= Solution
In first Born order the unknown potential enters linearly. Let $M_i$ multiply by the known incident field, $\mathcal G$ be the outgoing Green <integral> over the known support $D$, and $\mathcal R$ restrict the resulting field to the measurement set. With $y$ the measured scattered data,
$$
\boxed{y=AV,\qquad A=\mathcal R\mathcal G M_i.}
$$
The formal least-squares, <minimum-norm least-squares solution> is $V=A^\dagger y$, or $V=(A^*A)^{-1}A^*y$ on a domain where that inverse is meaningful. The <refractive index> is then
$$
\boxed{n(\mathbf r)=\sqrt{1+V(\mathbf r)/k^2},}
$$
choosing the physical branch approaching one outside the scatterer. At weak contrast $n-1\simeq V/(2k^2)$.
A noisy stable reconstruction uses <Tikhonov regularization>:
$$
V_\alpha^\delta=\operatorname*{arg\,min}_V\left\{\|AV-y^\delta\|_Y^2+\alpha\|V\|_X^2\right\},\qquad
\boxed{V_\alpha^\delta=(A^*A+\alpha I)^{-1}A^*y^\delta,\quad\alpha>0.}
$$
A smoothness penalty $\alpha\|LV\|^2$ may be substituted when appropriate, giving $A^*A+\alpha L^*L$ in the <normal equation for a linear inverse problem>, under the corresponding <coercivity> assumptions. The parameter balances noise amplification and smoothing bias.
The data geometry matters: knowing a <scattered field> on a measurement surface does not automatically mean knowing it throughout the support. A single incident wave and fixed-frequency far field sample only a shifted sphere of the potential's <Fourier transform>, and in general do not determine an arbitrary three-dimensional potential uniquely. <Tikhonov regularization> stabilizes a selected <minimum-norm least-squares solution>; it cannot create missing information or guarantee uniqueness from inadequate data. If the total field and its <derivatives> were known throughout $D$, the first-Born equation could instead be inverted formally as $V\simeq-(\Delta+k^2)\psi_s/\psi_i$ where $\psi_i\ne0$, but differentiation still amplifies measurement noise.
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