= Solution
For a nonzero everywhere attractive weak potential, $\gamma<0$, the pole has $k=i\kappa$ with $\kappa=m|\gamma|/\hbar^2>0$. The exterior waves decay as $e^{-\kappa|x|}$, giving one shallow <bound state> to leading weak-coupling order:
$$
\boxed{E_b\simeq-\frac{m}{2\hbar^2}\left(\int U(x)\,dx\right)^2.}
$$
Its size $\kappa^{-1}$ is much larger than the potential range because $\kappa a=O(\lambda)\ll1$. For an everywhere repulsive potential there is no negative-energy <normalizable> <bound state>: the kinetic and potential <quadratic forms> are both nonnegative. Its lower-half-plane pole is a <virtual state>, not a stable <bound state>. More complicated sign-changing potentials with a vanishing <integral> require higher-order analysis and are not covered by this simple leading result.
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