×

Inverse scattering for the 1-D Helmholtz equation. (English) Zbl 1356.34084

The aim of the paper is to give a more elementary proof in the inverse scattering problem for the equation \[ u''+\frac{k^{2}}{c^{2}}u=0 \text{ on } \mathbb{R}. \] The function \(c\) is a measurable, bounded and bounded from below by a positive constant function and \(c-1 \in L^{1}(\mathbb{R})\). The main result states that the function \(c\) is uniquely determined by the reflection coefficient \(R_{2}(k)\) of the scattering matrix. The proof uses a uniqueness result for Nevanlinna functions.
Let us mention that uniqueness in the inverse scattering problem for the more general equation \[ u''+qu=k^{2}wu, \] \(q\geq0\), \(w\) real and may change the sign, was proved in [C. Bennewitz et al., J. Differ. Equations 253, No. 8, 2380–2419 (2012; Zbl 1323.34096)].

MSC:

34L25 Scattering theory, inverse scattering involving ordinary differential operators
81U40 Inverse scattering problems in quantum theory

Citations:

Zbl 1323.34096
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aktosun, T., Klaus, M., van der Mee, C.: Inverse scattering in one-dimensional media. Integral Equ. Oper. Theory 30, 279-316 (1990) · Zbl 0916.34070 · doi:10.1007/BF01195585
[2] Aronszajn, N., Donogue, W.F.: On exponential representation of analytic functions in the upper half-plane with positive imaginary part. J. Anal. Math. 5, 321-388 (1956-1957) · Zbl 0999.35102
[3] Beltiţă, I.: Inverse scattering in a layered medium. Commun. Partial Differ. Equ. 26(9-10), 1739-1786 (2001) · Zbl 1134.35385
[4] Bennewitz, C., Brown, B.M., Weikard, R.: Scattering and inverse scattering for a left-definite Sturm-Liouville problem. J. Differ. Equ. 253(8), 2380-2419 (2012) · Zbl 1323.34096 · doi:10.1016/j.jde.2012.06.016
[5] Browning, B.L.: Time and Frequency Domain Scattering for One-Dimensional Wave Equation. Thesis, University of Washington, (1999) · Zbl 0851.34011
[6] Browning, B.L.: Time and frequency domain scattering for one-dimensional wave equation. Inverse Probl. 16, 1377-1403 (2000) · Zbl 0999.35102 · doi:10.1088/0266-5611/16/5/315
[7] Chen, Y., Rokhlin, V.: On the inverse scattering problem for the Helmholtz equation in one dimension. Inverse Probl. 8, 365-391 (1992) · Zbl 0760.34017 · doi:10.1088/0266-5611/8/3/002
[8] Deift, P., Trubowitz, E.: Inverse scattering on the line. Commun. Pure Appl. Math. 32, 121-251 (1979) · Zbl 0388.34005 · doi:10.1002/cpa.3160320202
[9] Faddeev, L.D.: The inverse problem in the quantum theory of scattering. Uspehi Mat. Nauk. 14, 72-104 (1959) · Zbl 0112.45101
[10] Melin, A.: Operator methods for the inverse scattering on the real line. Commun. Part. Differ. Equ. 10, 677-766 (1985) · Zbl 0585.35077 · doi:10.1080/03605308508820393
[11] Rosenblum, M., Rovnyak, J.: Birkhäuser Advanced Texts: Basler Lehrbücher. Topics in Hardy classes and univalent functions. Birkhäuser Verlag, Basel (1994) · Zbl 0816.30001
[12] Walter, W.R.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co, New York (1987) · Zbl 0925.00005
[13] Sylvester, J., Winebrenner, T., Gyles-Colwell, F.: Layer stripping for the Helmholtz equation. SIAM J. Appl. Math. 50, 736-754 (1996) · Zbl 0851.34011 · doi:10.1137/S0036139995280257
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.