二维逆问题/克莱因非均匀弦

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:

${\frac {1}{\rho (x)}}{\frac {\partial ^{2}f(x,\lambda )}{\partial x^{2}}}=\lambda f(x,\lambda ),$

其中 $\rho (x)={\frac {dm}{dx}}$ 是弦的密度，可能包括原子质量。可以用 ODE 的基本解来表达系数

$H(\lambda )={\frac {f'(0,\lambda )}{f(0,\lambda )}},$

其中，$f(l,\lambda )=0.$

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 

$\beta (\lambda )=\lambda H(-\lambda ^{2})$

是右半平面 $C^{+}$ 的解析自同构，即在实正半轴上为正。Herglotz 定理完全刻画了以下积分表示的此类函数

$\beta (\lambda )=\sigma _{\infty }\lambda +{\frac {\sigma _{0}}{\lambda }}+\int _{0}^{\infty }{\frac {\lambda (1+x^{2})d\sigma (x)}{\lambda ^{2}+x^{2}}},$

其中，

$\sigma$ 是闭合正半轴 $(0,\infty )$ 上的有界变差的正测度。

练习(**)。 使用上面的定理，变量变换和傅里叶变换来刻画具有旋转不变电导率的圆盘的 Dirichlet 到 Neumann 映射集。