二維逆問題/克萊因非均勻弦

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，可能包括原子質量。可以用 ODE 的基本解來表達係數

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

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 $C^{+}$ 的解析自同構，即在實正半軸上為正。Herglotz 定理完全刻畫了以下積分表示的此類函式

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

其中，

$\sigma$ 是閉合正半軸 $(0,\infty )$ 上的有界變差的正測度。

練習(**)。 使用上面的定理，變數變換和傅立葉變換來刻畫具有旋轉不變電導率的圓盤的 Dirichlet 到 Neumann 對映集。