arXiv subjects

Luca Fanelli

Publications and source records attributed to Luca Fanelli.

At least 19 recordsLinked to original sources

arXiv2026

Scattering Theory For 3D Cubic Damped Magnetic Schr\"odinger Equation

We consider the three-dimensional defocusing cubic nonlinear Schr\"odinger equation with variable coefficients, a magnetic potential, and a non-negative localized damping term, \[ i\partial_tu+(\nabla-iA)\cdot G(\nabla-iA)u+ia(x)u=|u|^2u, \qquad t>0,\quad x\in\mathbb R^3. \] No non-trapping condition is imposed on the metric $G$. Instead, the variable-coefficient region is assumed to be contained in the effective damping region. Under a one-centre condition on the tangential magnetic field, we prove global well-posedness for initial data in $H^{1+\varepsilon}$, uniform mass and energy bounds, and show the local energy decay. To obtain scattering, we impose a support condition on the full magnetic field inside the damping region. Under these stronger assumptions, the solution scatters to a free Schr\"odinger evolution in $H^s$ for every $0\le s<1$. The appendix discusses a separate constant-damping framework for abstract Hamiltonians.

math.AP↗

arXiv2026

Global factors for local shocks in a data-scarce environment: with an application to regional fiscal multipliers in Italy

We propose a novel econometric methodology for Structural Vector Autoregressions with external instruments (`proxy-SVARs' or `SVAR-IVs') in panel data characterized by strong cross-sectional dependence, dynamic heterogeneity, and limited availability of direct external instruments for the shocks of interest. For each unit, we specify a Factor-Augmented proxy-SVAR (`proxy-FA-SVAR') that incorporates factors summarizing cross-sectional information from the non-policy variables of the system. The effects of the policy shocks are then recovered indirectly by estimating unit-specific policy reaction functions through a Minimum Distance approach. Identification relies on global instruments for the non-policy shocks; that is, proxies common to all units in the panel, internally constructed from a separate SVAR estimated on factors for the policy and non-policy variables. These global instruments can be complemented with local (idiosyncratic) instruments constructed from auxiliary unit-level SVARs. Their joint use renders the proxy-FA-SVARs overidentified and therefore statistically testable. We illustrate the methodology by estimating government spending multipliers for Italian NUTS-2 regions using annual data. The global and local instruments for the regional output shocks are obtained from Blanchard-Perotti-type SVARs.

econ.EM↗

arXiv2026

Magnetic uncertainty in variable geometry

In this paper, we study Hardy-type uncertainty principles and unique continuation properties for linear covariant Schrodinger equations with variable coefficients in the presence of bounded electric and magnetic potentials. Under suitable smallness assumptions on the leading coefficients, we prove that any solution exhibiting super-quadratic exponential decay at two distinct times must vanish identically. Under an additional structural assumption on the coefficient matrix $G$, we further establish a Hardy-type result at the quadratic exponential scale. We also obtain an analogous uniqueness result for the heat equation with variable-coefficient magnetic perturbations. Our results unify and extend previous works in two directions: they recover the constant-coefficient covariant case treated by Barcelo-Fanelli-Gutierrez-Ruiz-Vilela when $G=I$, and the variable-coefficient non-magnetic case considered by Federico-Li-Yu when $A=0$. The proofs combine logarithmic convexity arguments with Carleman estimates adapted to variable-coefficient covariant Schr\"odinger and parabolic flows. Although our approach follows the general strategy introduced by Escauriaza-Kenig-Ponce-Vega, substantial new difficulties arise from the interaction between the variable metric and the magnetic structure, which requires new weight functions and refined commutator estimates.

math.AP↗

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↗

arXiv2026

Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups

We establish unweighted Hardy-type inequalities on step-two Carnot groups with one-dimensional vertical layer, with explicit lower bounds for the optimal Hardy constant. The approach is based on a quantitative integration-by-parts mechanism that replaces the non-horizontal Euler vector field by a suitably constructed horizontal vector field with controlled norm. As applications, we obtain fully explicit bounds in the Heisenberg group for both the Kor{\`a}nyi gauge and the Carnot--Carath{\'e}odory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor{\`a}nyi-type homogeneous norm.

math.AP↗

arXiv2026

Quantitative Landis-type result for Dirac operators

We study quantitative unique continuation at infinity for Dirac equations with bounded matrix-valued potentials. For the massless Dirac operator $\mathcal{D}_n$ in $\mathbb{R}^n$, we establish a Landis-type estimate showing that the vanishing order of any nontrivial bounded solution of $( \mathcal{D}_n + \mathbb{V} ) \varphi = 0$ satisfies a lower bound of order $\exp(-\kappa R^{2} (\log R)^{2})$ as $|x|=R\to \infty$; the quadratic growth in the exponent is sharp, in view of previous known results. Our proof follows a Bourgain--Kenig type approach based on a Carleman inequality for Dirac operators which relies on a local H\"older regularity result, which we also prove. In two dimension, we obtain improved quantitative estimates under symmetry assumptions on the potential $\mathbb{V}$ and for real-valued solutions. Finally, we also derive qualitative Landis-type results for Dirac equations with decaying potentials, including critical decay rates.

math.AP↗

arXiv2025

Bootstrap Diagnostic Tests

Violation of the assumptions underlying classical (Gaussian) limit theory often yields unreliable statistical inference. This paper shows that the bootstrap can detect such violations by delivering simple and powerful diagnostic tests that (a) induce no pre-testing bias, (b) use the same critical values across applications, and (c) are consistent against deviations from asymptotic normality. The tests compare the conditional distribution of a bootstrap statistic with the Gaussian limit implied by valid specification and assess whether the resulting discrepancy is large enough to indicate failure of the asymptotic Gaussian approximation. The method is computationally straightforward and only requires a sample of i.i.d. draws of the bootstrap statistic. We derive sufficient conditions for the randomness in the data to mix with the randomness in the bootstrap repetitions in a way such that (a), (b) and (c) above hold. We demonstrate the practical relevance and broad applicability of bootstrap diagnostics by considering several scenarios where the asymptotic Gaussian approximation may fail, including weak instruments, non-stationarity, parameters on the boundary of the parameter space, infinite variance data and singular Jacobian in applications of the delta method. An illustration drawn from the empirical macroeconomic literature concludes.

econ.EM↗

arXiv2024

Intertwining operators beyond the Stark Effect

The main mathematical manifestation of the Stark effect in quantum mechanics is the shift and the formation of clusters of eigenvalues when a spherical Hamiltonian is perturbed by lower order terms. Understanding this mechanism turned out to be fundamental in the description of the large-time asymptotics of the associated Schr\"odinger groups and can be responsible for the lack of dispersion in Fanelli, Felli, Fontelos and Primo [Comm. Math. Phys., 324(2013), 1033-1067; 337(2015), 1515-1533]. Recently, Miao, Su, and Zheng introduced in [Tran. Amer. Math. Soc., 376(2023), 1739--1797] a family of spectrally projected intertwining operators, reminiscent of the Kato's wave operators, in the case of constant perturbations on the sphere (inverse-square potential), and also proved their boundedness in $L^p$. Our aim is to establish a general framework in which some suitable intertwining operators can be defined also for non constant spherical perturbations in space dimensions 2 and higher. In addition, we investigate the mapping properties between $L^p$-spaces of these operators. In 2D, we prove a complete result, for the Schr\"odinger Hamiltonian with a (fixed) magnetic potential an electric potential, both scaling critical, allowing us to prove dispersive estimates, uniform resolvent estimates, and $L^p$-bounds of Bochner--Riesz means. In higher dimensions, apart from recovering the example of inverse-square potential, we can conjecture a complete result in presence of some symmetries (zonal potentials), and open some interesting spectral problems concerning the asymptotics of eigenfunctions.

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↗

arXiv2024

Error bounds for Physics Informed Neural Networks in Nonlinear Schr\"odinger equations placed on unbounded domains

We consider the subcritical nonlinear Schr\"odinger (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.

math.AP↗

arXiv2024

Uniform resolvent estimates, smoothing effects and spectral stability for the Heisenberg sublaplacian

We establish global bounds for solutions to stationary and time-dependent Schr\"odinger equations associated with the sublaplacian $\mathcal L$ on the Heisenberg group, as well as its pure fractional power $\mathcal L^s$ and conformally invariant fractional power $\mathcal L_s$. The main ingredient is a new abstract uniform weighted resolvent estimate which is proved by using the method of weakly conjugate operators -- a variant of Mourre's commutator method -- and Hardy's type inequalities on the Heisenberg group. As applications, we show Kato-type smoothing effects for the time-dependent Schr\"odinger equation, and spectral stability of the sublaplacian perturbed by complex-valued decaying potentials satisfying an explicit subordination condition. In the local case $s=1$, we obtain uniform estimates without any symmetry or derivative loss, which improve previous results.

math.AP↗

arXiv2024

Invalid proxies and volatility changes

When in proxy-SVARs the covariance matrix of VAR disturbances is subject to exogenous, permanent breaks that cause IRFs to change across volatility regimes, even strong, exogenous external instruments yield inconsistent estimates of the dynamic causal effects. However, if these volatility shifts are properly incorporated into the analysis through (testable) "stability restrictions", we demonstrate that the target IRFs are point-identified and can be estimated consistently under a necessary and sufficient rank condition. If the shifts in volatility are sufficiently informative, standard asymptotic inference remains valid even with (i) local-to-zero covariance between the proxies and the instrumented structural shocks, and (ii) potential failures of instrument exogeneity. Intuitively, shifts in volatility act similarly to strong instruments that are correlated with both the target and non-target shocks. We illustrate the effectiveness of our approach by revisiting a seminal fiscal proxy-SVAR for the US economy. We detect a sharp change in the size of the tax multiplier when the narrative tax instrument is complemented with the decline in unconditional volatility observed during the transition from the Great Inflation to the Great Moderation. The narrative tax instrument contributes to identify the tax shock in both regimes, although our empirical analysis raises concerns about its "statistical" validity.

econ.EM↗

arXiv2024

Quantitative Hardy inequality for magnetic Hamiltonians

In this paper we present a new method of proof of Hardy type inequalities for two-dimensional quantum Hamiltonians with a magnetic field of finite flux. Our approach gives a quantitative lower bound on the best constant in these inequalities both for Schr\"odinger and Pauli operators. Pauli operators with Aharonov-Bohm magnetic field are discussed as well.

math-ph↗

arXiv2023

Non-existence of radial eigenfunctions for the perturbed Heisenberg sublaplacian

We prove uniform resolvent estimates in weighted $L^2$-spaces for radial solutions of the sublaplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^d$. The proofs are based on the multipliers methods, and strongly rely on the use of suitable multipliers and of the associated Hardy inequalities. The constants in our inequalities are explicit and depend only on the dimension $d$. As application of the method, we obtain some suitable smallness and repulsivity conditions on a complex radial potential $V$ on $\mathbb{H}^d$ such that $\mathcal{L}+V$ has no radial eigenfunctions.

math.AP↗

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.

math.AP↗

arXiv2023

Unique Continuation Properties from one time for hyperbolic Schr\"odinger equations

In this paper, we investigate properties of unique continuation for hyperbolic Schr\"odinger equations with time-dependent complex-valued electric fields and time-independent real magnetic fields. We show that positive masses inside of a bounded region at a single time propagate outside the region and prove gaussian lower bounds for the solutions, provided a suitable average in space-time cylinders is taken.

math.AP↗

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.

math.AP↗

arXiv2022

An identification and testing strategy for proxy-SVARs with weak proxies

When proxies (external instruments) used to identify target structural shocks are weak, inference in proxy-SVARs (SVAR-IVs) is nonstandard and the construction of asymptotically valid confidence sets for the impulse responses of interest requires weak-instrument robust methods. In the presence of multiple target shocks, test inversion techniques require extra restrictions on the proxy-SVAR parameters other those implied by the proxies that may be difficult to interpret and test. We show that frequentist asymptotic inference in these situations can be conducted through Minimum Distance estimation and standard asymptotic methods if the proxy-SVAR can be identified by using `strong' instruments for the non-target shocks; i.e. the shocks which are not of primary interest in the analysis. The suggested identification strategy hinges on a novel pre-test for the null of instrument relevance based on bootstrap resampling which is not subject to pre-testing issues, in the sense that the validity of post-test asymptotic inferences is not affected by the outcomes of the test. The test is robust to conditionally heteroskedasticity and/or zero-censored proxies, is computationally straightforward and applicable regardless of the number of shocks being instrumented. Some illustrative examples show the empirical usefulness of the suggested identification and testing strategy.

econ.EM↗