Ordinary differential equations, property C for

Let

\begin{equation*} \square _ { m } u = \left( - \frac { d ^ { 2 } } { d x ^ { 2 } } + q _ { m } ( x ) \right) u, \end{equation*}

\begin{equation*} m = 1,2 , x \in \mathbf{R} _ { + } : = [ 0 , \infty ), \end{equation*}

and let $q _ { m } ( x )$ be a real-valued function,

\begin{equation*} q _ { m } ( x ) \in L _ { 1,1 } (\mathbf{ R} _ { + } ) : = \left\{ q : \int _ { 0 } ^ { \infty } x | q ( x ) | d x < \infty \right\}. \end{equation*}

Consider the problem

\begin{equation*} ( \text{l} _ { m } - k ^ { 2 } ) f _ { m } = 0, \end{equation*}

\begin{equation*} x \in \mathbf{R} _ { + } ,\; f _ { m } ( x , k ) = e ^ { i k x } + o ( 1 ) \;\text { as } x \rightarrow + \infty. \end{equation*}

This problem has a unique solution, which is called the Jost function.

Define also the solutions to the problem

\begin{equation*} \left( \text{l} _ { m } - k ^ { 2 } \right) \varphi _ { m } ( x , k ) = 0, \end{equation*}

\begin{equation*} x \in \mathbf{R} _ { + } ,\, \varphi _ { m } ( 0 , k ) = 0 ,\, \varphi _ { m } ^ { \prime } ( 0 , k ) = 1, \end{equation*}

and to the problem

\begin{equation*} \left( \text{l} _ { m } - k ^ { 2 } \right) \varphi _ { m } ( x , k ) = 0, \end{equation*}

\begin{equation*} x \in {\bf R} _ { + } , \psi _ { m } ( 0 , k ) = 1 , \psi _ { m } ^ { \prime } ( 0 , k ) = 0. \end{equation*}

Assume $h ( x ) \in L ^ { 2 } ( \mathbf{R} _ { + } )$ and

\begin{equation} \tag{a1} \int _ { 0 } ^ { \infty } h ( x ) f _ { 1 } ( x , k ) f _ { 2 } ( x , k ) d x = 0 , \forall k > 0. \end{equation}

If (a1) implies $f ( x ) \equiv 0$, then one says that the pair $\{ \operatorname{l}_{1}, \operatorname{l}_{2} \}$ has property $C _ { + }$.

Let $b > 0$ be an arbitrary fixed number, let $h ( x ) \in L ^ { 1 } ( \mathbf{R} _ { + } )$ and assume

\begin{equation} \tag{a2} \int _ { 0 } ^ { b } h ( x ) \varphi _ { 1 } ( x , k ) \varphi _ { 2 } ( x , k ) d x = 0 , \forall k > 0. \end{equation}

If (a2) implies $h ( x ) \equiv 0$, then one says that the pair $\{ l _ { 1 } , l _ { 2 } \}$ has property $C _ { \varphi }$.

Similarly one defines property $C _ { \psi }$.

It is proved in [a1] that the pair $\{ l _ { 1 } , l _ { 2 } \}$ has property $C _ { + }$ if $q _ { m } \in L _ { 1,1 }$, $m = 1,2$.

It is proved in [a2] that the pair $\{ l _ { 1 } , l _ { 2 } \}$ has properties $C _ { \varphi }$ and $C _ { \psi }$.

However, if $b = \infty$, then, in general, property $C _ { \varphi }$ fails to hold for a pair $\{ l _ { 1 } , l _ { 2 } \}$. This means that there exist a function $h ( x ) \not\equiv 0$, $h \in L ^ { 1 } ( \mathbf{R} _ { + } )$, and two potentials $q_1 , q _ { 2 } \in L _ { 1 ,1} $, such that (a1) holds for all $k > 0$.

In [a2] many applications of properties $C _ { + }$, $C _ { \varphi }$ and $C _ { \psi }$ to inverse problems are presented.

For instance, suppose that the $I$-function, defined as $I ( k ) : = f ^ { \prime } ( 0 , k ) / f ( k )$, is known for all $k > 0$, $f ( k ) : = f ( 0 , k )$ and $f ( x , k )$ is the Jost function corresponding to a potential $q ( x ) \in L _ { 1,1 }$.

The function $I ( k )$ is known as the impedance function [a4], and it can be measured in some problems of electromagnetic probing of the Earth. The inverse problem (IP) is: Given $I ( k )$ for all $k > 0$, can one recover $q ( x )$ uniquely?

This problem was solved in [a4], but in [a1] and [a2] a new approach to this and many other inverse problems is developed. This new approach is sketched below.

Suppose that there are two potentials, $q _ { 1 } ( x )$ and $q_2 ( x )$, which generate the same data $I ( k )$. Subtract from the equation $( l _ { 1 } - k ^ { 2 } ) f _ { 1 } = 0$ the equation $( l _ { 2 } - k ^ { 2 } ) f _ { 2 } = 0$, and denote $f _ { 1 } - f _ { 2 } : = f$, $q _ { 2 } - q _ { 1 } : = p ( x )$, to get $( l _ { 1 } - k ^ { 2 } ) f = p f _ { 2 }$. Multiply this equation by $f _ { 1 } ( x , k )$, integrate over $( 0 , \infty )$ and then by parts. The assumption

\begin{equation*} I _ { 1 } ( k ) = \frac { f _ { 1 } ^ { \prime } ( 0 , k ) } { f _ { 1 } ( k ) } = \frac { f _ { 2 } ^ { \prime } ( 0 , k ) } { f _ { 2 } ( k ) } = I _ { 2 } ( k ) \end{equation*}

implies $\int _ { 0 } ^ { \infty } p ( x ) f _ { 1 } ( x , k ) f _ { 2 } ( x , k ) d x = 0$, $\forall k > 0$.

Using property $C _ { + }$ one concludes $p ( x ) \equiv 0$, that is, $q _ { 1 } ( x ) = q _ { 2 } ( x )$. This is a typical scheme for proving uniqueness theorems using property $C$.

References

[a1]	A.G. Ramm, "Property C for ODE and applications to inverse scattering" Z. Angew. Anal. , 18 : 2 (1999) pp. 331–348
[a2]	A.G. Ramm, "Property C for ODE and applications to inverse problems" A.G. Ramm (ed.) P.N. Shivakumar (ed.) A.V. Strauss (ed.) , Operator Theory And Its Applications , Fields Inst. Commun. , 25 , Amer. Math. Soc. (2000) pp. 15–75
[a3]	A.G. Ramm, "Inverse scattering problem with part of the fixed-energy phase shifts" Comm. Math. Phys. , 207 : 1 (1999) pp. 231–247
[a4]	A.G. Ramm, "Recovery of the potential from $I$-function" Math. Rept. Acad. Sci. Canada , 9 (1987) pp. 177–182