RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Personal entry: Login: Password: Save password Enter Forgotten password? Register

 Mat. Sb. (N.S.), 1986, Volume 130(172), Number 3(7), Pages 349–385 (Mi msb1880)

This article is cited in 35 scientific papers (total in 35 papers)

The direct and inverse scattering problems for the one-dimensional perturbed Hill operator

N. E. Firsova

Abstract: The problem of scattering by a one-dimensional periodic lattice $p(x)$ with impurity potential $q(x)$ is considered. A stationary scattering matrix is constructed on the basis of the asymptotics of the scattered waves, its properties are studied, and it is shown to coincide with the nonstationary scattering operator defined in the usual way in the quasimomentum representation of the unperturbed operator $H_0$. The inverse scattering problem is also solved, i.e., the problem of recovering $q(x)$ on the basis of $p(x)$ and the scattering data. Here we follow the scheme going back to the well-known work of V. A. Marchenko and L. D. Faddeev. However, solution of the inverse problem in the presence of a periodic lattice required considerable modification of classical arguments. The theory of so-called “global” quasimomentum serves as analytic basis. Conditions on the scattering data are found which are necessary with a finite second moment and sufficient in order that there exist a unique impurity potential with given scattering characteristics and a finite first moment.
Bibliography: 28 titles.

Full text: PDF file (1894 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1987, 58:2, 351–388

Bibliographic databases:

UDC: 517.9+517.4
MSC: 34B25, 34B30
Received: 31.05.1984 and 12.09.1985

Citation: N. E. Firsova, “The direct and inverse scattering problems for the one-dimensional perturbed Hill operator”, Mat. Sb. (N.S.), 130(172):3(7) (1986), 349–385; Math. USSR-Sb., 58:2 (1987), 351–388

Citation in format AMSBIB
\Bibitem{Fir86} \by N.~E.~Firsova \paper The direct and inverse scattering problems for the one-dimensional perturbed Hill operator \jour Mat. Sb. (N.S.) \yr 1986 \vol 130(172) \issue 3(7) \pages 349--385 \mathnet{http://mi.mathnet.ru/msb1880} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=865766} \zmath{https://zbmath.org/?q=an:0627.34028|0616.34017} \transl \jour Math. USSR-Sb. \yr 1987 \vol 58 \issue 2 \pages 351--388 \crossref{https://doi.org/10.1070/SM1987v058n02ABEH003108} 

Linking options:
• http://mi.mathnet.ru/eng/msb1880
• http://mi.mathnet.ru/eng/msb/v172/i3/p349

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. E. Firsova, “On solution of the Cauchy problem for the Korteweg–de Vries equation with initial data the sum of a periodic and a rapidly decreasing function”, Math. USSR-Sb., 63:1 (1989), 257–265
2. Gasymov M., Levitan B., “On the Expansion on Products of Special Solutions of 2 Sturm-Liouville Equations”, 310, no. 4, 1990, 781–784
3. B. M. Levitan, A. B. Khasanov, “Estimation of the Cauchy function for finite-zone nonperiodic potentials”, Funct. Anal. Appl., 26:2 (1992), 91–98
4. E. L. Korotyaev, N. E. Firsova, “Diffusion in layered media at large time”, Theoret. and Math. Phys., 98:1 (1994), 72–99
5. Korotyaev E., “The Propagation of Waves in One-Dimensional Periodic Media”, Dokl. Akad. Nauk, 336:2 (1994), 171–174
6. Kargaev P., Korotyaev E., “Effective Masses for Hill Operator and Conformal-Maps”, Dokl. Akad. Nauk, 336:3 (1994), 312–315
7. F. Gesztesy, H. Holden, B. Simon, “Absolute summability of the trace relation for certain Schrödinger operators”, Comm Math Phys, 168:1 (1995), 137
8. Firsova N., “On the Time Decay of a Wave Packet in a One-Dimensional Finite Band Periodic Lattice”, J. Math. Phys., 37:3 (1996), 1171–1181
9. Boris N Zakhariev, Vladimir M Chabanov, Inverse Probl, 13:6 (1997), R47
10. Korotyaev E., “The Propagation of the Waves in Periodic Media at Large Time”, Asymptotic Anal., 15:1 (1997), 1–24
11. Korotyaev E., “The Estimates of Periodic Potentials in Terms of Effective Masses”, Commun. Math. Phys., 183:2 (1997), 383–400
12. Firsova N., “On the Global Quasimomentum in Solid State Physics”, Proceedings of the 1999 Londrina Winter School Mathematical Methods in Physics, eds. Bytsenko A., Williams F., World Scientific Publ Co Pte Ltd, 2000, 98–141
13. Fedotov A., Klopp F., “Geometric Tools of the Adiabatic Complex Wkb Method”, Asymptotic Anal., 39:3-4 (2004), 309–357
14. de Monvel A., Egorova I., “Transformation Operator for Jacobi Matrices with Asymptotically Periodic Coefficients”, J. Differ. Equ. Appl., 10:8 (2004), 711–727
15. Ag. Kh. Khanmamedov, “Direct and inverse scattering problems for the perturbed Hill difference equation”, Sb. Math., 196:10 (2005), 1529–1552
16. Scipio Cuccagna, “Stability of standing waves for NLS with perturbed Lamé potential”, Journal of Differential Equations, 223:1 (2006), 112
17. Khanmamedov A.Kh., “The Inverse Scattering Problem for the Difference Schrodinger Operator with Asymptotically Periodic Coefficients on the Half-Axis”, Dokl. Math., 74:1 (2006), 548–551
18. Bazargan J., “Direct and Inverse Scattering on the Line for One-Dimensional Schrodinger Equation”, Probl. At. Sci. Tech., 2007, no. 3, Part 2, 245–248
19. de Monvel A.B., Egorova I., Teschl G., “Inverse Scattering Theory for One-Dimensional Schrodinger Operators with Steplike Finite-Gap Potentials”, J. Anal. Math., 106 (2008), 271–316
20. Ag. Kh. Khanmamedov, “The Inverse Scattering Problem for a Perturbed Difference Hill Equation”, Math. Notes, 85:3 (2009), 441–452
21. I. Egorova, G. Teschl, “A Paley–Wiener theorem for periodic scattering with applications to the Korteweg–de Vries equation”, Zhurn. matem. fiz., anal., geom., 6:1 (2010), 21–33
22. Katrin Grunert, “The transformation operator for Schrödinger operators on almost periodic infinite-gap backgrounds”, Journal of Differential Equations, 250:8 (2011), 3534
23. Egorova I., Teschl G., “On the Cauchy Problem for the Kortewegde Vries Equation with Steplike Finite-Gap Initial Data II. Perturbations with Finite Moments”, J. Anal. Math., 115 (2011), 71–101
24. Mikikits-Leitner A., Teschl G., “Trace Formulas for Schrodinger Operators in Connection with Scattering Theory for Finite-Gap Backgrounds”, Spectral Theory and Analysis, Operator Theory Advances and Applications, 214, eds. Janas J., Kurasov P., Laptev A., Naboko S., Stolz G., Birkhauser Verlag Ag, 2011, 107–124
25. Korotyaev E., “Resonance Theory for Perturbed Hill Operator”, Asymptotic Anal., 74:3-4 (2011), 199–227
26. Korotyaev E.L., “Sharp Asymptotics of the Quasimomentum”, Asymptotic Anal., 80:3-4 (2012), 269–287
27. Korotyaev E.L., Schmidt K.M., “On the Resonances and Eigenvalues for a 1D Half-Crystal with Localised Impurity”, J. Reine Angew. Math., 670 (2012), 217–248
28. Katrin Grunert, “Scattering theory for Schrödinger operators on steplike, almost periodic infinite-gap backgrounds”, Journal of Differential Equations, 254:6 (2013), 2556
29. I. Egorova, Z. Gladka, T. L. Lange, G. Teschl, “Inverse Scattering Theory for Schrödinger Operators with Steplike Potentials”, Zhurn. matem. fiz., anal., geom., 11:2 (2015), 123–158
30. A. V. Vestyak, H. A. Matevossian, “Behavior of the Solution of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefficients”, Math. Notes, 100:5 (2016), 751–754
31. A. R. Bikmetov, I. Kh. Khusnullin, “Perturbation of Hill operator by narrow potentials”, Russian Math. (Iz. VUZ), 61:7 (2017), 1–10
32. A. V. Vestyak, H. A. Matevossian, “On the Behavior of the Solution of the Cauchy Problem for an Inhomogeneous Hyperbolic Equation with Periodic Coefficients”, Math. Notes, 102:3 (2017), 424–428
33. Pfirsch B., Sobolev A.V., “Formulas of Szego Type For the Periodic Schrodinger Operator”, Commun. Math. Phys., 358:2 (2018), 675–704
34. I. M. Guseinov, A. Kh. Khanmamedov, A. F. Mamedova, “Inverse scattering problem for the Schrödinger equation with an additional quadratic potential on the entire axis”, Theoret. and Math. Phys., 195:1 (2018), 538–547
35. H. A. Matevossian, A. V. Vestyak, O. N. Peshcherikova, “On the Behavior of Solutions of Initial Boundary-Value Problems for a Hyperbolic Equation with Periodic Coefficients”, Math. Notes, 104:5 (2018), 762–766
•  Number of views: This page: 415 Full text: 134 References: 31

 Contact us: math-net2020_01 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2020