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Date: 2026-01-30
Authors:
Cangiotti, Nicolò, Gallo, Ivan, Spitzkopf, David
Abstract
In this manuscript, we shall investigate the Nonlinear Magnetic Schrödinger Equation on noncompact metric graphs, focusing on the existence of ground states. We prove that the magnetic Hamiltonian is variationally equivalent to a non-magnetic operator with additional repulsive potentials supported on the graph's cycles. This effective potential is strictly determined by the Aharonov-Bohm flux through the topological loops. Leveraging this reduction, we extend classical existence criteria to the magnetic setting. As a key application, we characterize the ground state structure on the tadpole graph, revealing a mass-dependent phase transition. The ground states exist for sufficiently small repulsion in an intermediate regime of masses while sufficiently strong flux prevents the formation of ground states.
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Date: 2025-12-03
Authors:
Nicola, Fabio, Riccardi, Federico, Tilli, Paolo
Abstract
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.
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Date: 2025-10-21
Authors:
Federico Stra, Erling A. T. Svela, S. Ivan Trapasso
Abstract
We prove that, for any measurable phase space subset \(Ω\subset\mathbb{R}^{2d}\) with \(0<|Ω|<\infty\) and any \(1\le p < \infty\), the nonlinear concentration problem \[ \sup_{f \in L^2(\mathbb{R}^d)\setminus\{0\}}\frac{\|Wf\|_{L^p(Ω)}}{\|f\|_{L^2}^2}\] admits an optimizer, where \(Wf\) is the Wigner distribution of \(f\). The main obstruction is that \(Wf\) is covariant (not invariant) under time-frequency shifts, which impedes weak upper semicontinuity, so the effects of constructive interference must be taken into account. We close this compactness gap via concentration compactness for Heisenberg-type dislocations, together with a new asymptotic formula that quantifies the limiting contribution to concentration over \(Ω\) from asymptotically separated wave packets. When \(p=\infty\) we also identify the sharp constant \(2^d\) and show that it is attained. We also discuss some related extensions: For \(τ\)-Wigner distributions with \(τ\in (0,1)\) we isolate a chain phenomenon that obstructs the same strategy beyond the Wigner case (\(τ=1/2\)), while for the Born-Jordan distribution in \(d=1\) we obtain weak continuity, and thus existence of concentration optimizers for all \(1\le p<\infty\) (the \(p=\infty\) supremum equals \(π\) but is not attained).
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Date: 2025-08-13
Authors:
Patrik Wahlberg
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Date: 2025-08-12
Authors:
S. Ivan Trapasso
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Date: 2025-08-04
Authors:
Elena Cordero, Gianluca Giacchi, Edoardo Pucci, S. Ivan Trapasso
Abstract
Motivated by the phase space analysis of Schrödinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave packets. Quantitative bounds for the matrix coefficients arising in the Gabor wave packet decomposition of such operators are established, revealing precise exponential decay rates together with subtler dispersive and spreading phenomena. To this aim, we present several novel results concerning the time-frequency analysis of functions with controlled Gelfand-Shilov regularity, which are of independent interest. As a byproduct, we generalize Vemuri's Gaussian confinement results for the solutions of the quantum harmonic oscillator in two respects, namely by encompassing general exponential decay rates as well as arbitrary quadratic Schrödinger propagators. In particular, we extensively discuss some prominent models such as the harmonic oscillator, the free particle in a constant magnetic field and fractional Fourier transforms.
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Date: 2025-07-28
Authors:
Federico Riccardi
Abstract
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.
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Date: 2025-07-21
Authors:
Fabio Nicola
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Date: 2025-07-17
Authors:
Fabio Nicola, Federico Riccardi
Abstract
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.
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Date: 2025-07-07
Authors:
Aleksei Kulikov, Fabio Nicola, Joaquim Ortega‐Cerdà, Paolo Tilli
Abstract
We prove 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 on a number of results and conjectures on this subject, in particular on the Wehrl entropy conjecture of Lieb and Solovej. In this connection, we completely prove that conjecture for the group S U ( 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.
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Date: 2025-04-25
Authors:
Patrik Wahlberg
Abstract
We treat the optimal linear filtering problem for a sum of two second order uncorrelated generalized stochastic processes. This is an operator equation involving covariance operators. We study both the wide-sense stationary case and the non-stationary case. In the former case the equation simplifies into a convolution equation. The solution is the Radon–Nikodym derivative between non-negative tempered Radon measures, for signal and signal plus noise respectively, in the frequency domain. In the non-stationary case we work with pseudodifferential operators with symbols in Sjöstrand modulation spaces which admits the use of its spectral invariance properties.
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Date: 2025-04-17
Authors:
Sonia Mazzucchi, Fabio Nicola, S. Ivan Trapasso
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Date: 2025-04-04
Authors:
Patrik Wahlberg
Abstract
We define the Wigner distribution of a tempered generalized stochastic process that is complex-valued symmetric Gaussian. This gives a time-frequency generalized stochastic process defined on the phase space. We study its covariance and our main result is a formula for the Weyl symbol of the covariance operator, expressed in terms of the Weyl symbol of the covariance operator of the original generalized stochastic process.
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Date: 2025-02-22
Authors:
Riccardo Adami, Ugo Boscain, Dario Prandi, Lucia Tessarolo
Abstract
Let \(M\) be a 3-dimensional contact sub-Riemannian manifold and \(S\) a surface embedded in \(M\). Such a surface inherits a field of directions that becomes singular at characteristic points. The integral curves of such field define a characteristic foliation \(\mathscr{F}\). In this paper we study the Schrödinger evolution of a particle constrained on \(\mathscr{F}\). In particular, we relate the self-adjointness of the Schrödinger operator with a geometric invariant of the foliation. We then classify a special family of its self-adjoint extensions: those that yield disjoint dynamics.
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Date: 2025-02-14
Authors:
Federico Riccardi
Abstract
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.
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Date: 2025-02-05
Authors:
Riccardo Adami, Filippo Boni, Takaaki Nakamura, Alice Ruighi
Abstract
We study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of \(L^2\)-subcritical and \(L^2\)-critical nonlinearity. For a subcritical nonlinearity we prove the existence and the uniqueness of Ground States at any mass. If the mass exceeds an explicit threshold, then there exists a positive excited state too. For the critical nonlinearity we prove that Ground States exist only in a specific interval of masses, while in a different interval excited states exist. We provide the value of the optimal constant in the Gagliardo-Nirenberg estimate and describe in the dipole case the branches of the stationary states as the strength of the interaction varies. Since all stationary states are explicitly computed, ours is a solvable model involving a non-standard interplay of a nonlinearity with a point interaction.
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Date: 2025-01-22
Authors:
Maria Cristina Grimaldi, Sara Bozzer, Dick J. Sjöström, Linnéa Andersson, Tom Eirik Mollnes, Per H. Nilsson, Luca De Maso, Federico Riccardi, Michele Dal Bo, Daniele Sblattero, Paolo Macor
Abstract
Our findings provide a rationale for the use of targeted NPs as a DNA delivery system for the local expression of therapeutic proteins.
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Date: 2025-01-18
Authors:
Riccardo Adami, Ivan Gallo, David Spitzkopf
Abstract
We consider the subcritical nonlinear Schrödinger equation on non-compact quantum graphs with an attractive potential supported in the compact core, and investigate the existence and the nonexistence of Ground States, defined as minimizers of the energy at fixed \(L^2\)-norm, or mass. We finally reach the following picture: for small and large mass there are Ground States. Moreover, according to the metric features of the compact core of the graph and to the strength of the potential, there may be an interval of intermediate masses for which there are no Ground States. The study was inspired by the research on quantum waveguides, in which the curvature of a thin tube induces an effective attractive potential.
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Date: 2025-01-03
Authors:
Giovanni S. Alberti, Alessandro Felisi, Matteo Santacesaria, S. Ivan Trapasso
Abstract
This paper extends the sample complexity theory for ill-posed inverse problems developed in a recent work by the authors [`Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform', J. Eur. Math. Soc., to appear], which was originally focused on the sparse Radon transform. We demonstrate that the underlying abstract framework, based on infinite-dimensional compressed sensing and generalized sampling techniques, can effectively handle a variety of practical applications. Specifically, we analyze three case studies: (1) The reconstruction of a sparse signal from a finite number of pointwise blurred samples; (2) The recovery of the (sparse) source term of an elliptic partial differential equation from finite samples of the solution; and (3) A moderately ill-posed variation of the classical sensing problem of recovering a wavelet-sparse signal from finite Fourier samples, motivated by magnetic resonance imaging. For each application, we establish rigorous recovery guarantees by verifying the key theoretical requirements, including quasi-diagonalization and coherence bounds. Our analysis reveals that careful consideration of balancing properties and optimized sampling strategies can lead to improved reconstruction performance. The results provide a unified theoretical foundation for compressed sensing approaches to inverse problems while yielding practical insights for specific applications.
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Date: 2024-12-18
Authors:
Junior, Alexandre Arias, Patrik Wahlberg
Abstract
We study continuity of the multiplier operator \(e^{i q}\) acting on Gelfand–Shilov spaces, where \(q\) is a polynomial on \(\mathbf R^d\) of degree at least two with real coefficients. In the parameter quadrant for the spaces we identify a wedge that depends on the polynomial degree for which the operator is continuous. We also show that in a large part of the complement region the operator is not continuous in dimension one. The results give information on well-posedness for linear evolution equations that generalize the Schrödinger equation for the free particle.
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Date: 2024-12-14
Authors:
Fabio Nicola, Federico Riccardi, Paolo Tilli
Abstract
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.
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Date: 2024-08-20
Authors:
S. Ivan Trapasso
Abstract
We perform a phase space analysis of evolution equations associated with the Weyl quantization \(q^{\mathrm{w}}\) of a complex quadratic form \(q\) on \(\mathbb{R}^{2d}\) with non-positive real part. In particular, we obtain pointwise bounds for the matrix coefficients of the Gabor wave packet decomposition of the generated semigroup \(e^{tq^{\mathrm{w}}}\) if \(\mathrm{Re} (q) \le 0\) and the companion singular space associated is trivial. This result is then leveraged to achieve a comprehensive analysis of the phase regularity of \(e^{tq^{\mathrm{w}}}\) with \(\mathrm{Re} (q) \le 0\), thereby extending the \(L^2\) analysis of quadratic semigroups initiated by Hitrik and Pravda-Starov to general modulation spaces \(M^p(\mathbb{R}^d)\), \(1 \le p \le \infty\), with optimal explicit bounds.
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Date: 2024-07-22
Authors:
Sonia Mazzucchi, Fabio Nicola, S. Ivan Trapasso
Abstract
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.
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Date: 2024-06-18
Authors:
Fabio Nicola
Abstract
A Gabor orthonormal basis, on a locally compact Abelian (LCA) group A , is an orthonormal basis of L 2 ( A ) that consists of time-frequency shifts of some template f ∈ L 2 ( A ) . It is well known that, on 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 note we find all the Gabor orthonormal bases with this extremal property. To this end, we identify all the functions in L 2 ( A ) that are maximally localized in the time-frequency space in the above sense – an issue that 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.
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Date: 2024-06-10
Authors:
Fabio Nicola
Abstract
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.
