On 2D Inverse Problems/On inhomogeneous string of Krein

The following physical model of a vibrating inhomogeneous string (or string w/with beads) by Krein provides mechanical interpretation for the study of continued fractions, see [15]. The model is one-dimensional, and it arises as a 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 [12] and [21].

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

A fundamental theorem of Krein and Kac, see [12] & also [21], 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 real on the real line. The theorem of Herglotz completely characterizes such functions by the following integral representation, see [18]:

$\beta(\lambda) = \sigma_{\infty}\lambda + \frac{\sigma_0}{\lambda} + \int_0^{\infty}\frac{\lambda(1+x^2)d\sigma(x)}{\lambda^2+x^2},$

where,

$\sigma$ is a positive measure of bounded variation on the half-line $(0,\infty)$.

Exercise(**). Use the theorem above, Fourier transform and a change of variables to characterize the set of Dirichlet-to-Neumann maps for a unit disc with rotation independent conductivity.