On 2D Inverse Problems/On inhomogeneous string of Krein

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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),

\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},


\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.