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

The operator equation

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:

$\mathrm{\Delta }f=r^{1-N}\frac{\partial }{\partial r}\left(r^{N-1}\frac{\partial f}{\partial r}\right)+r^{-2}{\mathrm{\Delta }}_{S^{N-1}}f,$

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

$\mathrm{\Lambda }(\mathrm{\Lambda }-(N-2)Id)+{\mathrm{\Delta }}_{S^{N-1}}=0$.

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

${\mathrm{\Lambda }}^2=-{\mathrm{\Delta }}_{S^1}.$

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

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:

${\mathrm{\Lambda }}_G^2=L,$

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

$L=\left(\begin{array}{ccccc}2& -1& 0& \dots & -1\\ -1& 2& -1& \dots & 0\\ 0& -1& \ddots & \ddots & ⋮\\ ⋮& ⋮& \ddots & 2& -1\\ -1& 0& \dots & -1& 2\end{array}\right).$

Exercise(*). Prove that the entries of the cofactor matrix of ${\mathrm{\Lambda }}_G$ 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

$2\sin (\frac{k\pi }{2n+1}),k=1,2,\dots n$

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:

$\beta (z)=\cot (\frac{n\pi }{2n+1})z+\frac{1}{\cot (\frac{(n-1)\pi }{2n+1})z+\frac{1}{\ddots +\frac{1}{\cot (\frac{\pi }{2n+1})z}}}.$

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

$\tan (\frac{n\pi }{2n+1})=2\sum _k\sin (\frac{k\pi }{2n+1}).$

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

$\tan (\frac{l\pi }{2n+1})=2\sum _k(\pm )\sin (\frac{k\pi }{2n+1}),l=1,2,\dots n.$

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

Correct signs