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

Abstract

Abstract This work studies the three space dimensional focusing inhomogeneous Schrödinger equation with inverse square potential i ∂ t u − ( − Δ + λ | x | 2 ) u + | x | − 2 τ | u | 2 ( q − 1 ) u = 0 , u ( t , x ) : R × R 3 → C . $$ i\partial _{t} u-\biggl(-\Delta +\frac{\lambda}{ \vert x \vert ^{2}}\biggr)u + \vert x \vert ^{-2\tau} \vert u \vert ^{2(q-1)}u=0 , \qquad u(t,x):\mathbb{R}\times \mathbb{R}^{3}\to \mathbb{C}. $$ 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 τ $|x|^{-2\tau}$ for τ > 0 $\tau >0$ guarantees the existence of ground states for λ ≥ 0 $\lambda \geq 0$ , contrarily to the homogeneous case τ = 0 $\tau =0$ . 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 → L ∞ $L^{1}\to L^{\infty}$ for the free Schrödinger equation with inverse square potential for λ < 0 $\lambda <0$ , one restricts this work to the case λ ≥ 0 $\lambda \geq 0$ .

Keywords

• Nonlinear equations
• Inhomogeneous Schrödinger equation
• Inverse square potential
• Scattering