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

This article is cited in 37 scientific papers (total in 37 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.

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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}


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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  mathscinet  zmath
    2. Gasymov M., Levitan B., “On the Expansion on Products of Special Solutions of 2 Sturm-Liouville Equations”, 310, no. 4, 1990, 781–784  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath  isi
    4. E. L. Korotyaev, N. E. Firsova, “Diffusion in layered media at large time”, Theoret. and Math. Phys., 98:1 (1994), 72–99  mathnet  crossref  mathscinet  zmath  isi
    5. Korotyaev E., “The Propagation of Waves in One-Dimensional Periodic Media”, Dokl. Akad. Nauk, 336:2 (1994), 171–174  mathnet  mathscinet  zmath  isi
    6. Kargaev P., Korotyaev E., “Effective Masses for Hill Operator and Conformal-Maps”, Dokl. Akad. Nauk, 336:3 (1994), 312–315  mathnet  mathscinet  zmath  isi
    7. F. Gesztesy, H. Holden, B. Simon, “Absolute summability of the trace relation for certain Schrödinger operators”, Comm Math Phys, 168:1 (1995), 137  crossref  mathscinet  zmath  adsnasa
    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  crossref  mathscinet  zmath  adsnasa  isi
    9. Boris N Zakhariev, Vladimir M Chabanov, Inverse Probl, 13:6 (1997), R47  crossref  mathscinet  isi
    10. Korotyaev E., “The Propagation of the Waves in Periodic Media at Large Time”, Asymptotic Anal., 15:1 (1997), 1–24  mathscinet  zmath  isi
    11. Korotyaev E., “The Estimates of Periodic Potentials in Terms of Effective Masses”, Commun. Math. Phys., 183:2 (1997), 383–400  crossref  mathscinet  zmath  adsnasa  isi
    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  mathscinet  zmath  adsnasa  isi
    13. Fedotov A., Klopp F., “Geometric Tools of the Adiabatic Complex Wkb Method”, Asymptotic Anal., 39:3-4 (2004), 309–357  mathscinet  zmath  isi
    14. de Monvel A., Egorova I., “Transformation Operator for Jacobi Matrices with Asymptotically Periodic Coefficients”, J. Differ. Equ. Appl., 10:8 (2004), 711–727  crossref  mathscinet  zmath  isi
    15. Ag. Kh. Khanmamedov, “Direct and inverse scattering problems for the perturbed Hill difference equation”, Sb. Math., 196:10 (2005), 1529–1552  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. Scipio Cuccagna, “Stability of standing waves for NLS with perturbed Lamé potential”, Journal of Differential Equations, 223:1 (2006), 112  crossref
    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  crossref  zmath  isi
    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  isi
    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  crossref  mathscinet  zmath  isi
    20. Ag. Kh. Khanmamedov, “The Inverse Scattering Problem for a Perturbed Difference Hill Equation”, Math. Notes, 85:3 (2009), 441–452  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  mathscinet  zmath  elib
    22. Katrin Grunert, “The transformation operator for Schrödinger operators on almost periodic infinite-gap backgrounds”, Journal of Differential Equations, 250:8 (2011), 3534  crossref
    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  crossref  mathscinet  isi  elib
    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  mathscinet  isi
    25. Korotyaev E., “Resonance Theory for Perturbed Hill Operator”, Asymptotic Anal., 74:3-4 (2011), 199–227  crossref  mathscinet  zmath  isi  elib
    26. Korotyaev E.L., “Sharp Asymptotics of the Quasimomentum”, Asymptotic Anal., 80:3-4 (2012), 269–287  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    28. Katrin Grunert, “Scattering theory for Schrödinger operators on steplike, almost periodic infinite-gap backgrounds”, Journal of Differential Equations, 254:6 (2013), 2556  crossref
    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  mathnet  crossref  mathscinet
    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  mathnet  crossref  crossref  mathscinet  isi  elib
    31. A. R. Bikmetov, I. Kh. Khusnullin, “Perturbation of Hill operator by narrow potentials”, Russian Math. (Iz. VUZ), 61:7 (2017), 1–10  mathnet  crossref  isi
    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  mathnet  crossref  crossref  mathscinet  isi  elib
    33. Pfirsch B., Sobolev A.V., “Formulas of Szego Type For the Periodic Schrodinger Operator”, Commun. Math. Phys., 358:2 (2018), 675–704  crossref  isi
    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  mathnet  crossref  crossref  adsnasa  isi  elib
    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  mathnet  crossref  crossref  isi  elib
    36. A. Kh. Khanmamedov, M. G. Makhmudova, “Ob operatore preobrazovaniya dlya uravneniya Shredingera s dopolnitelnym lineinym potentsialom”, Funkts. analiz i ego pril., 54:1 (2020), 93–96  mathnet  crossref
    37. A. Kh. Khanmamedov, M. G. Makhmudova, “Inverse spectral problem for the Schrödinger equation with an additional linear potential”, Theoret. and Math. Phys., 202:1 (2020), 58–71  mathnet  crossref
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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