# On 2D Inverse Problems/Schrodinger equation

The conductivity equation ${\displaystyle \Delta _{\gamma }u=\nabla \cdot (\gamma \nabla u)=0}$

is equivalent to Schrodinger equation: ${\displaystyle (\Delta -q)(u{\sqrt {\gamma }})=0}$

for potential ${\displaystyle q={\frac {\Delta {\sqrt {\gamma }}}{\sqrt {\gamma }}}}$

For an analog of this system on e-networks, one defines the solution of the Schrodinger equation u on the nodes and the square of the conductivity on the edges by the following formula: ${\displaystyle \gamma ^{2}(a,b)=x(a)x(b).}$

Exercise (*). Express the DN operator for the Schrodinger equation in terms of the one for the conductivity equation on the same e-network.

(Hint). Let ${\displaystyle \Lambda _{q}=A-B(C+D_{q})^{-1}B^{T},}$

where,

${\displaystyle K={\begin{pmatrix}A&B\\B^{T}&C+D_{q}\end{pmatrix}}.}$

Then

${\displaystyle {\tilde {K}}={\begin{pmatrix}A+D_{y}&BD_{x}\\D_{x}B^{T}&D_{x}(C+D_{q})D_{x}\end{pmatrix}}}$

is the Laplace matrix of the network.

Exercise (**). Reduce the inverse problem for the Schrodinger equation on e-nerwork to the conductivity equation one w/signed conductivity.