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A new approach to the inverse scattering and spectral problems for the Sturm‐Liouville equation
Author(s) -
Ramm A.G.
Publication year - 1998
Publication title -
annalen der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.009
H-Index - 68
eISSN - 1521-3889
pISSN - 0003-3804
DOI - 10.1002/(sici)1521-3889(199811)7:4<321::aid-andp321>3.0.co;2-x
Subject(s) - inverse scattering problem , uniqueness , scattering , mathematical proof , inversion (geology) , inverse , inverse problem , inverse scattering transform , scattering theory , mathematics , class (philosophy) , uniqueness theorem for poisson's equation , mathematical analysis , physics , computer science , optics , geometry , paleontology , structural basin , artificial intelligence , biology
Abstract New proofs of the known uniqueness theorems for the one‐dimensional inverse spectral and scattering problems are given. Proof of the invertibility of all of the steps in the inversion procedures of Gelfand‐Levitan and Marchenko is given. The proposed method of investigation yields some new results, for example, a Marchenko‐type equation at x = 0 which holds on the whole axis, rather than on a half‐axis, as usual for the scattering theory on half‐axis. It also yields a new method, shorter and simpler than earlier published, for proving that the potential in the class L 1,1 , obtained by the Marchenko reconstruction procedure, generates the scattering data from which it was reconstructed.