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We study the Strichartz estimates for Schr\"odinger equation on a metric cone$X$, where the metric cone $X=C(Y)=(0,\infty)_r\times Y$ and the cross section$Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. The equippedmetric on $X$ is given by $g=dr^2+r^2h$, and let $\Delta_g$ be the Friedrichextension positive Laplacian on $X$ and $V=V_0 r^{-2}$ where$V_0\in\CC^\infty(Y)$ is a real function such that the operator$\Delta_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. Weestablish the full range of the global-in-time Strichartz estimate without lossfor the Sch\"odinger equation associated with the operator $\LL_V=\Delta_g+V_0r^{-2}$ including the endpoint estimate both in homogeneous and inhomogeneouscases. As an application, we study the well-posed theory and scattering theoryfor the Sch\"odinger equation with cubic nonlinearity on this setting.

Global-in-time Strichartz estimates and cubic Schrodinger equation on metric cone