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Piero D'Ancona

Publications and source records attributed to Piero D'Ancona.

At least 19 recordsLinked to original sources
arXiv2026

$L^p$ bounds for wave operators with critical electromagnetic potentials

We study the Møller wave operators for scaling critical electromagnetic Hamiltonians in the plane. For smooth transverse magnetic and angular electric potentials, with nonnegative angular operator and magnetic flux outside $\frac12\Z$, we prove that the wave operators relative to $-Δ$ exist, are unitary on $L^2$, and, together with their adjoints, are bounded on every $L^p$, $1<p<\infty$. We then specialize to the the Aharonov--Bohm model and we determine the exact ranges for the boundedness of their wave operators on weighted $L^p(\R^2,|x|^β\,dx)$ spaces, for both the Friedrichs and the Krein realizations. In the Friedrichs case, this gives in particular the already known boundedness on all $L^{p}$ spaces $1<p<\infty$, while boundedness fails at $p=1,\infty$. In the Krein case, both wave operators and adjoints are bounded precisely when $2/(2-η_α)<p<2/η_α$, where $η_α=\max\{α,1-α\}$ (here $α\in(0,1)$). Thus the boundary condition changes the admissible exponents.

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arXiv2026

Long Range Asymptotics for the Quadratic Aharonov-Bohm NLS

We study the long range behavior of solutions to $i\partial_tu=H_αu+λ|u|u$ on $\mathbb R^2$, where $H_α$ is the Friedrichs realization of the Aharonov-Bohm Hamiltonian with a single pole. The logarithmic phase of the long range ansatz may push a profile out of the domain of $H_α$. We characterize profiles that stay in the operator domain by the vanishing of boundary traces at 0 of order $\le \frac 12$; at half flux $α=\frac 12$, no nonzero trace survives. However, every profile in the full domain of $H_α$ with small $L^\infty$ amplitude determines a unique global solution with a modified final state, with a remainder rate $t^{-b}$ for all $0<b<1/2+ν_α$, $ν_α=\min\{α,1-α\}$. For profiles satisfying the vanishing trace condition, the rate improves to every $0<b<1$. This result is sharp in the sense that, if $α\neq \frac 12$, we can construct profiles with an error of size $t^{-1/2-ν_α}\log t$, ruling out all faster rates. The upper bound comes from a retarded Strichartz estimate for a residual that is not in $L^2$; the lower bound is an explicit calculation via Hankel transforms. For smoother profiles we also compute the first correction, which gives remainder rates with $1<b<2$.

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arXiv2026

Quantitative uniform resolvent estimates

We derive quantitative uniform resolvent estimates for Schrödinger operators on the half-line with inverse-square potentials, which provide a sharp behaviour in the limit of large coupling. Our approach is based on a matrix representation of the boundary value of a weighted resolvent. The partial wave decomposition then turns these one-dimensional channel estimates into explicit weighted resolvent estimates for the Laplacian, its inverse-square potential perturbations and for the magnetic Laplacian with an Aharonov--Bohm potential. We also obtain exact Simon-type identities for the imaginary parts of the weighted resolvents of these operators.

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arXiv2026

Dynamical Amrein-Berthier Uncertainty for Fractional Schrödinger Flows

We prove dynamical Amrein-Berthier uncertainty principles for fractional Schrödinger flows. For the free Hamiltonian $H=(-Δ)^α$ on $L^2(\mathbb{R}^n)$, with $α>\frac{1}{2}$, we show that two--time localization on finite measure sets $E,F$ forces the quantitative estimate \begin{equation*} \|u(t)\|_{L^{2}}\lesssim_{E,F,T,n,α} \|u(0)\|_{L^{2}(E^{c})} + \|u(T)\|_{L^{2}(F^{c})}, \qquad T\neq0,\ t\in \mathbb{R} \end{equation*} for $u(t)=e^{-itH}u(0)$ at every time. The threshold $α>\frac{1}{2}$ is tied to the stationary phase structure of the fractional kernel. If $α\ge1$ the sets can be arbitrary finite measure sets; if $\frac{1}{2}<α<1$ we impose the finiteness of a natural interaction energy \begin{equation*} \textstyle \mathcal{I}_γ(E,F) = \int_{\mathbb{R}^n \times \mathbb{R}^n} \mathbf{1}_{F}(x)|x-y|^{2γ}\mathbf{1}_{E}(y)\,dx\,dy<\infty, \qquad γ= \frac{n(1-α)}{2 α-1} \end{equation*} of the pair $(E,F)$, essentially equivalent to a sufficiently fast joint decay of the measure of the sets at infinity. In particular, compact support at two distinct times is impossible for a nonzero solution. We also prove corresponding results for one dimensional fractional Hamiltonians $(-\partial_x^2+V)^α$ under weighted scattering assumptions, and for higher order Hamiltonians $(-Δ)^m+V$ for suitable classes of decaying potentials $V$.

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arXiv2026

A Phase Space Criterion for Dynamical Amrein-Berthier Uncertainty

We prove a phase space criterion for dynamical Amrein-Berthier uncertainty principles. The abstract result says that, for a Fourier integral operator $A\in FIO(χ)$ associated with a tame canonical transformation $χ$, the localized operator $\mathbf{1}_E A\mathbf{1}_F$ is compact on $L^2(\mathbb {R}^d)$ whenever $χ$ satisfies a vertical non refocusing condition: high frequency covectors issued from a spatially localized region cannot return to a vertical direction over the observation region. In the linear symplectic case this condition is equivalent to the familiar nondegeneracy $\det B\neq0$ of the upper right block of the symplectic matrix. We apply this compactness theorem to Schrödinger propagators for Yajima--type Hamiltonians, including quadratic electric and linear magnetic growth, and obtain two--time Amrein--Berthier inequalities for compact localization sets at all nonrefocusing times. The result extends the compactness mechanism behind the dynamical Amrein-Berthier principle to a genuinely microlocal setting.

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arXiv2026

Uniform resolvent estimates for magnetic operators

We prove Kenig--Ruiz--Sogge type uniform resolvent estimates for selfadjoint magnetic Schrödinger operators $H=(i\partial+A(x))^2+V(x)$ on $\mathbb{R}^{n}$, $n\ge3$. Under suitable decay assumptions on the electric and magnetic potentials, and excluding a threshold resonance at zero, we show that for all $z \in \mathbb{C}\setminus[0,+\infty)$, \begin{equation*} \|(H-z)^{-1}ϕ\|_{L^{q}}\lesssim|z|^{θ(p,q)} (1+|z|^γ) \|ϕ\|_{L^{p}} \end{equation*} throughout the full free resolvent range $(\frac1p,\frac1q)\inΔ(n)$, where $θ(p,q)=\frac n2(\frac1p-\frac1q)-1$. Here $γ=\frac 12\frac{n-1}{n+1}$ under the basic magnetic decay hypothesis, or $γ=\frac{n-1}{4n}$ under a different decay assumption on $A(x)$; for the second case we use a weak endpoint estimate of Frank--Simon type \begin{equation*} \|R_{0}(z)ϕ\| _{L^{\frac{2n}{n-1},\infty}_{r}L^{2}_ω} \lesssim |z|^{-\frac12} \|ϕ\|_{L^{\frac{2n}{n+1},1}_{r}L^{2}_ω}. \end{equation*} The result extends the known electromagnetic estimates from fixed frequency and a smaller exponent region to all frequencies and the full Kenig--Ruiz--Sogge range. We also prove a variant with weaker local assumptions in a smaller range $Δ_1(n)$. As applications, we obtain $L^p-L^{p'}$ restriction type estimates for the density of the spectral measure of magnetic Schrödinger operators, and an eigenvalue enclosure result for complex scalar perturbations.

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arXiv2026

A dynamical Amrein-Berthier uncertainty principle

Given a selfadjoint magnetic Schrödinger operator \begin{equation*} H = ( i \partial + A(x) )^2 + V(x) \end{equation*} on $L^{2}(\mathbb{R}^n)$, with $V(x)$ strictly subquadratic and $A(x)$ strictly sublinear, we prove that the flow $u(t)=e^{-itH}u(0)$ satisfies an Amrein--Berthier type inequality \begin{equation*} \|u(t)\|_{L^{2}}\lesssim_{E,F,T,A,V} \|u(0)\|_{L^{2}(E^{c})} + \|u(T)\|_{L^{2}(F^{c})}, \qquad 0\le t\le T \end{equation*} for all compact sets $E,F \subset \mathbb{R}^{n}$. In particular, if both $u(0)$ and $u(T)$ are compactly supported, then $u$ vanishes identically. Under different assumptions on the operator, which allow for time--dependent coefficients, the result extends to sets $E,F$ of finite measure. We also consider a few variants for Schrödinger operators with singular coefficients, metaplectic operators, and we include applications to control theory.

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arXiv2025

Dispersive estimates for Dirac equations in Aharonov-Bohm magnetic fields: massless case

In this paper we study the dispersive properties of a two dimensional massless Dirac equation perturbed by an Aharonov--Bohm magnetic field. Our main results will be a family of pointwise decay estimates and a full range family Strichartz estimates for the flow. The proof relies on the use of a relativistic Hankel transform, which allows for an explicit representation of the propagator in terms of the generalized eigenfunctions of the operator. These results represent the natural continuation of earlier research on evolution equations associated to operators with magnetic fields with strong singularities (see \cite{DF, FFFP, FZZ} where the Schrödinger and the wave equations were studied). Indeed, we recall the fact that the Aharonov--Bohm field represents a perturbation which is critical with respect to the scaling: this fact, as it is well known, makes the analysis particularly challenging.

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arXiv2025

Dispersive and Strichartz estimates for Dirac equation in a cosmic string spacetime

In this work we study the Dirac equation on the cosmic string background, which models a one--dimensional topological defect in the spacetime. We first define the Dirac operator in this setting, classifying all of its selfadjoint extensions, and we give an explicit kernel for the propagator. Secondly, we prove dispersive estimates for the flow, with and without weights. Finally, we prove Strichartz estimates for the flow in a sharp restricted set of indices, which are different from the classical Euclidean ones.

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arXiv2023

Scattering for the defocusing NLS on the line with variable coefficients

We prove $H^{1}$ scattering for a defocusing NLS on the line with fully variable coefficients. The result is proved by adapting the Kenig--Merle scheme to a non translation invariant setting. In addition, we give an abstract version of the scheme which can be applied to other operators.

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arXiv2021

Global Strichartz estimates for an inhomogeneous Maxwell system

We show global-in-time Strichartz estimates for the isotropic Maxwell system with divergence free data. On the scalar permittivity and permeability we impose decay assumptions as $|x|\to\infty$ and a non-trapping condition. The proof is based on smoothing estimates in weighted $L^2$ spaces which follow from corresponding resolvent estimates for the underlying Helmholtz problem.

math.AP↗
arXiv2021

On the supercritical Schrödinger equation on the exterior of a ball

We consider the mixed problem on the exterior of the unit ball in $\mathbb{R}^{n}$, $n\ge2$, for a defocusing Schrödinger equation with a power nonlinearity $|u|^{p-1}u$, with zero boundary data. Assuming that the initial data are non radial, sufficiently small perturbations of \emph{large} radial initial data, we prove that for all powers $p>n+6$ the solution exists for all times, its Sobolev norms do not inflate, and the solution is unique in the energy class.

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arXiv2020

On the supercritical defocusing NLW outside a ball

We study a defocusing semilinear wave equation, with a power nonlinearity $|u|^{p-1}u$, defined outside the unit ball of $\mathbb{R}^{n}$, $n\ge3$, with Dirichlet boundary conditions. We prove that if $p>n+4$ and the initial data are nonradial perturbations of large radial data, there exists a global smooth solution. The solution is unique among energy class solutions satisfying an energy inequality. The main tools used are the Penrose transform and a pointwise decay estimate for the exterior linear wave equation perturbed with a large, time dependent potential.

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arXiv2020

Eigenvalue bounds for non-selfadjoint Dirac operators

In this work we prove that the eigenvalues of the $n$-dimensional massive Dirac operator $\mathscr{D}_0 + V$, $n\ge2$, perturbed by a possibly non-Hermitian potential $V$, are localized in the union of two disjoint disks of the complex plane, provided that $V$ is sufficiently small with respect to the mixed norms $L^1_{x_j} L^\infty_{\widehat{x}_j}$, for $j\in\{1,\dots,n\}$. In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on $V$, and in particular the spectrum is the same of the unperturbed operator, namely $σ(\mathscr{D}_0+V)=σ(\mathscr{D}_0)=\mathbb{R}$. The main tools we employ are an abstract version of the Birman-Schwinger principle, which include also the study of embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

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arXiv2019

A short proof of commutator estimates

The goal of this note is to give, at least for a restricted range of indices, a short proof of homogeneous commutator estimates for fractional derivatives of a product, using classical tools. Both $L^{p}$ and weighted $L^{p}$ estimates can be proved by the same argument. When the space dimension is 1, we obtain some new estimates in the unexplored range $1/3<r\le1/2$.

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arXiv2018

A limiting absorption principle for the Helmholtz equation with variable coefficients

We prove a limiting absorption principle for a generalized Helmholtz equation on an exterior domain with Dirichlet boundary conditions \begin{equation*} (L+λ)v=f, \qquad λ\in \mathbb{R} \end{equation*} under a Sommerfeld radiation condition at infinity. The operator $L$ is a second order elliptic operator with variable coefficients, the principal part is a small, long range perturbation of $-Δ$, while lower order terms can be singular and large. The main tool is a sharp uniform resolvent estimate, which has independent applications to the problem of embedded eigenvalues and to smoothing estimates for dispersive equations.

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arXiv2017

Sharp $L^p$ estimates for Schrödinger groups on spaces of homogeneous type

We prove an $L^{p}$ estimate $$ \|e^{-itL} φ(L)f\|_{p}\lesssim (1+|t|)^s\|f\|_p, \qquad t\in \mathbb{R}, \qquad s=n\left|\frac{1}{2}-\frac{1}{p}\right| $$ for the Schrödinger group generated by a semibounded, selfadjoint operator $L$ on a metric measure space $\mathcal{X}$ of homogeneous type (where $n$ is the doubling dimension of $\mathcal{X}$). The assumptions on $L$ are a mild $L^{p_{0}}\to L^{p_{0}'}$ smoothing estimate and a mild $L^{2}\to L^{2}$ off--diagonal estimate for the corresponding heat kernel $e^{-tL}$. The estimate is uniform for $ φ$ varying in bounded sets of $\mathscr{S}(\mathbb{R})$, or more generally of a suitable weighted Sobolev space. We also prove, under slightly stronger assumptions on $L$, that the estimate extends to $$ \|e^{-itL} φ(θL)f\|_{p}\lesssim (1+θ^{-1}|t|)^s\|f\|_p, \qquad θ>0, \quad t\in \mathbb{R}, $$ with uniformity also for $θ$ varying in bounded subsets of $(0,+\infty)$. For nonnegative operators uniformity holds for all $θ>0$.

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