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

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The operator equationEdit

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. For example, for a unit ball in N-dimensions, writing the Laplace equation in 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 settingEdit

To 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 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 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 1s and -1s.

 
Correct signs