arXiv subjects

Fabio Nicola

Publications and source records attributed to Fabio Nicola.

At least 19 recordsLinked to original sources

arXiv2026

The Fefferman-Phong uncertainty principle for representations of Lie groups and applications

The Fefferman-Phong uncertainty principle is a deep result concerning the order of magnitude of the bottom of the spectrum of second order pseudodifferential operators with nonnegative (Weyl) symbol. We show that a similar uncertainty principle holds for discrete series representations of connected Lie groups, where the concentration of the matrix coefficients is measured in terms of weighted $L^p$ norms, with weights in the local Muckenhoupt class $A_{\infty,{\rm loc}}$ associated with a subRiemannian left-invariant metric and a relatively invariant measure. The proof relies on a certain connection with lower bounds for left-invariant subLaplacians. As a consequence of this result (in the case of the Schrödinger representation of the reduced Heisenberg group), we provide an explicit formula for the order of magnitude of the bottom of the spectrum and also of the essential spectrum of semiclassical anti-Wick operators with nonnegative symbols of arbitrary order, hence providing the analog, for the anti-Wick quantization, of the above mentioned result by Fefferman and Phong. We consider symbols in standard symbol classes appearing in semiclassical analysis, and also in the Muckenhoupt classes (hence possibly non-smooth).

math.CA↗

arXiv2026

On the existence of extremizers for the sum of eigenvalues of Toeplitz operators

We prove that, among all measurable sets $Ω\subset\mathbb{C}$ of prescribed Lebesgue measure, there exists a set maximizing the sum of the first $K$ eigenvalues ($K\geq 1$) of the associated Toeplitz operator on the Fock space. In the Fock setting, the case $K=1$ is well known, the optimal sets being balls of prescribed measure, whereas for $K>1$ the existence of optimal sets appears to be new (maximizers are not known explicitly, and the optimality of balls remains conjectural). Moreover, under mild assumptions, our proof extends to localization operators associated with abstract wavelet transforms. In this broader setting, the result is new even for $K=1$. As an application, we prove the existence of optimal sets for the Donoho--Stark concentration problem and its generalization to orthonormal systems.

math.FA↗

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.

math.AP↗

arXiv2025

An elementary approach to Wehrl-type entropy bounds in quantitative form

We consider the problem of the stability (with sharp exponent) of the Lieb--Solovej inequality for symmetric $SU(N)$ coherent states, which was obtained only recently by the authors. Here, we propose an elementary proof of this result, based on reformulating the Wehrl-type entropy as a function defined on the unit sphere in $\mathbb{C}^d$, for some suitable $d$, and on some explicit (and somewhat surprising) computations.

math-ph↗

arXiv2025

The Wehrl-type entropy conjecture for symmetric $SU(N)$ coherent states: cases of equality and stability

Lieb and Solovej proved that, for the symmetric $SU(N)$ representations, the corresponding Wehrl-type entropy is minimized by symmetric coherent states. However, the uniqueness of the minimizers remained an open problem when $N\geq 3$. In this note, we complete the proof of the Wehrl entropy conjecture for such representations by showing that symmetric coherent states are, in fact, the only minimizers. We also provide an application to the maximum concentration of holomorphic polynomials and deduce a corresponding Faber-Krahn inequality. A sharp quantitative form of the bound by Lieb and Solovej is also proved.

math-ph↗

arXiv2025

The Hudson theorem in LCA groups and infinite quantum spin systems

The celebrated Hudson theorem states that the Gaussian functions in $\mathbb{R}^d$ are the only functions whose Wigner distribution is everywhere positive. Motivated by quantum information theory, D. Gross proved an analogous result on the Abelian group $\mathbb{Z}_d^n$, for $d$ odd - corresponding to a system of $n$ qudits - showing that the Wigner distribution is nonnegative only for the so-called stabilizer states. Extending this result to the thermodynamic limit of finite-dimensional systems naturally leads us to consider general $2$-regular LCA groups that possess a compact open subgroup, where the issue of the positivity of the Wigner distribution is currently an open problem. We provide a complete solution to this question by showing that if the map $x\mapsto 2x$ is measure-preserving, the functions whose Wigner distribution is nonnegative are exactly the subcharacters of second degree, up to translation and multiplication by a constant. Instead, if the above map is not measure-preserving, the Wigner distribution always takes negative values. We discuss in detail the particular case of infinite sums of discrete groups and infinite products of compact groups, which correspond precisely to infinite quantum spin systems. Further examples include $n$-adic systems, where $n\geq 2$ is an arbitrary integer (not necessarily a prime), as well as solenoid groups.

math-ph↗

arXiv2025

The isoperimetric inequality for partial sums of Toeplitz eigenvalues in the Fock space

We prove that, among all subsets $Ω\subset \mathbb{C}$ having circular symmetry and prescribed measure, the ball is the only maximizer of the sum of the first $K$ eigenvalues ($K\geq 1$) of the corresponding Toeplitz operator $T_Ω$ on the Fock space $\mathcal{F}$. As a byproduct, we prove that balls maximize any Schatten $p$-norm of $T_Ω$ for $p>1$ (and minimize the corresponding quasinorm for $p<1$), and that the second eigenvalue is maximized by a particular annulus. Moreover, we extend some of these results to general radial symbols in $L^p(\mathbb{C})$, with $p > 1$, characterizing those that maximize the sum of the first $K$ eigenvalues. We also show a symmetry breaking phenomenon for the second eigenvalue, when the assumption of circular symmetry is dropped.

math.FA↗

arXiv2025

Dynamical restriction for Schrödinger equations

We prove a dynamical restriction principle, asserting that every restriction estimate satisfied by the Fourier transform in $\mathbb{R}^d$ is also valid for the propagator of certain Schrödinger equations. We consider smooth Hamiltonians with an at most quadratic growth, and also a class of nonsmooth Hamiltonians, encompassing potentials that are Fourier transforms of complex (finite) Borel measures. Roughly speaking, if the initial datum belongs to $L^p(\mathbb{R}^d)$, for $p$ in a suitable range of exponents, the solution $u(t,\cdot)$ (for each fixed $t$, with the exception of certain particular values) can be meaningfully restricted to compact curved submanifolds of $\mathbb{R}^d$. The underlying property responsible for this phenomenon is the boundedness of the propagator $L^p\to(\mathcal{F}L^p)_{\rm loc}$, with $1\leq p\leq2$, which is derived from almost diagonalization and dispersive estimates in function spaces defined in terms of wave packet decompositions in phase space.

math.AP↗

arXiv2025

A life in Mathematical Analysis: a conversation with Luigi Rodino

This note is the transcription of an interview with Professor Luigi Rodino, on the occasion of the ISAAC-ICMAM Conference of Analysis in Developing Countries (December 2, 2024 - Bogotà), that was dedicated to him. Luigi Rodino is at present Emeritus Professor at the University of Turin, and a member of the Accademia delle Scienze di Torino.

math.HO↗

arXiv2025

Generalized moduli of continuity under irregular or random deformations via multiscale analysis

Motivated by the problem of robustness to deformations of the input for deep convolutional neural networks, we identify signal classes which are inherently stable to irregular deformations induced by distortion fields $τ\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)$, to be characterized in terms of a generalized modulus of continuity associated with the deformation operator. Resorting to ideas of harmonic and multiscale analysis, we prove that for signals in multiresolution approximation spaces $U_s$ at scale $s$, stability in $L^2$ holds in the regime $\|τ\|_{L^\infty}/s\ll 1$ - essentially as an effect of the uncertainty principle. Instability occurs when $\|τ\|_{L^\infty}/s\gg 1$, and we provide a sharp upper bound for the asymptotic growth rate. The stability results are then extended to signals in the Besov space $B^{d/2}_{2,1}$ tailored to the given multiresolution approximation. We also consider the case of more general time-frequency deformations. Finally, we provide stochastic versions of the aforementioned results, namely we study the issue of stability in mean when $τ(x)$ is modeled as a random field (not bounded, in general) with identically distributed variables $|τ(x)|$, $x\in\mathbb{R}^d$.

math.FA↗

arXiv2025

Phase space analysis of finite and infinite dimensional Fresnel integrals

The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class $M^{\infty,1}$ - a family of continuous and bounded functions, locally enjoying the mild regularity of the Fourier transform of an integrable function. This result broadly extends the current knowledge on the Fresnel integrability of Fourier transforms of finite complex measures, and relies upon ideas and techniques of Gabor wave packet analysis. We also discuss the problem of designing infinite-dimensional extensions of this result, obtaining the first, non-trivial concrete realization of a general framework of projective functional extensions introduced by Albeverio and Mazzucchi. As an interesting byproduct, we obtain the exact $M^{\infty,1} \to L^\infty$ operator norm of the free Schrödinger evolution operator.

math.FA↗

arXiv2024

Phase space analysis of higher-order dispersive equations with point interactions

We investigate nonlinear, higher-order dispersive equations with measure (or even less regular) potentials and initial data with low regularity. Our approach is of distributional nature and relies on the phase space analysis (via Gabor wave packets) of the corresponding fundamental solution - in fact, locating the modulation/amalgam space regularity of such generalized Fresnel-type oscillatory functions is a problem of independent interest in harmonic analysis.

math.AP↗

arXiv2024

The wave function of stabilizer states and the Wehrl conjecture

We focus on quantum systems represented by a Hilbert space $L^2(A)$, where $A$ is a locally compact Abelian group that contains a compact open subgroup. We examine two interconnected issues related to Weyl-Heisenberg operators. First, we provide a complete and elegant solution to the problem of describing the stabilizer states in terms of their wave functions, an issue that arises in quantum information theory. Subsequently, we demonstrate that the stabilizer states are precisely the minimizers of the Wehrl entropy functional, thereby resolving the analog of the Wehrl conjecture for any such group. Additionally, we construct a moduli space for the set of stabilizer states, that is, a parameterization of this set, that endows it with a natural algebraic structure, and we derive a formula for the number of stabilizer states when $A$ is finite. Notably, these results are novel even for finite Abelian groups.

math-ph↗

arXiv2024

The quantitative isoperimetric inequality for the Hilbert-Schmidt norm of localization operators

In this paper we study the Hilbert-Schmidt norm of time-frequency localization operators $L_Ω \colon L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$, with Gaussian window, associated with a subset $Ω\subset\mathbb{R}^{2d}$ of finite measure. We prove, in particular, that the Hilbert-Schmidt norm of $L_Ω$ is maximized, among all subsets $Ω$ of a given finite measure, when $Ω$ is a ball and that there are no other extremizers. Actually, the main result is a quantitative version of this estimate, with sharp exponent. A similar problem is addressed for wavelet localization operators, where rearrangements are understood in the hyperbolic setting.

math.CA↗

arXiv2023

The generalized Wehrl entropy bound in quantitative form

Lieb and Carlen have shown that mixed states with minimal Wehrl entropy are coherent states. We prove that mixed states with almost minimal Wehrl entropy are almost coherent states. This is proved in a quantitative sense where both the norm and the exponent are optimal and the constant is explicit. We prove a similar bound for generalized Wehrl entropies. As an application, a sharp quantitative form of the log-Sobolev inequality for functions in the Fock space is provided.

math-ph↗

arXiv2023

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

A Gabor orthonormal basis, on a locally compact Abelian (LCA) group $A$, is an orthonormal basis of $L^2(A)$ which consists of time-frequency shifts of some template $f\in L^2(A)$. It is well-known that, on $\mathbb{R}^d$, the elements of such a basis cannot have a good time-frequency localization. The picture is drastically different on LCA groups containing a compact open subgroup, where one can easily construct examples of Gabor orthonormal bases with $f$ maximally localized, in the sense that the ambiguity function of $f$ (i.e. the correlation of $f$ with its time-frequency shifts) has support of minimum measure, compatibly with the uncertainty principle. In this paper we find all the Gabor orthonormal bases with this extremal property. To this end, we identify all the functions in $L^2(A)$ which are maximally localized in the time-frequency space in the above sense -- an issue which is open even for finite Abelian groups. As a byproduct, on every LCA group containing a compact open subgroup, we exhibit the complete family of optimizers for Lieb's uncertainty inequality, and we also show previously unknown optimizers on a general LCA group.

math.FA↗

arXiv2023

A monotonicity theorem for subharmonic functions on manifolds

We provide a sharp monotonicity theorem about the distribution of subharmonic functions on manifolds, which can be regarded as a new, measure theoretic form of the uncertainty principle. As an illustration of the scope of this result, we deduce contractivity estimates for analytic functions on the Riemann sphere, the complex plane and the Poincaré disc, with a complete description of the extremal functions, hence providing a unified and illuminating perspective of a number of results and conjectures on this subject, in particular on the Wehrl entropy conjecture by Lieb and Solovej. In this connection, we completely prove that conjecture for SU(2), by showing that the corresponding extremals are only the coherent states. Also, we show that the above (global) estimates admit a local counterpart and in all cases we characterize also the extremal subsets, among those of fixed assigned measure.

math.CA↗

arXiv2022

The norm of time-frequency and wavelet localization operators

Time-frequency localization operators (with Gaussian window) $L_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, where $F$ is a weight in $\mathbb{R}^{2d}$, were introduced in signal processing by I. Daubechies in 1988, inaugurating a new, geometric, phase-space perspective. Sharp upper bounds for the norm (and the singular values) of such operators turn out to be a challenging issue with deep applications in signal recovery, quantum physics and the study of uncertainty principles. In this note we provide optimal upper bounds for the operator norm $\|L_F\|_{L^2\to L^2}$, assuming $F\in L^p(\mathbb{R}^{2d})$, $1<p<\infty$ or $F\in L^p(\mathbb{R}^{2d})\cap L^\infty(\mathbb{R}^{2d})$, $1\leq p<\infty$. It turns out that two regimes arise, depending on whether the quantity $\|F\|_{L^p}/\|F\|_{L^\infty}$ is less or greater than a certain critical value. In the first regime the extremal weights $F$, for which equality occurs in the estimates, are certain Gaussians, whereas in the second regime they are proved to be truncated Gaussians, degenerating in a multiple of a characteristic function of a ball for $p=1$. This phase transition through truncated Gaussians appears to be a new phenomenon in time-frequency concentration problems. For the analogous problem for wavelet localization operators -- where the Cauchy wavelet plays the role of the above Gaussian window -- a complete solution is also provided.

math.CA↗