  Let $\L$ be a Schr\"odinger operator of the form $\L=-\Delta+V$ acting on
$L^2(\mathbb R^n)$, $n\geq3$, where the nonnegative potential $V$ belongs to
the reverse H\"older class $B_q$ for some $q\geq n.$ Let ${\rm
BMO}_{{\mathcal{L}}}(\RR)$ denote the BMO space associated to the Schr\"odinger
operator $\L$ on $\RR$. In this article we show that for every $f\in {\rm
BMO}_{\mathcal{L}}(\RR)$ with compact support, then there exist $g\in
L^{\infty}(\RR)$ and a finite Carleson measure $\mu$ such that $$
  f(x)=g(x) + S_{\mu, {\mathcal P}}(x)
  $$ with $\|g\|_{\infty} +\||\mu\||_{c}\leq C \|f\|_{{\rm
BMO}_{\mathcal{L}}(\RR)},$ where
  $$
  S_{\mu, {\mathcal P}}=\int_{{\mathbb R}^{n+1}_+} {\mathcal P}_t(x,y) d\mu(y,
t),
  $$ and ${\mathcal P}_t(x,y)$ is the kernel of the Poisson semigroup
$\{e^{-t\sqrt{\L}}\}_{t> 0} $ on $L^2(\mathbb R^n)$. Conversely, if $\mu$ is a
Carleson measure, then $S_{\mu, {\mathcal P}}$ belongs to the space ${\rm
BMO}_{{\mathcal{L}}}(\RR)$. This extends the result for the classical
John--Nirenberg BMO space by Carleson \cite{C} (see also \cite{U,GJ,W}) to the
BMO setting associated to Schr\"odinger operators.
