On 2D Inverse Problems/Schrodinger equation

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The conductivity equation

\Delta_{\gamma}u = \nabla\cdot(\gamma\nabla u) = 0.

can be rewritten as the Schrodinger equation

(\Delta - q)(u\sqrt{\gamma}) = 0,


q = \frac{\Delta\sqrt{\gamma}}{\sqrt{\gamma}}.

For the analog of this system to work on networks, one can define the solution of the Schrodinger equation u on the nodes and the square of the solution on the edges by the following formula:

\gamma^2(a,b) = u(a)u(b).
Exercise (*). Express the Dirichlet-to-Neumann operator for the Schrodinger equation in terms of the Dirichlet-to-Neumann operator for the corresponding Laplace equation on the network with the same underlying graph.

(Hint). Let

\Lambda_q = A-B(C+D_q)^{-1}B^T,


K = 
A & B \\
B^T & C + D_q


\tilde{K} =
A+D_y & BD_x \\
D_x B^T & D_x(C+D_q)D_x

is the Laplace matrix of the network with

\Lambda(\tilde{K}) = A + D_y - B D_x (D_x (C+D_q) D_x)^{-1} D_x B^T = \Lambda_q + D_y,


x = - (C+D_q)^{-1}B^T1


y = -\Lambda_q 1.
Exercise (**). Reduce the inverse problem for Schrodinger operator to the inverse problem for the Laplace operator on the network w/same underlying graph (w/ possibly signed conductivity).