# On 2D Inverse Problems/The case of the unit disc

### The operator equation

editThe continuous Dirichlet-to-Neumann operator can be calculated explicitly for certain domains, such as a half-space, a ball and a cylinder and a shell with uniform conductivity. For example, for a unit ball inN-dimensions, writing theLaplace equationin spherical coordinates:

and, therefore, the Dirichlet-to-Neumann operator satisfies the following equation:

- .

In two-dimensions the equation takes a particularly simple form:

The study of material of this chapter is largely motivated by the question of Professor of Mathematics at the University of Washington Gunther Uhlmann: "Is there a discrete analog of the equation?"

### The network setting

editTo match the functional equation for the Dirichlet-to-Neumann operator of the unit disc with uniform conductivity, is to find the self-dual layered planar network with rotational symmetry. The Dirichlet-to-Neumann operator for such graph *G* is equal to:

where *-L* is equal to the Laplacian on the circle:

**Exercise(*).**Prove that the entries of the cofactor matrix of are ±1 w/the chessboard pattern.

The problem then reduces to calculating a Stieltjes continued fraction equalled to1at the non-zero eigenvalues ofL. For the(2n+1)-case, wherenis a natural number, the eigenvalues are0with the multiplicity one and

w/multiplicity two. The existence and uniqueness of such fraction with *n* levels follow from our results on layered networks, see [BIMS].

**Exercise (***).**Prove that the continued fraction is given by the following formula:

**Exercise 2 (*).**Use the previous exercise to prove the trigonometric formula:

**Exercise 3(**).**Find the right signs in the following trigonometric formula

Example: the following picture provides the solution for *n=8* w/white and black squares representing *1*s and *-1*s.