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

### Operator equation[edit]

The 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 *1*. For example, for a unit ball in *N*-dimensions, writing the Laplace equation in spherical coordinates one gets:

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?"

**Exercise (**)**: Prove that for the unit ball the Dirichlet-to-Neumann operator satisfies the quadratic equation above.

**Exercise (*)**: Prove that for the Dirichlet-to-Neumann operator of a half-space of *R*^{N} with uniform conductivity *1*,

### Network case[edit]

To match the functional equation for the Dirichlet-to-Neumann operator of the unit disc with conductivity *1*, one needs to find a self-dual layered planar network with rotational symmetry. The Dirichlet-to-Neumann map for such graph *G* is equal to:

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

**Exercise(*).** Prove that the cofactor matrix for is constant *1* up to sign w/the chessboard pattern.

The problem of finding the graph *G* then reduces to finding a Stieltjes continued fraction that is equal to *1* at the non-zero eigenvalues of *L*. For the *(2n+1)*-case, where *n* is a natural number, the eigenvalues are *0* with multiplicity *1* and

with multiplicity *2*. The existence and uniqueness of such fraction with *n* floors follows 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 squares meaning *1* and black squares meaning *-1*.