= Solution
In either exterior region of the <finite square well>, the <time-independent Schrödinger equation> reduces to $\psi''=k^2\psi$, since $E<0$. Its general solution is a <linear combination> of $e^{kx}$ and $e^{-kx}$. On the left, $e^{-kx}$ grows as $x\to-\infty$ and cannot be a <square-integrable function>, so its coefficient must be zero. On the right, the excluded growing term is $e^{kx}$. Hence the surviving tails are $Ae^{kx}$ for $x\leq-a$ and $De^{-kx}$ for $x\geq a$.
Inside the well, $E+V_0>0$ and the equation becomes $\psi''=-\kappa^2\psi$, whose general solution is $B\cos\kappa x+C\sin\kappa x$. Thus
$$
\boxed{\psi(x)=\begin{cases}Ae^{kx},&x\leq-a,\\B\cos\kappa x+C\sin\kappa x,&-a<x<a,\\De^{-kx},&x\geq a.\end{cases}}
$$
The coefficients are initially independent; <wavefunction matching at a finite potential step> imposes the remaining relations. Here the paper's $k$ is the exterior decay constant and $\kappa$ is the interior oscillation constant.
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