= Solution
At the <delta potential>, the bound equation becomes $\psi(x)=\beta\psi(0)e^{-\kappa|x|}/\kappa$. A nonzero solution therefore requires $\kappa=\beta$, giving
$$
\boxed{E_b=-\frac{m\alpha^2}{2\hbar^2},\qquad\psi_b(x)=\sqrt\beta e^{-\beta|x|}.}
$$
There is one bound level. The transmission \i[<amplitude>] $t(k)$ has its physical-sheet <pole> at $k=i\beta$, which produces exactly this decaying state. The real-axis <probability>'s rational continuation $T(k)=k^2/(k^2+\beta^2)$ has both $+i\beta$ and $-i\beta$ <Pole>[poles]; the lower <pole> is not an additional <bound state>. It is essential to distinguish the analytic <amplitude> from an even continuation of its modulus squared.
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