= Solution
Let $b=a/2$, $q=\sqrt{2m(U_0-|E|)}/\hbar$ and $\kappa=\sqrt{2m|E|}/\hbar$. Bound levels have $-U_0<E<0$. Up to <normalization>, parity gives
$$
\psi_e(x)=\begin{cases}A\cos(qx),&|x|\leq b,\\A\cos(qb)e^{-\kappa(|x|-b)},&|x|>b,\end{cases}
$$
$$
\psi_o(x)=\begin{cases}A\sin(qx),&|x|\leq b,\\A\operatorname{sgn}(x)\sin(qb)e^{-\kappa(|x|-b)},&|x|>b.\end{cases}
$$
The respective inverse squared <Normalization constant>[normalization constants] are $b+\sin(2qb)/(2q)+\cos^2(qb)/\kappa$ and $b-\sin(2qb)/(2q)+\sin^2(qb)/\kappa$. These include both exponential tails.
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