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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 &lt; \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 &gt; 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 $b &gt; 0$ 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 &gt; 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 [[#References|[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 [[#References|[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 $k &gt; 0$.

In [[#References|[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 $k &gt; 0$, $f ( k ) : = f ( 0 , k )$ 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 [[#References|[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 $k &gt; 0$, can one recover $q ( x )$ uniquely?

This problem was solved in [[#References|[a4]]], but in [[#References|[a1]]] and [[#References|[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 $\int _ { 0 } ^ { \infty } p ( x ) f _ { 1 } ( x , k ) f _ { 2 } ( x , k ) d x = 0$, $\forall k &gt; 0$.

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====
<table><tr><td valign="top">[a1]</td> <td valign="top">  A.G. Ramm,   "Property C for ODE and applications to inverse scattering"  ''Z. Angew. Anal.'' , '''18''' :  2  (1999)  pp. 331–348</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A.G. Ramm,   "Inverse scattering problem with part of the fixed-energy phase shifts"  ''Comm. Math. Phys.'' , '''207''' :  1  (1999)  pp. 231–247</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  A.G. Ramm,   "Recovery of the potential from $I$-function"  ''Math. Rept. Acad. Sci. Canada'' , '''9'''  (1987)  pp. 177–182</td></tr></table>