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.