MR2440939 Yutaka KAMIMURA Yutaka KAMIMURA Energy Dependent Inverse Scattering Funkcialaj Ekvacioj. Serio Internacia 51 2008 189--219 http://fe.math.kobe-u.ac.jp/FE/FullPapers/51-2/51_189.pdf http://www.ams.org/mathscinet-getitem?mr=MR2440939 The paper discusses the inverse scattering problem to recover the potentials of an energy dependent Schrödinger equation from the scattering data. A necessary and sufficient condition for a given function to be the scattering data is obtained. The potential can be recovered by a simple formula from the solution of a Marchenko type equation with a function determined from the scattering data. Energy dependent Schrödinger equation, Inverse scattering problem, Marchenko equation. 81U40, 34A55, 45E10. 51-189 2008 Energy Dependent Inverse Scattering Yutaka KAMIMURA Yutaka KAMIMURA 1 Agranovich, Z. S.; Marchenko, V. A. The Inverse Problem of Scattering Theory Gordon and Breach, New York 1963 MR0162497
2 Aktosun, T.; Klaus, M.; van der Mee, C. Wave scattering in one dimension with absorption J. Math. Phys. 39 1998 1957--1992 MR1614734
3 Aktosun, T.; Klaus, M.; van der Mee, C. Inverse scattering in one-dimensional nonconservative media Integral Equations Operator Theory 30 1998 279--316 MR1608677
4 Chadan, K.; Sabatier, P. C. Inverse Problems in Quantum Scattering Theory 2nd ed., Springer-Verlag, New York 1989 MR0985100
5 Cornille, H. Existence and uniqueness of crossing symmetric $N/D-$type equations corresponding to the Klein-Gordon equation J. Math. Phys. 11 1970 79--98
6 Jaulent, M. On an inverse scattering problem with an energy-dependent potential Ann. Inst. Henri Poincaré, Sect A 17 1972 363--378 MR0363217
7 Jaulent, M. Sur le problème inverse de la diffusion pour l'équation de Schrödinger radiale avec un potentiel dépendent de l'énergie C. R. Acad. Sc. Paris 280 1975 1467--1470 MR0375988
8 Jaulent, M. Inverse scattering problems in absorbing media J. Math. Phys. 17 1976 1351--1360 MR0416260
9 Jaulent, M.; Jean, C. The inverse $s$-wave scattering problem for a class of potentials depending on energy Comm. Math. Phys. 28 1972 177--220 MR0311227
10 Jaulent, M.; Jean, C. The inverse problem for the one-dimensional Schrödinger equation with an energy-dependent potential, I Ann. Inst. Henri Poincaré, Sect A 25 1976 105--118 MR0441125
11 Jaulent, M.; Jean, C. The inverse problem for the one-dimensional Schrödinger equation with an energy-dependent potential, II Ann. Inst. Henri Poincaré, Sect A 25 1976 119-137 MR0441126
12 Kamimura, Y. An inversion formula in energy dependent scattering J. Integral Equations Appl. 19 2007 473--512 MR2374166
13 Kamimura, Y. An inverse problem in advection-diffusion J. Phys., Conf. Ser. 54 2007 doi 012012
14 Kaup, D. J. A higher-order water-wave equation and the method for solving it Progr. Theoret. Phys. 54 1975 396--408 MR0434208
15 Marchenko, V. A. Sturm-Liouville Operators and Applications Operator Theory: Advances and Applications, 22, Birkhäuser-Verlag, Basel 1986 MR0897106 16 Paley, R. E. A. C.; Wiener, N Fourier Transforms in the Complex Domain Colloquium Publication, 19, Amer. Math. Soc. 1934 MR1451142
17 Sattinger, D. H.; Szmigielski, J. Energy dependent scattering theory Differential Integral Equations 8 1995 945--959 MR1325540
18 Sattinger, D. H.; Szmigielski, J. A Riemann-Hilbert problem for an energy dependent Schrödinger operator Inverse Problems 12 1996 1003--1025 MR1421662
19 van der Mee, C.; Pivovarchik, V Inverse scattering for a Schrödinger equation with energy dependent potential J. Math. Phys. 42 2001 158--181 MR1808771