On 2D Inverse Problems/The case of the unit disc
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 RN with uniform conductivity 1,
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.