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

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

Bibliographic databases:

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
\Bibitem{BoiPemPog92} \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 \mathnet{http://mi.mathnet.ru/tmf1522} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1233541} \zmath{https://zbmath.org/?q=an:0806.35131} \transl \jour Theoret. and Math. Phys. \yr 1992 \vol 93 \issue 2 \pages 1200--1224 \crossref{https://doi.org/10.1007/BF01083519} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1992LJ23200001} 

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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. Theoret. and Math. Phys., 99:2 (1994), 511–522
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
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
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
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
6. Prinari, B, “On some nondecaying potentials and related Jost solutions for the heat conduction equation”, Inverse Problems, 16:3 (2000), 589
7. Pelinovsky, DE, “Eigenfunctions and eigenvalues for a scalar Riemann–Hilbert problem associated to inverse scattering”, Communications in Mathematical Physics, 208:3 (2000), 713
8. Boiti, M, “Towards an inverse scattering theory for non-decaying potentials of the heat equation”, Inverse Problems, 17:4 (2001), 937
9. Boiti, M, “Extended resolvent and inverse scattering with an application to KPI”, Journal of Mathematical Physics, 44:8 (2003), 3309
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
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
12. Boiti, M, “Scattering transform for nonstationary Schrodinger equation with bidimensionally perturbed N-soliton potential”, Journal of Mathematical Physics, 47:12 (2006), 123510
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
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
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
16. St. Petersburg Math. J., 22:3 (2011), 473–483
17. A. K. Pogrebkov, “Hirota difference equation: Inverse scattering transform, Darboux transformation, and solitons”, Theoret. and Math. Phys., 181:3 (2014), 1585–1598
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
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