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In this paper, we study the scattering theory for the cubic inhomogeneous Schrodinger equations with inverse square potential $iu_t+Delta u-frac{a}{|x|^2}u=lambda |x|^{-b}|u|^2u$ with $a>-frac14$ and $0<b<1$ in dimension three. In the defocusing case (i.e. $lambda=1$), we establish the global well-posedness and scattering for any initial data in the energy space $H^1_a(mathbb R^3)$. While for the focusing case(i.e. $lambda=-1$), we obtain the scattering for the initial data below the threshold of the ground state, by making use of the virial/Morawetz argument as in Dodson-Murphy [Proc. Amer. Math. Soc.,145(2017), 4859-4867.] and Campos-Cardoso [arXiv: 2101.08770v1.] that avoids the use of interaction Morawetz estimate.
In this paper, we study the long time behavior of the solution of nonlinear Schrodinger equation with a singular potential. We prove scattering below the ground state for the radial NLS with inverse-square potential in dimension two $$iu_t+Delta u-
We consider the focusing energy critical NLS with inverse square potential in dimension $d= 3, 4, 5$ with the details given in $d=3$ and remarks on results in other dimensions. Solutions on the energy surface of the ground state are characterized. We
We extend the result of Farah and Guzman on scattering for the $3d$ cubic inhomogeneous NLS to the non-radial setting. The key new ingredient is a construction of scattering solutions corresponding to initial data living far from the origin.
In this article, we consider the focusing cubic nonlinear Schrodinger equation(NLS) in the exterior domain outside of a convex obstacle in $mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state of Killip
In this paper, we give a simple proof of scattering result for the Schrodinger equation with combined term $ipa_tu+Delta u=|u|^2u-|u|^4u$ in dimension three, that avoids the concentrate compactness method. The main new ingredient is to extend the scattering criterion to energy-critical.