Scattering threshold for a focusing inhomogeneous non-linear Schrödinger equation with inverse square potential

Abstract

This work studies the three space dimensional focusing inhomogeneous Schrödinger equation with inverse square potential i∂tu−(−Δ+λ|x|2)u+|x|−2τ|u|2(q−1)u=0,u(t,x):R×R3→C.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ i\\partial _{t} u-\\biggl(-\\Delta +\\frac{\\lambda}{ \\vert x \\vert ^{2}}\\biggr)u + \\vert x \\vert ^{-2\ au} \\vert u \\vert ^{2(q-1)}u=0 , \\qquad u(t,x):\\mathbb{R}\ imes \\mathbb{R}^{3}\ o \\mathbb{C}. $$\\end{document} The purpose is to investigate the energy scattering of global inter-critical solutions below the ground state threshold. The scattering is obtained by using the new approach of Dodson-Murphy, based on Tao’s scattering criteria and Morawetz estimates. This work naturally extends the recent paper by J. An et al. (Discrete Contin. Dyn. Syst., Ser. B 28(2): 1046–1067 2023). The threshold is expressed in terms the non-conserved potential energy. As a consequence, it can be given with a classical way with the conserved mass and energy. The inhomogeneous term |x|^{-2tau} for tau >0 guarantees the existence of ground states for lambda geq 0, contrarily to the homogeneous case tau =0. Moreover, the decay of the inhomogeneous term enables to avoid any radial assumption on the datum. Since there is no dispersive estimate of L^{1}to L^{infty} for the free Schrödinger equation with inverse square potential for lambda <0, one restricts this work to the case lambda geq 0.