Ordinary differential equations, property C for
Let
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and let 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 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
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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 ![]() |
Ordinary differential equations, property C for. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordinary_differential_equations,_property_C_for&oldid=50329