On 2D Inverse Problems/On inhomogeneous string of Krein

The following physical model of a vibrating inhomogeneous string (or string w/beads) by Krein provides physical/mechanical interpretation for the study of  Stieltjes continued fractions, see [GK]. The model is one-dimensional, but it arises as the restriction of n-dimensional inverse problems with rotational symmetry.

The string is represented by a non-decreasing positive mass function m(x) on a possibly infinite interval [0, l]. The right end of the string is fixed. The ratio of the forced oscillation to an applied periodic force @ the left end of the string is the function of frequency, called coefficient of dynamic compliance of the string, see [KK] and [I2].

The small vertical vibration of the string is described by the following differential equation:

${\frac {1}{\rho (x)}}{\frac {\partial ^{2}f(x,\lambda )}{\partial x^{2}}}=\lambda f(x,\lambda ),$

where $\rho (x)={\frac {dm}{dx}}$ is the density of the string, possibly including atomic masses. One can express the coefficient in terms of the fundamental solution of the ODE:

$H(\lambda )={\frac {f'(0,\lambda )}{f(0,\lambda )}},$

where, $f(l,\lambda )=0.$

The fundamental theorem of Krein and Kac, see [KK] & also [I2], essentially states that an analytic function $H(\lambda$) is the coefficient of dynamic compliance of a string if and only if the function 

$\beta (\lambda )=\lambda H(-\lambda ^{2})$

is an analytic automorphism of the right half-plane $C^{+}$, that is positive on the real positive ray. The Herglotz theorem completely characterizes such functions by the following integral representation:

$\beta (\lambda )=\sigma _{\infty }\lambda +{\frac {\sigma _{0}}{\lambda }}+\int _{0}^{\infty }{\frac {\lambda (1+x^{2})d\sigma (x)}{\lambda ^{2}+x^{2}}},$

where,

$\sigma$ is positive measure of bounded variation on the closed positive ray $(0,\infty )$.

Exercise(**). Use the theorem above, change of variables and the Fourier transform to characterize the set of Dirichlet-to-Neumann maps for a disc w/rotationally invariant conductivity.