Ordinary differential equations, property C for

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 .

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 -function" Math. Rept. Acad. Sci. Canada , 9 (1987) pp. 177–182