arXiv subjects

Sonae Hadama

Publications and source records attributed to Sonae Hadama.

11 recordsLinked to original sources

arXiv2026

Endpoint orthonormal Strichartz estimates

We establish the orthonormal Strichartz estimates in the abstract Keel--Tao framework, under precisely the same hypotheses as in their original theorem. As an important consequence, we resolve the endpoint conjecture for the Schr\"odinger equation in dimensions $d\ge 2$, which was first raised by Frank, Lewin, Lieb, and Seiringer. This consequence also yields an affirmative answer to the conjecture for the free transport equation, which was stated in dual form by Bennett, Bez, Guti\'errez, and Lee. As a further application, we also establish a refinement of the Strichartz estimate for a single function.

math.CA↗

arXiv2026

Small-data $L^2$ theory for the intermediate NLS and the Calogero--Moser derivative NLS

In this paper, we study a class of nonlinear Schr\"odinger equations (NLS) in a unified way. This class includes two important examples: the intermediate NLS (INLS) and the Calogero--Moser derivative NLS (CM-DNLS). Our main results are twofold. First, we prove small-data global well-posedness in $L^2(\mathbb{R})$ for a broad class of equations. This includes both focusing and defocusing CM-DNLS and the INLS for arbitrary choices of its parameters. Second, we prove small-data scattering in $L^2(\mathbb{R})$ under an additional assumption. This result covers both focusing and defocusing CM-DNLS and the INLS for specific choices of its parameters. Both the formulation of the problem, including the notion of solution, and the proofs rely crucially on a linear theory for Schr\"odinger equations with rough time-dependent potentials. This theory is also of independent interest, since we allow potentials so rough that the standard Duhamel formulation may not make sense. Our approach is perturbative and is built on the bilinear Strichartz estimate proved by Ozawa and Tsutsumi in 1998. In particular, our argument does not rely on integrability.

math.AP↗

arXiv2026

Applications of renormalisation to orthonormal Strichartz estimates and the NLS system on the circle

In this paper, we introduce a renormalisation procedure for the density associated with the system of nonlinear Schr\"odinger equations (NLSS) on a circle. We show that this renormalised density satisfies better orthonormal Strichartz estimates than the non-renormalised density, which was considered in Nakamura (2020). As an application, we determine the critical Schatten exponent below which the cubic renormalised NLSS on the circle is globally well-posed and above which it is ill-posed. Finally, we show that the improvement for orthonormal Strichartz estimates satisfied by the renormalised density on $\mathbb{T}^d$ for $d \ge 2$ is minimal.

math.AP↗

arXiv2026

Well-posedness in the full scaling-subcritical range for a class of nonlocal NLS on the line

In this paper, we study a class of one-dimensional nonlocal nonlinear Schr\"odinger equations on the line with nonlinearity given by a Fourier multiplier whose symbol has subcritical high-frequency growth. In terms of symbol order, this class is intermediate between the cubic nonlinear Schr\"odinger equation and the Calogero--Moser derivative nonlinear Schrd\"oinger equation. We prove local well-posedness in $L^2(\mathbb{R})$ throughout the full scaling-subcritical range. Due to derivative loss, the standard Duhamel integral is not directly meaningful for rough data. To avoid this problem, we first construct the propagator $S_V$ for rough time-dependent potentials $V$, and then prove an Ozawa-Tsutsumi type bilinear Strichartz estimate for the perturbed flow $S_V$. These linear theories yield a concrete construction of rough solutions without using any equation-specific algebraic structure. For real-valued symbols, mass is conserved, and the local solutions are therefore global.

math.AP↗

arXiv2025

Uniform dispersive estimates for the semi-classical Hartree equation with long-range interaction

In this paper, we consider the Hartree equation with smooth but long-range interaction in the semi-classical regime, in three-dimensional space. We show that the density function of small-data solution decays at the optimal rate. When the semi-classical parameter $\hbar \in (0,1]$ is fixed, our result is essentially covered by the recent work by Nguyen and You [arXiv:2408.15860]; however, the novelty of this paper is the uniformity with respect to $\hbar$. Namely, both smallness condition for initial data and bounds for the solution are independent of $\hbar$. Moreover, the argument in this paper provides a new proof of the modified scattering for the long-range nonlinear Schr\"{o}dinger equation with a Hartree type nonlinearity. Our proof relies on three main ingredients. First, we prove the boundedness of finite-time wave operators modified by phase corrections. Second, we show an $L^1$--$L^\infty$ dispersive estimate for the modified propagator. Third, we give various kinds of commutator estimates for density operators. By combining them, we can apply the usual bootstrap argument to obtain the main result.

math.AP↗

arXiv2025

Semi-classical limit of quantum scattering states for the nonlinear Hartree equation

This article concerns the long-time dynamics of quantum particles in the semi-classical regime. First, we show that for the nonlinear Hartree equation with short-range interaction potential, small-data solutions obey dispersion bounds and they scatter, where the smallness conditions and the bounds are independent of the small parameter $\hbar\in(0,1]$ representing the reduced Planck constant. Then, taking the semi-classical limit $\hbar\to0$, we prove that the Wigner transforms of such quantum scattering states converge weakly-* to the corresponding classical scattering states for the Vlasov equation. As a direct consequence, we establish small-data scattering for the Vlasov equation without assuming regularity on initial data. Our analysis is based on a new uniform dispersion estimate for the free Schr\"odinger flow, which is simple but crucial to include singular interaction potentials such as inverse power-law potential $\frac{1}{|x|^a}$ with $1<a<\frac{5}{3}$.

math.AP↗

arXiv2025

Scattering for the positive density Hartree equation

We study the asymptotic stability for large times of homogeneous stationary states for the nonlinear Hartree equation for density matrices in Rd for d\geq3. We can reach both the optimal Sobolev and Schatten exponents for the initial data, with a wide class of interaction potentials w (under the sole assumption that w is bounded, including in particular delta potentials). Our method relies on fractional Leibniz rules for density matrices to deal with the fractional critical Sobolev regularity s = d/2 -1 for odd d, as well as Christ-Kiselev lemmas in Schatten spaces.

math.AP↗

arXiv2024

Global well-posedness of the nonlinear Hartree equation for infinitely many particles with singular interaction

The nonlinear Hartree equation (NLH) in the Heisenberg picture admits steady states of the form $\gamma_f=f(-\Delta)$ representing quantum states of infinitely many particles. In this article, we consider the time evolution of perturbations from a large class of such steady states via the three-dimensional NLH. We prove that if the interaction potential $w$ has finite measure and initial states have finite relative entropy, then solutions preserve the relative free energy, and they exist globally in time. This result extends the important work of Lewin and Sabin [arXiv:1310.0603] to singular interaction cases.

math.AP↗

arXiv2023

Probabilistic Strichartz estimates in Schatten classes and their applications to the Hartree equation

In this paper, we consider the Strichartz estimates for orthonormal systems in the context of randomization. Frank, Lewin, Lieb, and Seiringer first proved the orthonormal Strichartz estimates. After that, many authors have studied this type of inequality. In this paper, we introduce two randomizations of operators and show that they allow us to treat strictly bigger Schatten exponents than the sharp exponents of the deterministic orthonormal Strichartz estimates. In the proofs, the orthogonality does not have any essential role, and randomness works instead of it. We also prove that our randomizations of operators never change the Schatten classes to which they originally belong. Moreover, we give some applications of our results to the Hartree and linearized Hartree equations for infinitely many particles. First, we construct local solutions to the Hartree equation with initial data in wide Schatten classes. By only using deterministic orthonormal Strichartz estimates, it is impossible to give any solution in our settings. Next, we consider the scattering problem of the linearized Hartree equation. One of our randomizations allows us to treat wide Schatten classes with some Sobolev regularities, and by the other randomization, we can remove all Sobolev regularities.

math.AP↗

arXiv2023

Asymptotic stability of a wide class of stationary solutions for the Hartree and Schr\"{o}dinger equations for infinitely many particles

We consider the Hartree and Schr\"{o}dinger equations describing the time evolution of wave functions of infinitely many interacting fermions in three-dimensional space. These equations can be formulated using density operators, and they have infinitely many stationary solutions. In this paper, we prove the asymptotic stability of a wide class of stationary solutions. We emphasize that our result includes Fermi gas at zero temperature. This is one of the most important steady states from the physics point of view; however, its asymptotic stability has been left open after Lewin and Sabin first formulated this stability problem and gave significant results in their seminal work [arXiv:1310.0604].

math.AP↗

arXiv2023

Asymptotic stability of a wide class of steady states for the Hartree equation for random fields

We study the Hartree equation describing the time evolution of the wave functions of infinitely many fermions interacting with each other. The Hartree equation can be formulated in terms of random fields. This formulation was introduced by de Suzzoni in [arXiv:1507.06180]. It has infinitely many steady states, and [arXiv:1811.03150, arXiv:2007.00472] have studied its asymptotic stability. However, they required somewhat strong conditions for steady states, and the stability of Fermi gas at zero temperature, one of the most important steady states from the physics point of view, was left open. In this paper, we prove the stability of steady states in a wide class, which includes Fermi gas at zero temperature in $d$-dimensional space when $d \ge 3$. This paper is the revised version. There are two major changes. The first one is that the assumptions for potential $w$ in Theorem 1.3 are weakened. The second one is that we removed Theorem 1.4 in the previous version because the author found some gaps which seemed difficult to fix.

math.AP↗