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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 $$iu_t+Delta u-frac{a u}{|x|^2}= -|u|^pu$$ when $2<p<infty$ and $a>0$. This work extends the result in [13, 14, 16] to dimension 2D. The key point is a modified version of Arora-Dodson-Murphys approach [2].
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
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 adapt the arguments in the recent work of Duyckaerts, Landoulsi, and Roudenko to establish a scattering result at the sharp threshold for the $3d$ focusing cubic NLS with a repulsive potential. We treat both the case of short-range potentials as p
In this paper we consider an inverse problem for the $n$-dimensional random Schr{o}dinger equation $(Delta-q+k^2)u = 0$. We study the scattering of plane waves in the presence of a potential $q$ which is assumed to be a Gaussian random function such
In this paper, we study the semilinear wave equations with the inverse-square potential. By transferring the original equation to a fractional dimensional wave equation and analyzing the properties of its fundamental solution, we establish a long-tim