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Let $\mathcal{L}$ be a Schr\"odinger operator of the form $\mathcal{L} =-\Delta+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ Let$L^{p,\lambda}(\mathbb{R}^{n})$, $0\le \lambda<n$ denote the Morrey space on$\mathbb{R}^{n}$. In this paper, we will show that a function $f\inL^{2,\lambda}(\mathbb{R}^{n})$ is the trace of the solution of ${\mathbbL}u=u_{t}+{\mathcal{L}}u=0, u(x,0)= f(x),$ where $u$ satisfies a Carleson-typecondition \begin{eqnarray*} \sup_{x_B, r_B}r_B^{-\lambda}\int_0^{r_B^2}\int_{B(x_B, r_B)} |\nabla u(x,t)|^2 {dx dt} \leq C<\infty. \end{eqnarray*} Conversely, this Carleson-type condition characterizesall the ${\mathbb L}$-carolic functions whose traces belong to the Morrey space$L^{2,\lambda}(\mathbb{R}^{n})$ for all $0\le \lambda<n$. This result extendsthe analogous characterization founded by Fabes and Neri for the classical BMOspace of John and Nirenberg.

Characterization of temperatures associated to Schr\"odinger operators with initial data in Morrey spaces