Inverse Scattering Problems with the Potential Known on an Interior Subinterval
DOI:
https://doi.org/10.15407/mag15.02.225Анотація
Розглянуто обернену задачу для одновимірного оператора Шредінгера на прямій у випадку, коли потенціал є дійсно значним, інтегрованим та має скінчений перший момент. Показано, що цей потенціал на прямій однозначно визначений змішаними даними розсіювання, які містять матрицю розсіювання, заданий на скінченому інтервалі потенціал та одну вузлову точку на заданому інтервалі для кожної власної функції.
Mathematics Subject Classification: 34A55, 34L25, 34L40.
Ключові слова:
рівняння Шредінгера, обернена задача розсіювання, відновлення потенціалу за частковими даними.Посилання
T. Aktosun, M. Klaus, and C. van der Mee, On the Riemann–Hilbert problem for the one-dimensional Schrö dinger equation, J. Math. Phys. 34 (1993), 2651–2690. https://doi.org/10.1063/1.530180
T. Aktosun, Bound-states and inverse scattering for the Schrödinger equation in one dimension, J. Math. Phys. 35 (1994), 6231–6236. https://doi.org/10.1063/1.530671
T. Aktosun, Inverse Schrödinger scattering on the line with partial knowledge of the potential, SIAM J. Appl. Math. 56 (1996), 219–231. https://doi.org/10.1137/S0036139994273995
T. Aktosun and R. Weder, Inverse scattering with partial knowledge of the potential, J. Math. Anal. Appl. 270 (2002), 247–266. https://doi.org/10.1016/S0022-247X(02)00070-7
M. Braun, S. Sofianos and R. Lipperheide, One-dimensional Marchenko inversion in the presence of bound states, Inverse Problems 11 (1995), L1–L3. https://doi.org/10.1088/0266-5611/11/3/001
X. Chen, Y.H. Cheng, and C.K. Law, Reconstructing potentials from zeros of one eigenfunction, Trans. Amer. Math. Soc. 363 (2011), 4831–4851. https://doi.org/10.1090/S0002-9947-2011-05408-5
P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Partial Differential Equations 32 (1979), 121–251. https://doi.org/10.1002/cpa.3160320202
I.M. Gel’fand and B.M. Levitan, On the determination of a differential equation from its spectral function, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 309– 360 (Russian).
B. Grebert and R. Weder, Reconstruction of a potential on the line that is a priori known on the half line, SIAM J. Appl. Math. 55 (1995), 242–254. https://doi.org/10.1137/S0036139993254656
O.H. Hald and J.R. McLaughlin, Inverse problems: recovery of BV coefficients from nodes, Inverse Problems 14 (1998), 245–273. https://doi.org/10.1088/0266-5611/14/2/003
O.H. Hald and J.R. McLaughlin, Solutions of inverse nodal problems, Inverse Problems 5 (1989), 307–347. https://doi.org/10.1088/0266-5611/5/3/008
A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, New York, 1996. https://doi.org/10.1007/978-1-4612-5338-9
B.M. Levitan, The determination of a Sturm–Liouville operator from one or from two spectra, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 185–199, 215–216 (Russian).
B.M. Levitan, Inverse Sturm–Liouville problem, VNU Science Press, Utrecht, 1987. https://doi.org/10.1515/9783110941937
V.A. Marchenko, Sturm–Liouville Operator and Applications, Birkhauser, Basel, 1986. https://doi.org/10.1007/978-3-0348-5485-6
J.R. McLaughlin, Inverse spectral theory using nodal points as data-a uniqueness result, J. Differential Equation 73 (1988), 354–362. https://doi.org/10.1016/0022-0396(88)90111-8
N.N. Novikova and V.M. Markushevich, Uniqueness of the solution of the onedimensional problem of scattering for potentials located on the positive semiaxis, Comput. Seismol. 18 (1987), 164–172.
D.L. Pursey and T. Weber, Formulations of certain Gelfand-Levitan and Marchenko equations, Phys. Rev. A 50 (1994), 325–347. https://doi.org/10.1103/PhysRevA.50.4472
W.T. Reid, Sturmian Theory for Ordinary Differential Equations, Springer-Verlag, New York, 1980. https://doi.org/10.1007/978-1-4612-6110-0
W. Rundell and P. Sacks, On the determination of potentials without bound state data, J. Comput. Appl. Math. 55 (1994), 325–347. https://doi.org/10.1016/0377-0427(94)90037-X
P. Sacks, Reconstruction of steplike potentials, Wave Motion 18 (1993), 21–30. https://doi.org/10.1016/0165-2125(93)90058-N
G. Wei and H.K. Xu, On the missing bound state data of inverse spectral-scattering problems on the half-line, Inverse Probl. Imaging 9 (2015), 239–255. https://doi.org/10.3934/ipi.2015.9.239
