# Ordinary differential equations, property C for

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Let  and let be a real-valued function, Consider the problem  This problem has a unique solution, which is called the Jost function.

Define also the solutions to the problem  and to the problem  Assume and (a1)

If (a1) implies , then one says that the pair has property .

Let be an arbitrary fixed number, let and assume (a2)

If (a2) implies , then one says that the pair has property .

Similarly one defines property .

It is proved in [a1] that the pair has property if , .

It is proved in [a2] that the pair has properties and .

However, if , then, in general, property fails to hold for a pair . This means that there exist a function , , and two potentials , such that (a1) holds for all .

In [a2] many applications of properties , and to inverse problems are presented.

For instance, suppose that the -function, defined as , is known for all , and is the Jost function corresponding to a potential .

The function 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 for all , can one recover 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, and , which generate the same data . Subtract from the equation the equation , and denote , , to get . Multiply this equation by , integrate over and then by parts. The assumption implies , .

Using property one concludes , that is, . This is a typical scheme for proving uniqueness theorems using property .

How to Cite This Entry:
Ordinary differential equations, property C for. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordinary_differential_equations,_property_C_for&oldid=18905
This article was adapted from an original article by A.G. Ramm (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article