Global-in-time Strichartz estimates and cubic Schrödinger equation in a conical singular space

Abstract

In this paper, we study Strichartz estimates for the Schrödinger equation on a metric cone $X$, where $X=C(Y)=(0,\infty {)}_r\times Y$ and the cross section $Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. For the metric $g$ on $X$ given by $g=d{r}^2+r^{2}h$, let ${\Delta }_g$ be the positive Friedrichs extension Laplacian on $X$ and $V=V_0{r}^{-2}$ where $V_0\in\CC^\infty(Y)$ is a real function such that the operator $P:={\Delta }_h+V_0+(n-2{)}^2/4$ is a strictly positive operator on $L^2(Y)$. We establish the full range of global-in-time Strichartz estimates without loss for the Schrödinger equation associated with the operator $\LL_V=Δ_g+V_0 r^{-2}$ including the endpoint estimate both in homogeneous and inhomogeneous cases. A new finding reveals that the range of admissible pairs at ${\dot{H}}^s$-level is influenced by the smallest eigenvalue of the operator $P$. This additionally proves the conjecture in Wang [Ann. Inst. Fourier 2006] and generalizes the results of Ford [Comm. Math. Phys. 2010] and Baskin-Marzuola-Wunsch [Contemp. Math. 2014]. As an application, we show the well-posedness theory and scattering theory for the Schrödinger equation with a cubic nonlinearity on this setting which verifies a conjecture in Baskin-Marzuola-Wunsch [Contemp. Math. 2014].

54 pages; The double endpoint inhomogeneous Stricahrtz estimate and the diffractive geometry are updated in the new version. Comments are welcome!