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Lorenzo Tentarelli

Publications and source records attributed to Lorenzo Tentarelli.

At least 19 recordsLinked to original sources
arXiv2026

NLS ground states on a hybrid plane

We study existence, nonexistence, and qualitative properties of ground states for a focusing, subcritical Nonlinear Schrödinger Equation on a hybrid plane, consisting of a half-line attached to a plane. Ground states are normalized minimizers of the associated energy, given by Nonlinear Schrödinger energies with contact interactions on the half-line and on the plane, plus a quadratic coupling term. At fixed mass, existence holds if the contact interaction on the half-line is not too repulsive, or the interaction on the plane is sufficiently attractive, or the coupling is strong enough. Nonexistence occurs when both interactions are sufficiently repulsive and the coupling is weak. Moreover, we discuss how the coupling affects the support and the symmetry properties of such ground states. These are the first results for a Nonlinear Schrödinger Equation on a mixed-dimensional manifold.

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arXiv2026

Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions

We investigate normalized solutions for doubly nonlinear Schrödinger equations on the real line with a defocusing standard nonlinearity and a focusing nonlinear point interaction of $δ$-type at the origin. We provide a complete characterization of existence and uniqueness for normalized solutions and for energy ground states for every value of the nonlinearity powers. We show that the interplay between a defocusing standard and a focusing point nonlinearity gives rise to new phenomena with respect to those observed with single nonlinearities, standard combined nonlinearities, and combined focusing standard and pointwise nonlinearities.

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arXiv2025

Dirac solitons in one-dimensional nonlinear Schrödinger equations

In this paper we study a family of one-dimensional stationary cubic nonlinear Schrödinger (NLS) equations with periodic potentials and linear part displaying Dirac points in the dispersion relation. By introducing a suitable periodic perturbation, one can open a spectral gap around the Dirac-point energy. This allows to construct standing waves of the NLS equation whose leading-order profile is a modulation of Bloch waves by means of the components of a spinor solving an appropriate cubic nonlinear Dirac (NLD) equation. We refer to these solutions as Dirac solitons. Our analysis thus provides a rigorous justification for the use of the NLD equation as an effective model for the original NLS equation.

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arXiv2025

Doubly nonlinear Schrödinger normalized ground states on 2D grids: existence results and singular limits

We investigate the existence and the singular limit of normalized ground states for focusing doubly nonlinear Schrödinger equations with both standard and concentrated nonlinearities on two-dimensional square grids. First, we provide existence and non-existence results for such ground states depending on the values of the nonlinearity powers and on the structure of the set of vertices where the concentrated nonlinearities are located. Second, we prove that suitable piecewise-affine extensions of such states converge strongly in $H^1(\R^2)$ to ground states of corresponding doubly nonlinear models defined on the whole plane as the length of the edges in the grid tends to zero. This convergence is proved both for limit models with standard nonlinearities only and for models combining standard and singular nonlinearities concentrated on a line or on a strip.

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arXiv2025

Non-uniqueness of normalized NLS ground states on polygons with homogeneous Neumann boundary conditions

We provide a non-uniqueness result for normalized ground states of nonlinear Schrödinger equations with pure power nonlinearity on polygons with homogeneous Neumann boundary conditions, defined as global minimizers of the associated energy functional among functions with prescribed mass. Precisely, for nonlinearity powers slightly smaller than the $L^2$-critical exponent, we prove that there always exists at least one value of the mass for which normalized ground states are not unique.

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arXiv2024

The search for NLS ground states on a hybrid domain: Motivations, methods, and results

We discuss the problem of establishing the existence of the Ground States for the subcritical focusing Nonlinear Schrödinger energy on a domain made of a line and a plane intersecting at a point. The problem is physically motivated by the experimental realization of hybrid traps for Bose-Einstein Condensates, that are able to concentrate the system on structures close to the domain we consider. In fact, such a domain approximates the trap as the temperature approaches the absolute zero. The spirit of the paper is mainly pedagogical, so we focus on the formulation of the problem and on the explanation of the result, giving references for the technical points and for the proofs.

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arXiv2023

Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity

We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on $\mathbb{S}^2$. Precisely, local well-posedness is proved for any $C^2$ power-nonlinearity, while global well-posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point-concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas.

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arXiv2023

A general review on the NLS equation with point-concentrated nonlinearity

The paper presents a complete (to the best of the author's knowledge) overview on the existing literature concerning the NLS equation with point-concentrated nonlinearity. Precisely, it mainly covers the following topics: definition of the model, weak and strong local well-posedness, global well-posedness, classification and stability (orbital and asymptotic) of the standing waves, blow-up analysis and derivation from the standard NLS equation with shrinking potentials. Also some related problem is mentioned.

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arXiv2022

Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

We investigate the ground states for the focusing, subcritical nonlinear Schrödinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass. We prove that ground states exist for every positive mass and show a logarithmic singularity at the defect. Moreover, up to a multiplication by a constant phase, they are positive, radially symmetric, and decreasing along the radial direction. In order to overcome the obstacles arising from the uncommon structure of the energy space, that complicates the application of standard rearrangement theory, we move to the study of the minimizers of the action functional on the Nehari manifold and then establish a connection with the original problem. An ad hoc result on rearrangements is given to prove qualitative features of the ground states.

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arXiv2022

Complete ionization for a non-autonomous point interaction model in d = 2

We consider the two dimensional Schrödinger equation with time dependent delta potential, which represents a model for the dynamics of a quantum particle subject to a point interaction whose strength varies in time. First, we prove global well-posedness of the associated Cauchy problem under general assumptions on the potential and on the initial datum. Then, for a monochromatic periodic potential (which also satisfies a suitable no-resonance condition) we investigate the asymptotic behavior of the survival probability of a bound state of the time-independent problem. Such probability is shown to have a time decay of order $\mathcal{O}(\log t/t)^2$, up to lower order terms.

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arXiv2022

Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect

We study the ground states for the Schrödinger equation with a focusing nonlinearity and a point interaction in dimension three. We establish that ground states exist for every value of the mass; moreover they are positive, radially symmetric, decreasing along the radial direction, and show a Coulombian singularity at the location of the point interaction. Remarkably, the existence of the ground states is independent of the attractive or repulsive character of the point interaction.

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arXiv2021

Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover

We discuss the model robustness of the infinite two-dimensional square grid with respect to symmetry breakings due to the presence of defects, that is, lacks of finitely or infinitely many edges. Precisely, we study how these topological perturbations of the square grid affect the so-called dimensional crossover identified in [Adami et al. 2019]. Such a phenomenon has two evidences: the coexistence of the one and the two-dimensional Sobolev inequalities and the appearence of a continuum of $L^2$-critical exponents for the ground states at fixed mass of the nonlinear Schrödinger equation. From this twofold perspective, we investigate which classes of defects do preserve the dimensional crossover and which classes do not.

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arXiv2021

On the nonlinear Dirac equation on noncompact metric graphs

The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., $ψ^{p-2}ψ$) on noncompact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. Precisely, we prove local well-posedness for the associated Cauchy problem in the operator domain and, for infinite $N$-star graphs, the existence of standing waves bifurcating from the trivial solution at $ω=mc^2$, for any $p>2$. In the Appendix we also discuss the nonrelativistic limit of the Dirac-Kirchhoff operator.

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arXiv2020

Stability of the standing waves of the concentrated NLSE in dimension two

In this paper we will continue the analysis of two dimensional Schrödinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the stability features.

math.AP↗
arXiv2019

Blow-up for the pointwise NLS in dimension two: absence of critical power

We consider the Schrödinger equation in dimension two with a fixed, pointwise, focusing nonlinearity and show the occurrence of a blow-up phenomenon with two peculiar features: first, the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves. Second, there is no critical power nonlinearity, i.e. for every power there exist blow-up solutions. This last property is uncommon among the conservative Schrödinger equations with local nonlinearity.

math.AP↗
arXiv2019

$L^2$-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features

Carrying on the discussion initiated in (Dovetta-Tentarelli'18), we investigate the existence of ground states of prescribed mass for the $L^2$-critical NonLinear Schrödinger Equation (NLSE) on noncompact metric graphs with localized nonlinearity. Precisely, we show that the existence (or nonexistence) of ground states mainly depends on a parameter called reduced critical mass, and then we discuss how the topological and metric features of the graphs affect such a parameter, establishing some relevant differences with respect to the case of the extended nonlinearity studied by (Adami-Serra-Tilli'17). Our results rely on a thorough analysis of the optimal constant of a suitable variant of the $L^2$-critical Gagliardo-Nirenberg inequality.

math.AP↗
arXiv2019

Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one

The paper discusses nonlinear singular perturbations of delta type of the fractional Schrödinger equation $\imath\partial_tψ=\left(-\triangle\right)^sψ$, with $s\in(\frac{1}{2},1]$, in dimension one. Precisely, we investigate local and global well posedness (in a strong sense), conservations laws and existence of blow-up solutions and standing waves.

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