Last modified on 22 September 2014, at 22:29

# On 2D Inverse Problems/Schrodinger equation

The conductivity equation

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

can be rewritten as the Schrodinger equation

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

where,

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

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

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

where,

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

Then

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

is the Laplace matrix of the network with

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

w/

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

and

$y = -\Lambda_q 1.$
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).
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