Operators with Infinite Mass Boundary Conditions in Sectors

We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac-Hardy inequality and exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate our results giving a description of the spectrum of these operators.

math.AP↗

arXiv2019

Self-Adjointness of two dimensional Dirac operators on corner domains

We investigate the self-adjointness of the two-dimensional Dirac operator $D$, with quantum-dot and Lorentz-scalar $δ$-shell boundary conditions, on piecewise $C^2$ domains with finitely many corners. For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space $H^{1/2}$, the formal form domain of the free Dirac operator. The main part of our paper consists of a description of the domain of $D^*$ in terms of the domain of $D$ and the set of harmonic functions that verify some mixed boundary conditions. Then, we give a detailed study of the problem on an infinite sector, where explicit computations can be made: we find the self-adjoint extensions for this case. The result is then translated to general domains by a coordinate transformation.

math.AP↗

arXiv2019

Boundary triples for the Dirac operator with Coulomb-type spherically symmetric perturbations

We determine explicitly a boundary triple for the Dirac operator $H:=-iα\cdot \nabla + mβ+ \mathbb V(x)$ in $\mathbb R^3$, for $m\in\mathbb R$ and $\mathbb V(x)= |x|^{-1} ( ν\mathbb{I}_4 +μβ-i λα\cdot{x}/|x|\,β)$, with $ν,μ,λ\in \mathbb R$. Consequently we determine all the self-adjoint realizations of $H$ in terms of the behaviour of the functions of their domain in the origin. When $\sup_{x} |x||\mathbb V(x)| \leq 1$, we discuss the problem of selecting the distinguished extension requiring that its domain is included in the domain of the appropriate quadratic form.

math.AP↗

arXiv2019

Dirac operators and shell interactions: a survey

In this survey we gather recent results on Dirac operators coupled with $δ$-shell interactions. We start by discussing recent advances regarding the question of self-adjointness for these operators. Afterward we switch to an approximation question: can these operators be recovered as limits of Dirac operators coupled with squeezing potentials ? We also discuss spectral features of these models. Namely, we recall the main spectral consequences of a resolvent formula and conclude the survey by commenting a result of asymptotic nature for the eigenvalues in the gap of a Dirac operator coupled with a Lorentz-scalar interaction.

math-ph↗

arXiv2018

A Hardy-type inequality and some spectral characterizations for the Dirac-Coulomb operator

We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials $\mathbf V$ of Coulomb type: we characterise its eigenvalues in terms of the Birman-Schwinger principle and we bound its discrete spectrum from below, showing that the \emph{ground-state energy} is reached if and only if $\mathbf V$ verifies some {rigidity} conditions. In the particular case of an electrostatic potential, these imply that $\mathbf V$ is the Coulomb potential.

math.AP↗

arXiv2018

Self-Adjoint Extensions for the Dirac Operator with Coulomb-Type Spherically Symmetric Potentials

We describe the self-adjoint realizations of the operator $H:=-iα\cdot \nabla + mβ+ \mathbb V(x)$, for $m\in\mathbb R $, and $\mathbb V(x)= |x|^{-1} ( ν\mathbb{I}_4 +μβ-i λα\cdot{x}/{|x|}\,β)$, for $ν,μ,λ\in \mathbb R$. We characterize the self-adjointness in terms of the behaviour of the functions of the domain in the origin, exploiting Hardy-type estimates and trace lemmas. Finally, we describe the distinguished extension.

math.AP↗

arXiv2017

The relativistic spherical $δ$-shell interaction in $\mathbb{R}^3$: spectrum and approximation

This note revolves on the free Dirac operator in $\mathbb{R}^3$ and its $δ$-shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by finding sharp constants and minimizers of some precise inequalities related to an uncertainty principle. On the other hand, we prove that the domains given by Dittrich, Exner and Šeba [Dirac operators with a spherically symmetric $δ$-shell interaction, J. Math. Phys. 30.12 (1989), 2875-2882] and by Arrizabalaga, Mas and Vega [Shell interactions for Dirac operators, J. Math. Pures et Appl. 102.4 (2014), 617-639] for the realization of an electrostatic spherical shell interaction coincide. Finally, we explore the spectral relation between the shell interaction and its approximation by short range potentials with shrinking support, improving previous results in the spherical case.

math.AP↗

arXiv2017

Spectral asymptotics for $δ$-interactions on sharp cones

We investigate the spectrum of three-dimensional Schrödinger operators with $δ$-interactions of constant strength supported on circular cones. As shown in earlier works, such operators have infinitely many eigenvalues below the threshold of the essential spectrum. We focus on spectral properties for sharp cones, that is when the cone aperture goes to zero, and we describe the asymptotic behavior of the eigenvalues and of the eigenvalue counting function. A part of the results are given in terms of numerical constants appearing as solutions of transcendental equations involving modified Bessel functions.

math.SP↗

arXiv2016

Klein's Paradox and the Relativistic $δ$-shell Interaction in $\mathbb{R}^3$

Under certain hypothesis of smallness of the regular potential $\mathbf{V}$, we prove that the Dirac operator in $\mathbb{R}^3$ coupled with a suitable re-scaling of $\mathbf{V}$ converges in the strong resolvent sense to the Hamiltonian coupled with a $δ$-shell potential supported on $Σ$, a bounded $C^2$ surface. Nevertheless, the coupling constant depends non-linearly on the potential $\mathbf{V}$: the Klein's Paradox comes into play.

math.AP↗