Inverse scattering, half-axis case

The direct scattering problem on the half-axis consists of finding the solution to the problem

(a1)

(a2)

(a3)

Here, is to be determined. The function is called the phase shift. The coefficient is called the scattering potential. It is assumed to be a real-valued function in the class

where the bar stands for complex conjugation. The solution to (a1) which satisfies the relation , as , is called the Jost solution. The function is called the Jost function. One has

If , then exists and is unique, is analytic in and has at most finitely many zeros in , all of which are simple and of the form , , . The numbers are the eigenvalues of the self-adjoint operator

which is determined by the Dirichlet boundary condition at in the Hilbert space , (cf. also Dirichlet boundary conditions). In physics, are called the bound states. The positive numbers are called the norming constants. The function is called the -matrix (cf. Scattering matrix). The triple is called the scattering data.

The inverse scattering problem consists of finding given .

The point can also be a zero of . It is called a resonance at . If , then . The basic results of inverse scattering theory are (see [a5], [a6]):

1) The uniqueness theorem: ; that is, the scattering data determine uniquely.

2) The reconstruction theorem: If , corresponding to a , is given, then can be reconstructed by the Marchenko method, as follows:'

<tbody> </tbody> 1 Calculate 2 Solve the following equation for : (a4) This equation is uniquely solvable and is called the Marchenko equatio. 3 Calculate .

3) The characterization theorem: For to be the scattering data corresponding to a it is necessary and sufficient that the following conditions hold:

i) , ; , , , ;

ii) , or ;

iii)

Here, .

Note that if , and if . The mapping is a homeomorphism between and the space of the scattering data equipped with the norm (see [a4], [a5] [a6]).

One can prove (see [a6], [a13]) the diagram

each step of which is invertible. Here, and are defined above. This result guarantees, in particular, that the potential recovered by the Marchenko method generates the original scattering data (provided that or satisfies the characterization conditions).

Other methods for solving the inverse scattering problem on the half-axis are based on the solution of the inverse problem of recovery of from the spectral function () and the Krein method ([a1], [a3], [a5] [a6], [a15]).

The scattering data are in one-to-one correspondence with the spectral function [a6] [a7], [a13]. Recovery of given the spectral function is discussed in [a1], [a3], [a5], [a6].

The original work of M.G. Krein [a2] and its review in [a1] do not contain proofs. A detailed presentation of Krein's theory with complete proofs is given in [a15] for the first time. Also, a proof of consistency of Krein's method is given in [a15]. In [a2] (and in [a1]) there is no discussion of the consistency of Krein's method. By the consistency of an inversion method one means a proof of the implication (the reconstructed potential generates the data from which it was reconstructed).

Below, Krein's method is described under the simplifying assumption (no bound states and no resonance at ). The general case is treated in [a15].'

<tbody> </tbody> 1 Given , , , one finds , then calculates and 2 Given , one solves the equation for and finds , . , and calculates . Alternatively,

In Step 1, one can find by a different method: Solve the Riemann problem

(a5)

If , this problem has the unique solution . One has , .

Note that the data allow one to find a unique by solving the Riemann problem (a5) with the additional conditions: has simple zeros at the points if and, if , has, in addition, a simple zero at . Thus, the data is equivalent to the data .

An inverse problem of recovery of from incomplete scattering data but with an a priori assumption that has compact support is investigated in [a8] [a9]. It is proved that if is compactly supported and if is known for a sequence which has a finite limit point inside , then is determined uniquely. An algorithm for finding a compactly supported from (that is, from ) known for all is given in [a8]. A uniqueness theorem for the problem of finding a compactly supported from the knowledge of , , is proved in [a13].

In [a7], [a12] an algorithm for recovery of from the -function is given, where the -function is identical with the Weyl function.

For to belong to it is necessary and sufficient [a6] that

(a6)

where

If , for , is compactly supported, then is an entire function of exponential type . Its zeros in are called resonances.

If , , , then there are infinitely many resonances [a6].

There exists a , for , where is arbitrary small, which generates infinitely many purely imaginary resonances [a6].

If , for and does not change sign in an interval , where is arbitrarily small, then generates only finitely many purely imaginary resonances (a6).

If , then the following estimate (see [a5]) is useful:

The Jost solution can be written as , where is the kernel of the transformation operator. If , then

where is a constant. The function solves the Volterra-type equation

If and for , then for , for , and for . Since , it follows that is an entire function of order and type , and is meromorphic on the whole complex -plane (cf. also Meromorphic function).

Conversely, if the scattering data correspond to a (necessary and sufficient conditions for this were given above) and generate (by solving the Riemann problem mentioned above) the function which is an entire function of exponential type , then for , (see [a6]).

References

[a1]	K. Chadan, P. Sabatier, "Inverse problems in quantum scattering theory" , Springer (1989)
[a2]	M. Krein, "Theory of accelerants and -matrices of canonical differential systems" Dokl. Akad. Nauk. USSR , III : 6 (1956) pp. 1167–1170 (In Russian)
[a3]	B. Levitan, "Inverse Sturm–Liouville problems" , VNU Press (1987)
[a4]	V. Marchenko, "Stability in the inverse problem of scattering theory" Mat. Sb. , 77 (1968) pp. 139–162 (In Russian)
[a5]	V. Marchenko, "Sturm–Liouville operators and applications" , Birkhäuser (1986)
[a6]	A.G. Ramm, "Multidimensional inverse scattering problems" , Longman/Wiley (1992)
[a7]	A.G. Ramm, "Recovery of the potential from I-function" Math. Rept. Acad. Sci. Canada , 9 (1987) pp. 177–182
[a8]	A.G. Ramm, "Recovery of compactly supported spherically symmetric potentials from the phase shift of -wave" A.G. Ramm (ed.) , Spectral and Scattering Theory , Plenum (1998) pp. 111–130
[a9]	A.G. Ramm, "Compactly supported spherically symmetric potentials are uniquely determined by the phase shift of -wave" Phys. Lett. A , 242 : 4–5 (1998) pp. 215–219
[a10]	A.G. Ramm, "Recovery of a quarkonium system from experimental data" J. Phys. A , 31 : 15 (1998) pp. L295–L299
[a11]	A.G. Ramm, "Inverse scattering problem with part of the fixed-energy phase shifts" Comm. Math. Phys. , 207 : 1 (1999) pp. 231–247
[a12]	A.G. Ramm, "Property C for ODE and applications to inverse scattering" Z. Angew. Anal. , 18 : 2 (1999) pp. 331–348
[a13]	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 Applications , Fields Inst. Commun. , 25 , Amer. Math. Soc. (2000) pp. 15–75
[a14]	A.G. Ramm, W. Scheid, "An approximate method for solving inverse scattering problem with fixed-energy data" J. Inverse Ill-Posed Probl. , 7 : 6 (1999) pp. 561–571
[a15]	A.G. Ramm, "Krein's method in inverse scattering" , Operator Theory and Applications , Amer. Math. Soc. (2000) pp. 441–456