Ordinary differential equations, property C for
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
If (a1) implies , then one says that the pair has property .
Let be an arbitrary fixed number, let and assume
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?
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 .
|[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|
Ordinary differential equations, property C for. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordinary_differential_equations,_property_C_for&oldid=18905