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TMF, 1992, Volume 93, Number 2, Pages 181–210 (Mi tmf1522)  

This article is cited in 18 scientific papers (total in 18 papers)

Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schrödinger problem and KPI equation

M. Boitia, F. Pempinellia, A. K. Pogrebkovb, M. K. Polivanov

a Lecce University
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The resolvent operator of the Linear Problem is determined as full Green function continued in the complex domain in two variables. An analog of the known Hilbert Identity is derived. We demonstrate the role of this identity in the study of two-dimensional scattering. Considering the Nonstationary Schrödinger Equation as an example we show that all types of solutions of the Linear Problem as well as Spectral Data known in the literature are given as specific values of this unique function — resolvent. New form of Inverse Problem is formulated.

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English version:
Theoretical and Mathematical Physics, 1992, 93:2, 1200–1224

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Received: 28.09.1992

Citation: M. Boiti, F. Pempinelli, A. K. Pogrebkov, M. K. Polivanov, “Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schrödinger problem and KPI equation”, TMF, 93:2 (1992), 181–210; Theoret. and Math. Phys., 93:2 (1992), 1200–1224

Citation in format AMSBIB
\by M.~Boiti, F.~Pempinelli, A.~K.~Pogrebkov, M.~K.~Polivanov
\paper Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schr\"odinger problem and KPI equation
\jour TMF
\yr 1992
\vol 93
\issue 2
\pages 181--210
\jour Theoret. and Math. Phys.
\yr 1992
\vol 93
\issue 2
\pages 1200--1224

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    This publication is cited in the following articles:
    1. Theoret. and Math. Phys., 99:2 (1994), 511–522  mathnet  crossref  mathscinet  zmath  isi
    2. T. I. Garagash, A. K. Pogrebkov, “Scattering problem for the differential operator $\partial_x\partial_y+1+a(x,y)\partial_y+ b(x,y)$”, Theoret. and Math. Phys., 102:2 (1995), 117–132  mathnet  crossref  mathscinet  zmath  isi
    3. A. K. Pogrebkov, T. I. Garagash, “On a solution of the Cauchy problem for the Boiti–Leon–Pempinelli equation”, Theoret. and Math. Phys., 109:2 (1996), 1369–1378  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. A. K. Pogrebkov, M. C. Prati, “An Ablowitz–Ladik system with a discrete potential: I. Extended resolvent”, Theoret. and Math. Phys., 119:1 (1999), 407–419  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Bäcklund and Darboux Transformations for the Nonstationary Schrödinger Equation”, Proc. Steklov Inst. Math., 226 (1999), 42–62  mathnet  mathscinet  zmath
    6. Prinari, B, “On some nondecaying potentials and related Jost solutions for the heat conduction equation”, Inverse Problems, 16:3 (2000), 589  crossref  mathscinet  zmath  adsnasa  isi
    7. Pelinovsky, DE, “Eigenfunctions and eigenvalues for a scalar Riemann–Hilbert problem associated to inverse scattering”, Communications in Mathematical Physics, 208:3 (2000), 713  crossref  mathscinet  zmath  adsnasa  isi
    8. Boiti, M, “Towards an inverse scattering theory for non-decaying potentials of the heat equation”, Inverse Problems, 17:4 (2001), 937  crossref  mathscinet  zmath  adsnasa  isi
    9. Boiti, M, “Extended resolvent and inverse scattering with an application to KPI”, Journal of Mathematical Physics, 44:8 (2003), 3309  crossref  mathscinet  zmath  adsnasa  isi
    10. M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Spectral Theory of the Nonstationary Schrodinger Equation with a Two-Dimensionally Perturbed Arbitrary One-Dimensional Potential”, Theoret. and Math. Phys., 144:2 (2005), 1100–1116  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Spectral Theory of the Nonstationary Schrödinger Equation with a Bidimensionally Perturbed One-Dimensional Potential”, Proc. Steklov Inst. Math., 251 (2005), 6–48  mathnet  mathscinet  zmath
    12. Boiti, M, “Scattering transform for nonstationary Schrodinger equation with bidimensionally perturbed N-soliton potential”, Journal of Mathematical Physics, 47:12 (2006), 123510  crossref  mathscinet  zmath  adsnasa  isi
    13. Boiti, M, “On the extended resolvent of the nonstationary Schrodinger operator for a Darboux transformed potential”, Journal of Physics A-Mathematical and General, 39:8 (2006), 1877  crossref  mathscinet  zmath  adsnasa  isi
    14. A. K. Pogrebkov, “Commutator identities on associative algebras and the integrability of nonlinear evolution equations”, Theoret. and Math. Phys., 154:3 (2008), 405–417  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    15. M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Building an extended resolvent of the heat operator via twisting transformations”, Theoret. and Math. Phys., 159:3 (2009), 721–733  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    16. St. Petersburg Math. J., 22:3 (2011), 473–483  mathnet  crossref  mathscinet  zmath  isi
    17. A. K. Pogrebkov, “Hirota difference equation: Inverse scattering transform, Darboux transformation, and solitons”, Theoret. and Math. Phys., 181:3 (2014), 1585–1598  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    18. A. K. Pogrebkov, “Commutator identities on associative algebras, the non-Abelian Hirota difference equation and its reductions”, Theoret. and Math. Phys., 187:3 (2016), 823–834  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
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