On the local smoothing for the Schrödinger equation

Abstract:

We prove a family of identities that involve the solution $u$ to the following Cauchy problem: \begin{equation*} \textbf {i} \partial _t u + \Delta u=0, u(0)=f(x), (t, x)\in {\mathbf R}_t\times {\mathbf R}^n_x, \end{equation*} and the $\dot H^\frac 12({\mathbf R}^n)$-norm of the initial datum $f$. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate proved by Constantin and Saut (1989), Sjölin (1987) and Vega (1988) and a uniqueness criterion for the solutions to the Schrödinger equation.