The Schrodinger equation provides a link between the local and spectral/global properties of solutions of Laplace-Beltrami equation.

The inverse boundary problem for the Schrodinger equation can be reduced to the Calderon problem due to the identities below that hold for graphs and surfaces. Suppose *u* on satisfies the Laplace equation in the domain,

Then

where,

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:

**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

where,

Then

is the Laplace matrix of the network with

w/

**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).