arXiv subjects

Lucrezia Cossetti

Publications and source records attributed to Lucrezia Cossetti.

17 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.

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arXiv2025

A unified approach to Hardy-type inequalities with Bessel pairs

In this paper, we provide suitable characterisations of pairs of weights $(V,W),$ known as Bessel pairs, that ensure the validity of weighted Hardy-type inequalities. The abstract approach adopted here makes it possible to establish such inequalities also going beyond the classical Euclidean setting and also within a more general $L^p$ framework. As a byproduct of our method, we obtain explicit expressions for the maximizing functions and, in certain specific situations, we show that the associated constants are sharp. We emphasise that our approach unifies, generalises and improves several existing results in the literature.

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arXiv2025

On the point spectrum of electromagnetic Dirac operators

In this work, we develop the method of multipliers for electromagnetic Dirac operators and establish sufficient conditions on the magnetic and electric fields that guarantee the absence of point spectrum. In the massless case, our approach covers Coulomb-type potentials of the form $V(x) = \frac{1}{|x|} \big(\nu \mathbb{I} + \mu \beta + i \delta \beta \big(\boldsymbol{\alpha}\cdot \frac{x}{|x|} \big) \big).$ We also adapt the method to show absence of embedded eigenvalues above a threshold which depends on the asymptotic behaviour of the magnetic and electric fields.

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arXiv2025

Error bounds for Physics Informed Neural Networks in Nonlinear Schrödinger equations placed on unbounded domains

We consider the subcritical nonlinear Schrödinger (NLS) in dimension one posed on the unbounded real line. Several previous works have considered the deep neural network approximation of NLS solutions from the numerical and theoretical point of view in the case of bounded domains. In this paper, we introduce a new PINNs method to treat the case of unbounded domains and show rigorous bounds on the associated approximation error in terms of the energy and Strichartz norms, provided a reasonable integration scheme is available. Applications to traveling waves, breathers and solitons, as well as numerical experiments confirming the validity of the approximation are also presented as well.

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arXiv2024

The virial theorem and the method of multipliers in spectral theory

We provide a link between the virial theorem in functional analysis and the method of multipliers in theory of partial differential equations. After giving a physical insight into the techniques, we show how to use them to deduce the absence of eigenvalues and other spectral properties of electromagnetic quantum Hamiltonians. We focus on our recent developments in non-self-adjoint settings, namely on Schroedinger operators with matrix-valued potentials, relativistic operators of Pauli and Dirac types, and complex Robin boundary conditions.

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arXiv2023

Uniform resolvent estimates and absence of eigenvalues of biharmonic operators with complex potentials

We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover critical Rellich-type potentials too. As a byproduct we obtain uniform resolvent estimates in weighted spaces. Some of the results are new also in the self-adjoint setting.

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arXiv2023

On uniqueness of solutions to the surface electromigration equation

In this paper we investigate on uniqueness properties of solutions to the surface electromigration (SEM) equation, which is a generalisation of the more classical Zakharov-Kuznetsov equation of plasma physics with non-local perturbation terms. We will show that if the difference of two solutions has a sufficiently strong spatial decay at two different instants of time, then the two solutions coincide on the whole interval of time.

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arXiv2021

Improved Hardy-Rellich inequalities

We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves the inequality, and may also provide an estimate which does not hold in the free case. The main examples are related to the introduction of a magnetic field: this is a manifestation of the diamagnetic phenomenon, which has been observed by Laptev and Weidl in \cite{LW1999} for the Hardy inequality, later by Evans and Lewis in \cite{EL2005} for the Rellich inequality; however, to the best of our knowledge, the so called Hardy-Rellich inequality has not yet been investigated in this regards. After showing the optimal inequality, we prove that the best constant is not attained by any function in the domain of the estimate.

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arXiv2021

A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equations

In this work we investigate the L^p-L^q-mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we obtain an L^p-L^q-type Limiting Absorption Principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations. Moreover, we provide an improved version of the Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems.

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arXiv2021

Spectral enclosures for the damped elastic wave equation

In this paper we investigate spectral properties of the damped elastic wave equation. Deducing a correspondence between the eigenvalue problem of this model and the one of Lamé operators with non self-adjoint perturbations, we provide quantitative bounds on the location of the point spectrum in terms of suitable norms of the damping coefficient.

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arXiv2021

Eigenvalue bounds and spectral stability of Lamé operators with complex potentials

This paper is devoted to providing quantitative bounds on the location of eigenvalues, both discrete and embedded, of non self-adjoint Lamé operators of elasticity $-Δ^\ast + V$ in terms of suitable norms of the potential $V$. In particular, this allows to get sufficient conditions on the size of the potential such that the point spectrum of the perturbed operator remains empty. In three dimensions we show full spectral stability under suitable form-subordinated perturbations: we prove that the spectrum is purely continuous and coincides with the non negative semi-axis as in the free case.

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arXiv2019

Absence of eigenvalues of Dirac and Pauli Hamiltonians via the method of multipliers

By developing the method of multipliers, we establish sufficient conditions on the magnetic field and the complex, matrix-valued electric potential, which guarantee that the corresponding system of Schrödinger operators has no point spectrum. In particular, this allows us to prove analogous results for Pauli operators under the same electromagnetic conditions and, in turn, as a consequence of the supersymmetric structure, also for magnetic Dirac operators.

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arXiv2019

Absence of eigenvalues of non-self-adjoint Robin Laplacians on the half-space

By developing the method of multipliers, we establish sufficient conditions which guarantee the total absence of eigenvalues of the Laplacian in the half-space, subject to variable complex Robin boundary conditions. As a further application of this technique, uniform resolvent estimates are derived under the same assumptions on the potential. Some of the results are new even in the self-adjoint setting, where we obtain quantum-mechanically natural conditions.

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arXiv2019

On the unique continuation property of solutions to the two-dimensional Zakharov-Kuznetsov equation

The purpose of the current paper is twofold: to some extent it is intended as a review of the recent optimal result in [4] concerning the unique continuation property of solutions to the two-dimensional Zakharov-Kuznetsov equation. On the other hand, the main core of the work is devoted to providing an alternative proof of the aforementioned result. The importance of this original contribution relies on the fact that, unlike the approach used in [4], the strategy adopted here is not sensitive of the two dimensional setting of the problem and therefore could be adapted to higher dimensional Zakharov-Kuznetsov equations for which, as far as we know, a proof of an analogous optimal unique continuation principle is still missing. For sake of clearness we focus here on the 2D case only, the higher dimensional analysis will be discussed somewhere else.

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arXiv2019

Bounds on eigenvalues of perturbed Lamé operators with complex potentials

Several recent papers have focused their attention in proving the correct analogue to the Lieb-Thirring inequalities for non self-adjoint operators and in finding bounds on the distribution of their eigenvalues in the complex plane. This paper provides some improvement in the state of the art in this topic. Precisely, we address the question of finding quantitative bounds on the discrete spectrum of the perturbed Lamé operator of elasticity $-Δ^\ast + V$ in terms of $L^p$-norms of the potential. Original results within the self-adjoint framework are provided too.

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arXiv2018

Uniqueness results for Zakharov-Kuznetsov equation

In this paper we study uniqueness properties of solutions to the Zakharov-Kuznetsov equation of plasma physic. Given two sufficiently regular solutions $u_1, u_2,$ we prove that, if $u_1-u_2$ decays fast enough at two distinct times, then $u_1\equiv u_2.$

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arXiv2016

Uniform resolvent estimates and absence of eigenvalues for Lamé operators with potentials

We consider the $0$-order perturbed Lamé operator $-Δ^\ast + V(x)$. It is well known that if one considers the free case, namely $V=0,$ the spectrum of $-Δ^\ast$ is purely continuous and coincides with the non-negative semi-axis. The first purpose of the paper is to show that, at least in part, this spectral property is preserved in the perturbed setting. Precisely, developing a suitable multipliers technique, we will prove the absence of point spectrum for Lamé operator with potentials which satisfy a variational inequality with suitable small constant. We stress that our result also covers complex-valued perturbation terms. Moreover the techniques used to prove the absence of eigenvalues enable us to provide uniform resolvent estimates for the perturbed operator under the same assumptions about $V$.

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