Related papers: Unique continuation of Schr\"odinger-type equation…

In 2013, R.L. Frank and E. Lenzmann [R.L. Frank, E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math. 210 (2) (2013) 261-318] study the following problem: \begin{align*} (-\Delta)^su + u…

In this paper we obtain minimal support properties of solutions of Schr\"odinger equations. We improve previously known conditions on the potential for which the measure of the support of solutions cannot be too small. We also use these…

We study the Cauchy problem for the non-linear Schr\"odinger equation with singular potentials. For point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-L^{p} spaces. Specific…

We prove that if a solution of the discrete time-dependent Schr\"odinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shr\"odinger operator and for operators with…

Let D be a self-adjoint differential operator of Dirac type acting on sections in a vector bundle over a closed Riemannian manifold M. Let H be a closed D-invariant subspace of the Hilbert space of square integrable sections. Suppose D…

We consider the following nonlinear Schr\"odinger equations with critical growth: \begin{equation} - \Delta u + V(|y|)u=u^{\frac{N+2}{N-2}},\quad u>0 \ \ \mbox{in} \ \mathbb {R}^N, \end{equation} where $V(|y|)$ is a bounded positive radial…

We investigate uniqueness of solutions to Schr\"odinger-type elliptic equations posed on infinite graphs. Solutions are assumed to belong to suitable weighted $\ell^p$ spaces where $p\geq 1$ and the weight is related to both the potential…

These notes are intended as an introduction to the question of unique continuation for the wave operator, and some of its applications. The general question is whether a solution to a wave equation in a domain, vanishing on a subdomain has…

The Calder\'on problem for the fractional Schr\"odinger equation was introduced in the work \cite{GSU}, which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a…

In this paper, we prove a sharp uniqueness result for the singular Schr\"odinger equation with an inverse square potential. This will be done without assuming geometrical restrictions on the observation region. The proof relies on a recent…

We investigate unique continuation inequalities for solutions of the Schr\"odinger equations associated with special Hermite operators. Our main result establishes that if the solution remains small at two distinct time points outside sets…

We provide a simple and direct proof of a strong-type unique continuation principle for the fractional $p$-Laplacian $(-\Delta_p)^s$ for a range of $s$ and $p$. The result extends to strong solutions of the fractional nonlinear…

We study quantitative unique continuation for second order elliptic equations with lower-order terms of H\"older regularity via a weighted frequency function method. We establish quantitative three-ball inequalities and corresponding…

We deal with the existence of positive solutions for the following fractional Schr\"odinger equation $$ \varepsilon ^{2s} (-\Delta)^{s} u + V(x) u = f(u) \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$,…

Let $u$ be a solution of $\Delta u=Vu$ on $\mathbb{R}^d$, where $V$ be continuous, nonnegative and bounded. We prove that the condition $$\int_{r_j\leq|x|\leq r_j+1}|u(x)|^2dx\to 0,$$ along any sequence $(r_j)$, $r_j\nearrow+\infty$,…

In a central potential the usual resolution of the Schr\"odinger equation in spherical coordinates consists in determining the solutions R(r) or u(r) of the radial equations considered as the radial parts of the Schr\"odinger equation.…

This is the second part of a two-paper series studying the nonlinear Schr\"odinger equation with quasi-periodic initial data. In this paper, we focus on the quasi-periodic Cauchy problem for the derivative nonlinear Schr\"odinger equation.…

In this paper, we study the following semilinear Schr\"odinger equation $$ -\epsilon^2\triangle u+ u+ V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}), $$ where $N\geq 2$ and $\epsilon>0$ is a small parameter. The function $V$ is bounded in…

The uniqueness of the positive ground state solutions of fractional Shrodinger equations with a harmonic potential has not been covered by the breakthrough method developed in [1, 2]. It has remained an open question for years. [3] and [5]…

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…