We prove that for Cauchy data in L 1 ( R n ) {L^1}({{\mathbf {R}}^n}) , the solution of a Schrödinger evolution equation with constant coefficients of order 2 m 2m is uniformly bounded for t ≠ 0 t \ne 0 , with bound ( 1 + | t | − c ) (1 + |t{|^{ - c}}) , where c c is an integer, c > n / 2 m − 1 c > n/2m - 1 . Moreover it belongs to L q ( R n ) {L^q}({{\mathbf {R}}^n}) if q > q ( m , n ) q > q(m,n) , with its L p {L^p} norm bounded by ( | t | c ′ + | t | − c ) (|t{|^{c’}} + |t{|^{ - c}}) , where c ′ c’ is an integer, c ′ > n / q c’ > n/q . A maximal local decay result is proved. Interpolating between L 1 {L^1} and L 2 {L^2} , we derive ( L p , L q ) ({L^p},{L^q}) estimates. On the other hand, we prove that for Cauchy data in L p ( R n ) {L^p}({{\mathbf {R}}^n}) , such a Cauchy problem is well posed as a distribution in the t t -variable with values in L p ( R n ) {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 ( R n ) {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 ( R n ) {L^p}({{\mathbf {R}}^n}) .