= Solution
Write $\nu=m\gamma/\hbar^2$, so $t(k)\simeq k/(k+i\nu)$. Its nontrivial pole is
$$
\boxed{k_*=-i\nu=-\frac{im\gamma}{\hbar^2},\qquad
E_*=-\frac{m\gamma^2}{2\hbar^2}.}
$$
For negative $\gamma$ this is on the positive imaginary $k$ axis, the physical bound-state sheet. For positive $\gamma$ it lies on the negative imaginary axis and is a virtual-state pole reached by <analytic continuation> through the continuum cut. These different sheets can have the same negative numerical <energy>; specifying only $E_*$ would conceal the distinction. As a function of $E$, the <momentum> <square root> also introduces the continuum threshold <branch point> at zero. If $\gamma=0$, numerator and denominator cancel at this order and no pole is predicted by this leading approximation.
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