Dispersion and Strichartz estimates for some finite rank perturbations of the Laplace operator.
The authors establish the \((L^1, L^\infty)\) dispersive inequality and the Strichartz estimates for finite rank perturbations in \(\mathbb{R}^d\), \(d \geq 3\). More specifically, the authors assume that the finite rank perturbation consists of a superposition of \(N\) translates of a rank one operator \(| \psi \rangle \langle \psi | \), where \(\psi\) has sufficient smoothness and decay (one roughly needs \(\langle x \rangle^{d/2}\) to lie in \(H^{\lfloor d/2 \rfloor+2}\)). Then, if the rank one obstacles are sufficiently far separated in physical space, one obtains dispersive and Strichartz estimates with bounds which depend on \(N\). Local \(L^2\) decay bounds are also proven. The bounds grow as a power of \(N\) (but assuming that the separation in physical space is greater than another power of \(N\)). The power of \(N\) in the dispersive inequality is rather complicated, but the Strichartz bound for \(L^q_t L^r_x\) norms of free solutions grows like \(N^{2/q}\). The authors conjecture both bounds are sharp, which would mean that the Strichartz estimate cannot be deduced directly from the dispersive inequality. One of the main tools in the proof is the fact that the resolvents for rank one perturbations can be computed explicitly, and these can be used to approximate the finite rank perturbations if the obstacles are sufficiently far from each other.
scientific article; zbMATH DE number 1901780