View source for Ordinary differential equations, property C for

Let

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and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o1300804.png" /> be a real-valued function,

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Consider the problem

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

Define also the solutions to the problem

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and to the problem

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Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008012.png" /> and

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If (a1) implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008014.png" />, then one says that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008015.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008017.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008018.png" /> be an arbitrary fixed number, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008019.png" /> and assume

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If (a2) implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008021.png" />, then one says that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008022.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008024.png" />.

Similarly one defines property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008026.png" />.

It is proved in [[#References|[a1]]] that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008027.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008028.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008030.png" />.

It is proved in [[#References|[a2]]] that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008031.png" /> has properties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008033.png" />.

However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008034.png" />, then, in general, property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008035.png" /> fails to hold for a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008036.png" />. This means that there exist a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008038.png" />, and two potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008039.png" />, such that (a1) holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008040.png" />.

In [[#References|[a2]]] many applications of properties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008043.png" /> to inverse problems are presented.

For instance, suppose that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008045.png" />-function, defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008046.png" />, is known for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008049.png" /> is the Jost function corresponding to a potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008050.png" />.

The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008051.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008052.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008053.png" />, can one recover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008054.png" /> 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008056.png" />, which generate the same data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008057.png" />. Subtract from the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008058.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008059.png" />, and denote <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008061.png" />, to get <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008062.png" />. Multiply this equation by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008063.png" />, integrate over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008064.png" /> and then by parts. The assumption

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implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008067.png" />.

Using property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008068.png" /> one concludes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008069.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008070.png" />. This is a typical scheme for proving uniqueness theorems using property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008071.png" />.

====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130080/o13008072.png" />-function"  ''Math. Rept. Acad. Sci. Canada'' , '''9'''  (1987)  pp. 177–182</TD></TR></table>