相关论文: Uniqueness Properties of Solutions to Schr\"odinge…

We present some a - priori bounds from above and from below for solutions to a class of conformally invariant Schroedinger equations. As a by - product we deduce some new uniqueness results.

We discuss spacetime singularities of a solution to the Schr\"odinger equation with a metric perturbation and a sublinear potential. The quasi-homogeneous wave front set, due to Lascar (1977), of a solution is characterized by that of the…

We present general results on exponential decay of finite energy solutions to stationary nonlinear Schr\"odinger equations.

A method is presented to construct exactly solvable nonlinear extensions of the Schr\"odinger equation. The method explores a correspondence which can be established under certain conditions between exactly solvable ordinary Schr\"odinger…

We study uniqueness for solutions to the Cauchy problem associated with the parabolic Schr\"odinger equation on complete noncompact Riemannian manifolds, under suitable integral conditions on the solution. We show that, under suitable…

We revisit the following nonlinear Schr\"odinger system \begin{align*}\begin{cases} -\epsilon^{2}\Delta u +P(x) u= \mu_1 u^3 +\beta uv^2, &~\text{in}\;\mathbb {R}^3,\\ -\epsilon^{2}\Delta v+Q(x) v= \mu_2 v^3 +\beta u^2v,…

We study the local behavior of solutions of the stationary Schr\" od\-inger equation with singular potentials, establishing a local decomposition into a homogeneous harmonic polynomial and a lower order term. Combining a corollary to this…

We utilize the theory of de Branges spaces to show when certain Schr\"odinger operators with strongly singular potentials are uniquely determined by their associated spectral measure. The results are applied to obtain an inverse uniqueness…

We consider equations of the type: \[\partial_t \omega = \omega R(\omega),\] for general linear operators $R$ in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localized…

We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.

We address the open problem of existence of singularities for the complex Ginzburg-Landau equation. Using a combination of rigourous results and numerical computations, we describe a countable family of self-similar singularities. Our…

A Carleman estimate and the unique continuation property of solutions for a multi-terms time fractional diffusion equation up to order $\alpha\,\,(0<\alpha<2)$ and general time dependent second order strongly elliptic time elliptic operator…

We study a variational problem for the first Schrodinger eigenvalue on closed Riemannian surfaces. More precisely, we explore concentration-compactness properties of sequences formed by its extremal potentials.

The paper is concerned with a nonlinear system of two coupled fractional Schr\"odinger equations with both attractive intraspecies and attractive interspecies interactions in $\mathbb{R}$. By analyzing an associated $L^2$-constrained…

For the first time, a nonlinear Schr\"odinger equation of the general form is considered, depending on time and two spatial variables, the potential and dispersion of which are specified by two arbitrary functions. This equation naturally…

We are concerned with the two-power nonlinear Schr\"odinger-type equations with non-local terms. We consider the framework of Sobolev-Lorentz spaces which contain singular functions with infinite-energy. Our results include global…

We investigate the dynamical behavior of continuous and discrete Schrodinger systems exhibiting parity-time (PT) invariant nonlinearities. We show that such equations behave in a fundamentally different fashion than their nonlinear…

In this paper, a new analysis for existence, uniqueness, and regularity of solutions to a time-dependent Kohn-Sham equation is presented. The Kohn-Sham equation is a nonlinear integral Schroedinger equation that is of great importance in…

We consider a nonlinear Schr{\"o}dinger equation set in the whole space with a single power of interaction and an external source. We first establish existence and uniqueness of the solutions and then show, in low space dimension, that the…

In this article we prove a reducibility result for the linear Schr\"odinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less or equal than $1/2$. As far as we know, this is the first…