Strong and Weak-Type Dispersive Estimates for the Energy-Critical Nonlinear Schr\"odinger Equation with an Inverse-Square Potential

Abstract

We prove dispersive estimates for the three-dimensional defocusing energy-critical nonlinear Schr\"odinger equation associated with $\mathcal L_a=-\Delta+a|x|^{-2}$. For nonnegative potentials, we extend the known finite-$p$ theory to the endpoint $L^1\to L^\infty$. For negative potentials in the global well-posedness range, set $\sigma=\frac12-\sqrt{\frac14+a}$. We obtain the free strong decay rate for $2<p<3/\sigma$ and the limiting Lorentz estimate from $L^{(3/\sigma)',1}$ to $L^{3/\sigma,\infty}$. The restriction $p<3/\sigma$ is sharp for strong Lebesgue decay. The proof combines a finite-interval bootstrap, nonlinear real interpolation on bounded energy sets, and endpoint Sobolev--Lorentz estimates adapted to $\mathcal L_a$.