Someone

\begin{center}
\begin{tikzpicture}[domain=-6:6,samples=300,xscale=1.1,yscale=0.7]
\draw[very thin] (-6.5,0) -- (6.5,0) node[right] {Position};
\draw[very thin] (0,-5) -- (0,2) node[above] {Energy};
\draw[thick,<->] plot (\x,{-16*(\x^2+1)/(\x^4+8)}) node[below] {$U(x)$};
\draw[<->] (-6,-2) -- (6,-2) node[below] {$E=-W/8$};
\draw (-1*2^0.5,-0.2)--(-1*2^0.5,+0.2) node[above] {$-\sqrt{2}d$};
\draw (+1*2^0.5,-0.2)--(+1*2^0.5,+0.2) node[above] {$+\sqrt{2}d$};
\draw (-2*2^0.5,-0.2)--(-2*2^0.5,+0.2) node[above] {$-2\sqrt{2}d$};
\draw (+2*2^0.5,-0.2)--(+2*2^0.5,+0.2) node[above] {$+2\sqrt{2}d$};
%\draw[very thin, dashed] (-2*2^0.5,-2)--(-2*2^0.5,-4.5) node[] {$x=-2\sqrt{2}d$};
%\draw[very thin, dotted] (+2*2^0.5,-2)--(+2*2^0.5,-4.5) node[] {$x=+2\sqrt{2}d$};
\draw[very thin, dashed] (-4,-4) -- (4,-4) node[below] {$-W/4$};
\end{tikzpicture}
\end{center}