  Let $L$ be a Schr\"odinger operator of the form $L=-\Delta+V$ acting on
$L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse
H\"older class $B_q$ for some $q\geq n.$ In this article we will show that a
function $f\in L^{2, \lambda}({\mathbb{R}^n}), 0<\lambda<n$ is the trace of the
solution of ${\mathbb L}u=-u_{tt}+L u=0, u(x,0)= f(x),$ where $u$ satisfies a
Carleson type condition \begin{eqnarray*}
  \sup_{x_B, r_B} r_B^{-\lambda}\int_0^{r_B}\int_{B(x_B, r_B)} t|\nabla
u(x,t)|^2 {dx dt} \leq C <\infty. \end{eqnarray*} Its proof heavily relies on
investigate the intrinsic relationship between the classical Morrey spaces and
the new Campanato spaces $\mathscr{L}_L^{2,\lambda}({\mathbb{R}^n})$ associated
to the operator $L$, i.e. $$\mathscr{L}_L^{2,\lambda}(\mathbb{R}^n)=
{L}^{2,\lambda}(\mathbb{R}^n). $$ Conversely, this Carleson type condition
characterizes all the ${\mathbb L}$-harmonic functions whose traces belong to
the space $L^{2, \lambda}({\mathbb{R}^n})$ for all $ 0<\lambda<n$. This extends
the previous results of [FJN, DYZ, JXY].
