# 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 [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:

where

is the density of the string, possibly including atomic masses. One can express the coefficient in terms of the fundamental solution of the ODE:

where,

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

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:

where,

is a positive measure of bounded variation on the half-line .

**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.