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:
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:
The fundamental theorem of Krein and Kac, see [KK] & also [I2], essentially states that an analytic function ) is the coefficient of dynamic compliance of a string if and only if the function
is an analytic automorphism of the right half-plane , that is positive on the real positive ray. The Herglotz theorem completely characterizes such functions by the following integral representation:
is positive measure of bounded variation on the closed positive ray .
- 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.