Abstract
The inverse scattering problem is considered for the radials-wave Schrödinger equation with the energy-dependent potentialV +(E,x)=U(x)+2\(\sqrt E \) Q(x). (Note that this problem is closely related to the inverse problem for the radials-wave Klein-Gordon equation of zero mass with a static potential.) Some authors have already studied it by extending the method given by Gel'fand and Levitan in the caseQ=0. Here, a more direct approach generalizing the Marchenko method is used. First, the Jost solutionf +(E,x) is shown to be generated by two functionsF +(x) andA +(x,t). After introducing the potentialV −(E,x)=U(x)−2\(\sqrt E \) Q(x) and the corresponding functionsF −(x) andA −(x,t), fundamental integral equations are derived connectingF +(x),F −(x),A +(x,t) andA −(x,t) with two functionsz +(x) andz −(x);z +(x) andz −(x) are themselves easily connected with the binding energiesE + n and the scattering “matrix”S +(E),E>0 (the input data of the inverse problem). The inverse problem is then reduced to the solution of these fundamental integral equations. Some specific examples are given. Derivation of more elaborate results in the case of real potentials, and applications of this work to other inverse problems in physics will be the object of further studies.
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References
For a survey of this inverse problem, see Faddeyev, L. D.: The inverse problem in the quantum theory of scattering. J. Math. Phys.4, 72 (1963).
Newton, R. G.: Scattering theory of waves and particles, Chapter 20. New York: Mc Graw-Hill Book Company, 1966.
Gel'fand, I. M., Levitan, B. M.: On the determination of a differential equation from its spectral function. Izvest. Akad. Nauk S.S.S.R.15, 309 (1951); translated in Am. Math. Soc. Transl.1, 253 (1955).
For the Marchenko method, see the book of Agranovich, Z. S., Marchenko, V. A.: The inverse problem of scattering theory. New York: Gordon and Breach 1963.
In the case of a complex energy-independent potential, this assumption has already been used by Gasymov, M. G.: Doklad. Akad. Nauk S.S.S.R.165, 261 (1965), see also Bertero, M., Dillon, G.: An outline of scattering theory for absorptive potentials. Nuovo Cimento2 A, 1024 (1971).
The first paper on the subject is that of Corinaldesi, E.: Construction of potentials from phase shift and binding energies of relativistic equations. Nuovo Cimento11, 468 (1954). See also a recent paper of Degasperis, A.: On the inverse problem for the Klein-Gordons-wave equation. J. Math. Phys.11, 551 (1970), in which other references are given.
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See, for example, De Alfaro, V., Regge, T.: Potential scattering, Chapter 3. Amsterdam: North Holland Publ. Comp. 1965.
See Chapter I, § 2, of Ref. [3].
For the caseQ=0, see Chapter 4 of Ref. [7].
See Chapter I, § 4, of Ref. [3].
See the paper of Bertero, M., Dillon, G.: Quoted in Ref. [4], formula (3.6).
Titchmarsh, E. C.: Introduction to the theory of Fourier integrals, p. 128. Oxford 1937.
See Chapter 1, § 3, of Ref. [3].
See, for example, Bochner, S., Chandrasekharan, K.: Fourier transforms, Annals of Mathematics Studies, Princeton University Press (1960), Theorem 60.
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See Chapter III of Ref. [3].
See formulas (3.5), (3.7) and (3.8) of Ref. [6].
For the proof, see Ref. [3], Chapter III, Lemma 3.2.1.
See the paper of Faddeyev, L. D.: Quoted in Ref. [1], Theorem 12.1.
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Physique Mathématique et Théorique, Equipe de recherche associée au C.N.R.S.
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Jaulent, M., Jean, C. The inverses-wave scattering problem for a class of potentials depending on energy. Commun.Math. Phys. 28, 177–220 (1972). https://doi.org/10.1007/BF01645775
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DOI: https://doi.org/10.1007/BF01645775