Related papers: Semi-classical resolvent estimates for short-range…
We consider nonselfadjoint perturbations of semiclassical harmonic oscillators. Under appropriate dynamical assumptions, we establish some spectral estimates such as upper bounds on the resolvent near the real axis when no geometric control…
In this article, we analyze the propagation of Wigner measures of a family of solutions to a system of semi-classical pseudodifferential equations presenting eigenvalues crossings on hypersurfaces. We prove the propagation along classical…
We study the following fractional Schr\"{o}dinger equation \begin{equation*}\label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} where $s\in(0,1)$. Under some conditions on $f(u)$, we show that…
The present article is devoted to the semi-parametric estimation of multivariate expectiles for extreme levels. The considered multivariate risk measures also include the possible conditioning with respect to a functional covariate,…
The note is devoted to estimates for convolutions appearing in some class of stochastic Volterra equations. Two maximal inequalities and exponential tail estimate are proved by the fractional method of infinite dimensional stochastic…
We consider the energy critical four dimensional semi-linear heat equation \[ \partial_{t}v-\Delta v-v^{3}=0, \quad(t,x)\in \mathbb{R}\times \mathbb{R}^4. \] Formal computation of Filippas et al. (R. Soc. Lond. Proc. 2000) conjectures the…
The semi-classical regime of standing wave solutions of a Schr\"odinger equation in presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is show that there exists a…
Let $X$ be a connected Riemannian manifold such that the resolvent of the free Laplacian $(\Delta-z)^{-1}, z\in\C\setminus\R^{+},$ has a meromorphic continuation through $\R^{+}$. The poles of this continuation are called resonances. When…
We study the existence of stationnary positive solutions for a class of nonlinear Schroedinger equations with a nonnegative continuous potential V. Amongst other results, we prove that if V has a positive local minimum, and if the exponent…
In the paper we consider the following quasilinear Schr\"odinger--Poisson system in the whole space $\mathbb R^{3}$ $$ \begin{cases} - \varepsilon^2 \Delta u + (V + \phi) u = u |u|^{p - 1} \newline - \Delta \phi - \beta \Delta_4 \phi = u^2,…
We study the semirelativistic Hamiltonian operator composed of the relativistic kinetic energy and a static harmonic-oscillator potential in three spatial dimensions and construct, for bound states with vanishing orbital angular momentum,…
We give examples of semiclassical Schr\"odinger operators with exponentially large cutoff resolvent norms, even when the supports of the cutoff and potential are very far apart. The examples are radial, which allows us to analyze the…
We shall study the following initial value problem: \begin{equation}{\bf i}\partial_t u - \Delta u + V(x) u=0, \hbox{} (t, x) \in {\mathbf R} \times {\mathbf R}^n, \end{equation} $$u(0)=f,$$ where $V(x)$ is a real short--range potential,…
We prove $\ell^2(\mathbb{Z}^n)-$estimate of long $r$-variational seminorm for the family of discrete averages associated to simplices.
The present paper addresses questions on resonances for a $1$D Schr\"{o}dinger operator with truncated periodic potential. Precisely, we consider the half-line operator $H^{\mathbb N}=-\Delta +V$ and $H^{\mathbb N}_L = -\Delta + V…
This paper is devoted to study a class of nonlinear fractional Schr\"{o}dinger equations: \begin{equation*} (-\Delta)^{s}u+V(x)u=f(x,u), \quad \text{in}\: \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $\ N>2s$, $(-\Delta)^{s}$ stands…
For an ideal $I\subseteq\mathbb{R}[x]$ given by a set of generators, a new semidefinite characterization of its real radical $I(V_\mathbb{R}(I))$ is presented, provided it is zero-dimensional (even if $I$ is not). Moreover we propose an…
If $L$ is a semisimple Lie algebra of vector fields on R^N with a split Cartan subalgebra C, then it is proved that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical…
A sufficient condition of regularity for solutions to the Navier-Stokes equations is proved. It generalizes the so-called $L_{3,\infty}$-case.
Consider operators $L^{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$. Assume that $V\in C^{1,1}(\Omega)$ and $V$ satisfies $V(x) \leq \overline{a} \mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and a second…