×

Reconstruction of the impedance Schrödinger equation from the modulus of the reflection coefficients. (English) Zbl 1360.35324

Summary: We study the problem of determining the electromagnetic properties of a stratified medium given only the amplitude of the reflected waves. Under some assumptions, the problem is formulated as the inverse scattering problem for a Schrödinger operator in impedance form. We show that a unique reconstruction from the modulus of the reflection coefficients is in general impossible and explain the cause of non-uniqueness. Several augmented sets of data allowing a unique reconstruction are suggested, and the corresponding numerical examples are provided.

MSC:

35R30 Inverse problems for PDEs
35J10 Schrödinger operator, Schrödinger equation
35P25 Scattering theory for PDEs
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Zuev, I. V.; Tikhonravov, A. V., Uniqueness of the determination of the parameters of a layered medium from the reflection energy coefficient, Zh. Vychisl. Mat. Mat. Fiz., 33, 3, 428-438 (1993), (in Russian); Engl. transl.: Comput. Math. Math. Phys. 33 (3) (1993) 387-395 · Zbl 0881.65125
[2] Klibanov, M. V.; Sacks, P. E.; Tikhonravov, A. V., The phase retrieval problem, Inverse Problems, 11, 1, 1-28 (1995) · Zbl 0821.35150
[3] Aktosun, T.; Sacks, P. E., Inverse problem on the line without phase information, Inverse Problems, 14, 2, 211-224 (1998) · Zbl 0902.34011
[4] Levitan, B. M., Inverse Sturm-Liouville Problems (1984), Nauka Publ.: Nauka Publ. Moscow, (in Russian); Engl. transl., VNU Science Press, Utrecht, 1987 · Zbl 0575.34001
[5] Marchenko, V. A., Sturm-Liouville Operators and their Applications (1977), Naukova Dumka Publ.: Naukova Dumka Publ. Kiev, (in Russian); Engl. transl., Birkhäuser Verlag, Basel, 1986 · Zbl 0399.34022
[6] Chadan, K.; Sabatier, P. C., Inverse Problems in Quantum Scattering Theory (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0681.35088
[7] Newton, R. G., (Inverse Schrödinger Scattering in Three Dimensions. Inverse Schrödinger Scattering in Three Dimensions, Texts and Monographs in Physics (1989), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0697.35005
[8] Albeverio, S.; Hryniv, R.; Mykytyuk, Ya., Inverse scattering for discontinuous impedance Schrödinger operators: a model example, J. Phys. A, 44, 8 (2011), ID 345204 · Zbl 1226.81285
[9] Weidmann, J., (Spectral Theory of Ordinary Differential Operators. Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, vol. 1258 (1987), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0647.47052
[10] Teschl, G., (Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators. Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, Graduate Studies in Mathematics, vol. 99 (2009), AMS: AMS Providence, Rhode Island) · Zbl 1166.81004
[11] Gelfand, I. M.; Levitan, B. M., On the determination of a differential equation from its spectral function, Izv. Akad. Nauk SSSR Ser. Mat., 15, 4, 309-360 (1951), (in Russian) · Zbl 0044.09301
[12] Marchenko, V. A., Some questions of the theory of one-dimensional linear differential operators of the second order, I, Tr. Mosk. Mat. Obs., 1, 327-420 (1952), (in Russian) · Zbl 0048.32501
[13] Krein, M. G., On determination of the potential of a particle from its \(S\)-function, Dokl. Akad. Nauk SSSR (NS), 105, 433-436 (1955) · Zbl 0065.43301
[14] Deift, P.; Trubowitz, E., Inverse scattering on the line, Comm. Pure Appl. Math., 32, 2, 121-251 (1979) · Zbl 0388.34005
[15] Melin, A., Operator methods for inverse scattering on the real line, Comm. Partial Differential Equations, 10, 7, 677-766 (1985) · Zbl 0585.35077
[16] Faddeev, L. D., The inverse problem in the quantum theory of scattering, Uspekhi Mat. Nauk, 14, 57-119 (1959), (in Russian); Engl. transl. in J. Math. Phys. 4 (1963) 72-104 · Zbl 0091.21902
[17] Ware, J. A.; Aki, K., Continuous and discrete inverse scattering problems in a stratified elastic medium. I. Plane waves at normal incidence, J. Acoust. Soc. Am., 45, 911-921 (1969) · Zbl 0197.23102
[18] Frayer, C.; Hryniv, R. O.; Mykytyuk, Ya. V.; Perry, P. A., Inverse scattering for Schrödinger operators with Miura potentials: I. Unique Riccati representatives and ZS-AKNS systems, Inverse Problems, 25, 11, 25 (2009) · Zbl 1181.35330
[19] Hryniv, R. O.; Mykytyuk, Ya. V.; Perry, P. A., Inverse scattering for Schrödinger operators with Miura potentials: II. Different Riccati representatives, Comm. Partial Differential Equations, 36, 9, 1587-1623 (2011) · Zbl 1232.34113
[20] Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., 53, 4, 249-315 (1974) · Zbl 0408.35068
[21] Zakharov, V. E.; Shabat, A. B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Phys. JETP, 34, 1, 62-69 (1972), Translated from Èksper. Teoret. Fiz. 61 (1) (1971) 118-134 (in Russian)
[22] Hinton, D. B.; Jordan, A. K.; Klaus, M.; Shaw, J. K., Inverse scattering on the line for a Dirac system, J. Math. Phys., 32, 11, 3015-3030 (1991)
[23] Glasko, V. B.; Khudak, Yu. I., Additive representation of characteristics of layered media and questions of uniqueness of the solutions of inverse problems, Zh. Vychisl. Mat. Mat. Fiz., 20, 2, 482-490 (1980), (in Russian); Engl. transl. USSR Comput. Math. Math. Phys. 20 (2) (1980) 213-222 · Zbl 0446.35061
[24] Sabatier, P. C., For an impedance scattering theory, (Nonlinear Evolutions (Balaruc-les-Bains, 1987) (1988), World Sci. Publ.: World Sci. Publ. Teaneck, NJ), 727-749
[25] Sabatier, P. C.; Dolveck-Guilpard, B., On modeling discontinuous media. One-dimensional approximations, J. Math. Phys., 29, 861-868 (1988) · Zbl 0709.34024
[26] Aktosun, T.; Klaus, M.; van der Mee, C., Inverse wave scattering with discontinuous wave speed, J. Math. Phys., 36, 2880-2928 (1995) · Zbl 0843.34080
[27] Aktosun, T.; Klaus, M.; van der Mee, C., Recovery of discontinuities in a non-homogeneous medium, Inverse Problems, 12, 1-25 (1996) · Zbl 0848.34070
[28] Grinberg, N., Inverse scattering problem for an elastic layered medium, Inverse Problems, 7, 567-576 (1991) · Zbl 0738.73020
[29] Gladwell, G. M.L., Inverse Problems in Scattering: An Introduction (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0796.35170
[30] Sacks, P., Reconstruction of steplike potentials, Wave Motion, 18, 1, 21-30 (1993) · Zbl 0803.34075
[31] S. Albeverio, R. Hryniv, Ya. Mykytyuk, Inverse scattering for impedance Schrödinger operators, I. Step-like impedance lattice (submitted).; S. Albeverio, R. Hryniv, Ya. Mykytyuk, Inverse scattering for impedance Schrödinger operators, I. Step-like impedance lattice (submitted). · Zbl 1382.34094
[32] Besicovitch, A. S., Almost Periodic Functions (1955), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 0065.07102
[33] Katznelson, Y., (An Introduction to Harmonic Analysis. An Introduction to Harmonic Analysis, Cambridge Mathematical Library (2004), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 1055.43001
[34] Cameron, R. H., Analytic functions of absolutely convergent generalized trigonometric sums, Duke Math. J., 3, 4, 682-688 (1937) · Zbl 0018.21001
[35] A. Beurling, Sur les intégrals de Fourier absolument convergentes et leur application à une transformation fonctionnelle, in: Neuvieme Congrès des Mathématique Scandinaves, 1938.; A. Beurling, Sur les intégrals de Fourier absolument convergentes et leur application à une transformation fonctionnelle, in: Neuvieme Congrès des Mathématique Scandinaves, 1938. · JFM 65.0483.02
[36] Shestakov, A. I., The inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients, Sibirsk. Mat. Zh., 44, 5, 1142-1162 (2003), (in Russian); Engl. transl.: Sib. Math. J. 44 (5) (2003) 891-907 · Zbl 1042.34026
[37] Nazarchuk, Z. T.; Synyavskyy, A. T., Determination of layered structure parameters by a scattering matrix recovered from known reflection coefficients, Radiofizika i Radioastronomia, 15, 3, 295-313 (2010), (in Ukrainian); Engl. transl.: Radio Physics and Radio Astronomy 2 (1) (2011) 47-62
[38] Fejér, L., Über trigonometrische Polynome, J. Reine Angew. Math., 146, 53-82 (1916) · JFM 45.0406.02
[39] Burge, R. E.; Fiddy, M. A.; Greenaway, A. H.; Ross, G., The phase problem, Proc. R. Soc. Lond. Ser. A, 350, 1661, 191-212 (1976) · Zbl 0339.44004
[40] Tikhonravov, A. V.; Klibanov, M. V.; Zuev, I. V., Numerical study of the phaseless inverse scattering problem in thin-film optics, Inverse Problems, 11, 1, 251-270 (1995) · Zbl 0827.35141
[41] Conway, J., Functions of One Complex Variable (1978), Springer-Verlag: Springer-Verlag New York, Heidelberg, Berlin
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.