Related papers: Unique continuation properties for one dimensional…
Recent (scale-free) quantitative unique continuation estimates for spectral subspaces of Schr\"odinger operators are extended to allow singular potentials such as certain $L^p$-functions. The proof is based on accordingly adapted Carleman…
The aim of this paper is to review and compare the spectral properties of (the closed extension of) --$\Delta$ + U (V $\ge$ 0) and --$\Delta$ + iV in L 2 (R^d) for C $\infty$ real potentials U or V with polynomial behavior. The case with…
We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then…
In this paper we consider the linear Schrodinger equation (LSE) on a regular tree with the last generation of edges of infinite length and analyze some unique continuation properties. The first part of the paper deals with the LSE on the…
This work mainly focuses on the spatial decay properties of solutions to the Zakharov-Kuznetsov equation. In earlier studies for the two- and three-dimensional cases, it was established that if the initial condition $u_0$ verifies $\langle…
We consider the following system of Schr\"odinger equations \begin{equation*}\left.\begin{cases} -\Delta U + \lambda U = \alpha_0 U^3+ \beta UV^2 -\Delta V + \mu(y) V = \alpha_1 V^3+\beta U^2V \end{cases}\right. \text{in} \quad…
We study the Schr\"odinger equations $-\Delta u + V(x)u = f(x,u)$ in $\mathbb{R}^N$ and $-\Delta u - \lambda u = f(x,u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$. We assume that $f$ is superlinear but of subcritical growth and…
We are interested in the long time asymptotic behavoir of solutions to the scalar Zakharov system \[ i u_{t} + \Delta u = nu,\] \[n_{tt} - \Delta n= \Delta |u|^2\] and the Klein-Gordon Zakharov system \[ u_{tt} - \Delta u + u = - nu,\] \[…
We prove unique continuation principles for solutions of evolution Schr\"odinger equations with time dependent potentials. These correspond to uncertainly principles of Paley-Wiener type for the Fourier transform. Our results extends to a…
In this paper we study the game $p-$Laplacian on a tree, that is, $$ u(x)=\frac{\alpha}2\left\{\max_{y\in \S(x)}u(y) + \min_{y\in \S(x)}u(y)\right\} + \frac{\beta}{m}\sum_{y\in \S(x)} u(y), $$ here $x$ is a vertex of the tree and $S(x)$ is…
This article is concerned with the unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain $\Omega$ prescribed with some regularity and growth conditions. Our result…
In this paper, we consider a nonlinear Fuchsian type partial differential equation of the second order in the complex domain. Under a very weak assumption, we show the uniqueness of the solution. The result is applied to the problem of…
The goal of this article is to obtain observability estimates for Schr{\"o}dinger equations in the plane R 2. More precisely, considering a 2$\pi$Z 2-periodic potential V $\in$ L $\infty$ (R 2), we prove that the evolution equation…
We study various qualitative and quantitative (global) unique continuation properties for the fractional discrete Laplacian. We show that while the fractional Laplacian enjoys striking rigidity properties in the form of (global) unique…
In this paper, we establish a novel unique continuation property for two-dimensional anisotropic elasticity systems with partial information. More precisely, given a homogeneous elasticity system in a domain, we investigate the unique…
We consider an asymptotically linear Schr\"odinger equation $-\Delta u + V(x)u = \lambda u + f(x,u), \ x\in R^N$, and show that if $\lambda_0$ is an isolated eigenvalue for the linearization at infinity, then under some additional…
In this paper, we study the focusing nonlinear Schr\"odinger equation with exponential nonlinearities \[ i \partial_t u + \Delta u = - \left(e^{4\pi |u|^2} - 1 - 4\pi \mu |u|^2 \right) u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R}…
We show novel types of uniqueness and rigidity results for Schr\"odinger equations in either the nonlinear case or in the presence of a complex-valued potential. As our main result we obtain that the trivial solution $u=0$ is the only…
Recent developments in quantum physics make heavy use of so-called "quantum trajectories." Mathematically, this theory gives rise to "stochastic Schr\"odinger equations", that is, perturbation of Schr\"odinger-type equations under the form…
We consider the defocusing nonlinear Schr{\"o}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast…