Consider the Schrödinger equation: $$\mathrm{i}\partial _tu=Hu$$ over $$\mathbb {R}^n$$ , where H is a self-adjoint operator on $$L^2(\mathbb {R}^n)$$ which sum of $$-\Delta $$ and some potential. This paper aims to study observability for above equation, including observable sets time. We mention that measurable subset $$E\subset \mathbb called an set at time $$T>0$$ if there constant $$C>0$$ ...

We study a new family of sign-changing solutions to the stationary nonlinear Schr\"odinger equation $$ -\Delta v +q =|v|^{p-2} v, \qquad \text{in $\mathbb{R}^3$,} with $2<p<\infty$ and $q \ge 0$. These are spiraling in sense that they not axially symmetric but invariant under screw motion, i.e., share symmetry properties helicoid. In addition existence results, we provide information on shape s...

Quantum trajectories are Markov processes that describe the time evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they defined as solutions so-called Stochastic Schrödinger Equations, which nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to study invariant measures trajectories. Particularly, we p...

In this short note, we present some decay estimates for nonlinear solutions of 3d quintic, cubic NLS, and 2d quintic NLS (nonlinear Schrödinger equations).

We study the Hamiltonian formalism for second order and fourth nonlinear Schr\"{o}dinger equations. In case of equation, we consider cubic logarithmic nonlinearities. Since Lagrangians generating these equations are degenerate, follow Dirac-Bergmann to construct their corresponding Hamiltonians. obtain consistent motion, imposes some set constraints which contribute total along with Lagrange mu...

This paper investigates two class of Schrödinger equations with mixed source terms, containing, respectively, a local/non-local non-linearity and an inhomogeneous perturbation. In the energy sub-critical regime, one obtains some sharp thresholds global/non-global existence dichotomy.

In this paper, we consider the cost of null controllability for a large class of linear equations of parabolic or dispersive type in one space dimension in small time. By extending the work of Tenenbaum and Tucsnak in New blow-up rates for fast controls of Schrödinger and heat equations, we are able to give precise upper bounds on the time-dependance of the cost of fast controls when the time o...