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:

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

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

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

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

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

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:

${\displaystyle \Lambda _{G}^{2}=L,}$

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

${\displaystyle L={\begin{pmatrix}2&-1&0&\ldots &-1\\-1&2&-1&\ldots &0\\0&-1&\ddots &\ddots &\vdots \\\vdots &\vdots &\ddots &2&-1\\-1&0&\ldots &-1&2\\\end{pmatrix}}.}$
Exercise(*). Prove that the entries of the cofactor matrix of ${\displaystyle \Lambda _{G}}$ 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

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

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:
${\displaystyle \beta (z)=\cot({\frac {n\pi }{2n+1}})z+{\cfrac {1}{\cot({\frac {(n-1)\pi }{2n+1}})z+{\cfrac {1}{\ddots +{\cfrac {1}{\cot({\frac {\pi }{2n+1}})z}}}}}}.}$
Exercise 2 (*). Use the previous exercise to prove the trigonometric formula:
${\displaystyle \tan({\frac {n\pi }{2n+1}})=2\sum _{k}\sin({\frac {k\pi }{2n+1}}).}$
Exercise 3(**). Find the right signs in the following trigonometric formula
${\displaystyle \tan({\frac {l\pi }{2n+1}})=2\sum _{k}(\pm )\sin({\frac {k\pi }{2n+1}}),l=1,2,\ldots n.}$

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

Correct signs