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/with beads) by Krein provides mechanical interpretation for the study of continued fractions, see [GK]. 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 [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.

A 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 real on the real line. The theorem of Herglotz 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 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.