= Solution
Take the same $e^{-i\omega t}$ convention and an outgoing field. Define the <scattering potential> with the positive-contrast sign,
$$
Q(\mathbf r)=k^2[n(\mathbf r)^2-1],\qquad
(\Delta+k^2)\psi=-Q\psi.
$$
The <outgoing Green function for the three-dimensional Helmholtz equation> is
$$
G_k(\mathbf r,\mathbf r')=\frac{e^{ik|\mathbf r-\mathbf r'|}}{4\pi|\mathbf r-\mathbf r'|},\qquad
(\Delta+k^2)G_k=-\delta.
$$
Thus the <Lippmann-Schwinger equation> has a plus sign:
$$
\psi=\psi_i+T\psi,\qquad
(T\phi)(\mathbf r)=\int_VG_k(\mathbf r,\mathbf r')Q(\mathbf r')\phi(\mathbf r')\,d^3r'.
$$
Replacing the total field inside the integral by the incident field gives the <Born approximation for scalar wave scattering>
$$
\boxed{\psi_s^{(1)}(\mathbf r)=\frac{k^2}{4\pi}\int_V
[n(\mathbf r')^2-1]\frac{e^{ik|\mathbf r-\mathbf r'|}}{|\mathbf r-\mathbf r'|}
e^{ik\widehat{\mathbf r}_0\cdot\mathbf r'}\,d^3r'.}
$$
More generally, iterating the integral equation gives the <Born series> $\psi_s=\sum_{j\geq1}T^j\psi_i$; its first term is the boxed expression. A sufficient convergence condition is $\|T\|<1$ on the chosen space of fields in $V$, and accurate single-scattering truncation needs the neglected terms to be small. Weak refractive contrast is useful, but for a sufficiently large or resonant scatterer it is not by itself a guarantee of negligible <multiple scattering>.
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