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 $$ i\partial _{t} u-\biggl(-\Delta +\frac{\lambda}{ \vert x \vert ^{2}}\biggr)u + \vert x \vert ^{-2\tau} \vert u \vert ^{2(q-1)}u=0 , \qquad u(t,x):\mathbb{R}\times \mathbb{R}^{3}\to \mathbb{C}. $$ i ∂ t u − ( − Δ + λ | x | 2 ) u + | x | − 2 τ | u | 2 ( q − 1 ) u = 0 , u ( t , x ) : R × R 3 → C . 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 $|x|^{-2\tau}$ | x | − 2 τ for $\tau >0$ τ > 0 guarantees the existence of ground states for $\lambda \geq 0$ λ ≥ 0 , contrarily to the homogeneous case $\tau =0$ τ = 0 . Moreover, the decay of the inhomogeneous term enables to avoid any radial assumption on the datum. Since there is no dispersive estimate of $L^{1}\to L^{\infty}$ L 1 → L ∞ for the free Schrödinger equation with inverse square potential for $\lambda <0$ λ < 0 , one restricts this work to the case $\lambda \geq 0$ λ ≥ 0 .