Contemporary Mathematics Volume 122, 1991 WIENER-HOPF FACTORIZATION IN MULTIDIMENSIONAL INVERSE SCHRODINGER SCATTERING 1 Tuncay Aktosun2 and Cornelis van der Mee3 ABSTRACT. We consider a Riemann-Hilbert problem arising in the study of the in- verse scattering for the multidimensional Schrodinger equation with a potential having no spherical symmetry. It is shown that under certain conditions on the potential, the corresponding scattering operator admits a Wiener-Hopf factorization. The solution of the Riemann-Hilbert problem can be obtained using a similar factorization for the unitar- ily dilated scattering operator. We also study the connection between the Wiener-Hop£ factorization and the Newton-Marchenko integral operator. 1. RIEMANN-HILBERT PROBLEM IN QUANTUM SCATTERING. Consider the n-dimensional Schrodinger equation ( n ~ 2) (1.1) where x E R n, ~ is the Laplacian, k2 is energy, and V ( x) is the potential. In nonrelativistic quantum mechanics the behavior of a particle in the force field of V ( x) is governed by ( 1.1). Mathematics Subject Classification (1991). 35Q15, 81U05, 35R30, 47A40. 1 This paper is in final form and no version of it will be submitted for publication elsewhere. The authors are indebted to Roger Newton for his help. 2 The author is supported by the National Science Foundation under grant DMS 9001903. 3 The author is supported by the National Science Foundation under grant DMS 8823102. © 1991 American Mathematical Society 0271-4132/91 $1.00 + $.25 per page http://dx.doi.org/10.1090/conm/122/1135851
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