User:DVD206/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 Stieltjes continued fractions. The model is one-dimensional, but 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.

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 [10], & also [19] 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.

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 conductivity depending only on radius.