$L^ p$ estimates for Schrödinger evolution equations

Abstract:

We prove that for Cauchy data in ${L^1}({{\mathbf {R}}^n})$, the solution of a Schrödinger evolution equation with constant coefficients of order $2m$ is uniformly bounded for $t \ne 0$, with bound $(1 + |t{|^{ - c}})$, where $c$ is an integer, $c > n/2m - 1$. Moreover it belongs to ${L^q}({{\mathbf {R}}^n})$ if $q > q(m,n)$, with its ${L^p}$ norm bounded by $(|t{|^{c’}} + |t{|^{ - c}})$, where $c’$ is an integer, $c’ > n/q$. A maximal local decay result is proved. Interpolating between ${L^1}$ and ${L^2}$, we derive $({L^p},{L^q})$ estimates. On the other hand, we prove that for Cauchy data in ${L^p}({{\mathbf {R}}^n})$, such a Cauchy problem is well posed as a distribution in the $t$-variable with values in ${L^p}({{\mathbf {R}}^n})$, and we compute the order of the distribution. We apply these two results to the study of Schrödinger equations with potential in ${L^p}({{\mathbf {R}}^n})$. We give an estimate of the resolvent operator in that case, and prove an asymptotic boundedness for the solution when the Cauchy data belongs to a subspace of ${L^p}({{\mathbf {R}}^n})$.