Complex Geometrical Optics Solutions

Complex geometrical optics (CGO) solutions are the main tool behind the uniqueness and reconstruction results for the Calderón Problem.

Reduction to Schrödinger form. For smooth $\gamma$, the substitution $u={\gamma }^{-1/2}w$ turns $\nabla \cdot (\gamma \nabla u)=0$ into

$$(-\Delta +q)w=0,q=\frac{\Delta \sqrt{\gamma }}{\sqrt{\gamma }}.$$

By Boundary Determination, ${\Lambda }_{\gamma }$ determines the DtN map of the Schrödinger operator, so it suffices to recover $q$.

CGO solutions. Let $\zeta \in {\mathbb{C}}^n$ with $\zeta \cdot \zeta =0$, for example $\zeta =\tau (\omega +i{\omega }^{\perp })$ with orthogonal unit vectors. Then $e^{x\cdot \zeta }$ is harmonic, and one looks for solutions of the form

$$w(x)=e^{x\cdot \zeta }(1+r(x,\zeta )),\Vert r\Vert =\mathcal{O}(|\zeta {|}^{-1})\text{ as }|\zeta |\to \infty .$$

These solutions grow exponentially in one direction and oscillate in another.

Uniqueness argument ($n\ge 3$). An integral identity (Alessandrini’s identity) gives ${\int }_{\Omega }(q_1-q_2)w_1{w}_2\mathrm{d}x=0$ whenever the DtN maps agree. In $n\ge 3$ one can choose ${\zeta }_1+{\zeta }_2=i\xi$ for any $\xi \in {\mathbb{R}}^n$ with $|{\zeta }_j|\to \infty$. Then $w_1{w}_2\to e^{ix\cdot \xi }$, the Fourier transform of $q_1-q_2$ vanishes, and so $q_1=q_2$.

In two dimensions there is not enough freedom to choose such $\zeta$. Nachman instead used $\bar{\partial }$-methods in the complex spectral parameter (see D-bar Method).

References

1. J. Sylvester, G. Uhlmann (1987). A Global Uniqueness Theorem for an Inverse Boundary Value Problem. Ann. of Math. 125(1), 153–169. doi:10.2307/1971291
2. A. I. Nachman (1988). Reconstructions From Boundary Measurements. Ann. of Math. 128(3), 531–576. doi:10.2307/1971435
3. G. Uhlmann (2009). Electrical impedance tomography and Calderón’s problem. Inverse Problems 25(12), 123011. doi:10.1088/0266-5611/25/12/123011