  We consider the Schr\"{o}dinger operator $\mathcal{L}=-\Delta+V$ on $\mathbb
R^d$, $d\geq3$, where the nonnegative potential $V$ belongs to the reverse
H\"{o}lder class $RH_s$ for some $s\geq d/2$. A real-valued function $f\in
L^1_{\mathrm{loc}}(\mathbb R^d)$ belongs to the (BMO) space
$\mathrm{BMO}_{\rho,\theta}(\mathbb R^d)$ with $0<\theta<\infty$ if
\begin{equation*} \|f\|_{\mathrm{BMO}_{\rho,\theta}}
:=\sup_{B(x_0,r)}\bigg(1+\frac{r}{\rho(x_0)}\bigg)^{-\theta}\bigg(\frac{1}{|B(x_0,r)|}\int_{B(x_0,r)}\big|f(x)-f_{B}\big|\,dx\bigg),
\end{equation*} where the supremum is taken over all balls
$B(x_0,r)\subset\mathbb R^d$, $\rho(\cdot)$ is the critical radius function in
the Schr\"{o}dinger context and \begin{equation*}
f_{B}:=\frac{1}{|B(x_0,r)|}\int_{B(x_0,r)}f(y)\,dy. \end{equation*} A
real-valued function $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ belongs to the
(Lipschitz) space $\mathrm{Lip}_{\beta}^{\rho,\theta}(\mathbb R^d)$ with
$0<\beta<1$ and $0<\theta<\infty$ if \begin{equation*}
\|f\|_{\mathrm{Lip}_{\beta}^{\rho,\theta}}
:=\sup_{B(x_0,r)}\bigg(1+\frac{r}{\rho(x_0)}\bigg)^{-\theta}
\bigg(\frac{1}{|B(x_0,r)|^{1+\beta/d}}\int_{B(x_0,r)}\big|f(x)-f_{B}\big|\,dx\bigg).
\end{equation*} It can be easily seen that $\mathrm{BMO}_{\rho,\theta}(\mathbb
R^d)$ (or $\mathrm{Lip}_{\beta}^{\rho,\theta}(\mathbb R^d)$) is a function
space which is larger than the classical BMO (or Lipschitz) space. In this
paper, we give some new characterizations of BMO and Lipschitz spaces
associated with the Schr\"{o}dinger operator $\mathcal{L}$. We extend some
previous works of Bongioanni--Harboure--Salinas and Liu--Sheng to the weighted
case. The classes of weights considered here are larger than the classical
Muckenhoupt classes.
