arXiv subjects

Federico Riccardi

Publications and source records attributed to Federico Riccardi.

8 recordsLinked to original sources

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↗

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

An optimal estimate for the norm of wavelet localization operators

In this paper we prove an optimal estimate for the norm of wavelet localization operators with Cauchy wavelet and weight functions that satisfy two constraints on different Lebesgue norms. We prove that multiple regimes arise according to the ratio of these norms: if this ratio belongs to a fixed interval (which depends on the Lebesgue exponents) then both constraints are active, while outside this interval one of the constraint is inactive. Furthermore, we characterize optimal weight functions.

math.FA↗

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

A new optimal estimate for the norm of time-frequency localization operators

In this paper we provide an optimal estimate for the operator norm of time-frequency localization operators with Gaussian window $L_{F,φ} : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$, under the assumption that $F \in L^p(\mathbb{R}^{2d}) \cap L^q(\mathbb{R}^{2d})$ for some $p$ and $q$ in $(1,+\infty)$. We are also able to characterize optimal weight functions, whose shape turns out to depend on the ratio $\|F\|_q / \|F\|_p$. Roughly speaking, if this ratio is "sufficiently large" or "sufficiently small" optimal weight functions are certain Gaussians, while if it is in the intermediate regime the optimal functions are no longer Gaussians. As an application, we extend Lieb's uncertainty inequality to the space $L^p + L^q$.

math.CA↗