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Faddeev eigenfunctions for two-dimensional Schrödinger operators via the Moutard transformation

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Abstract

We demonstrate how the Moutard transformation of two-dimensional Schrödinger operators acts on the Faddeev eigenfunctions on the zero-energy level and present some explicitly computed examples of such eigenfunctions for smooth rapidly decaying potentials of operators with a nontrivial kernel and for deformed potentials corresponding to blowup solutions of the Novikov-Veselov equation.

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References

  1. I. A. Taimanov and S. P. Tsarev, Theor. Math. Phys., 157, 1525–1541 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  2. L. D. Faddeev, Sov. Phys. Dokl., 10, 1033–1035 (1965).

    ADS  Google Scholar 

  3. L. D. Faddeev, J. Soviet Math., 5, 334–396 (1976).

    Article  MATH  Google Scholar 

  4. R. G. Novikov and G. M. Henkin, Russ. Math. Surveys, 42, 109–180 (1987).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. P. G. Grinevich and R. G. Novikov, Phys. Lett. A, 376, 1102–1106 (2012); arXiv:1110.3157v1 [math-ph] (2011).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. P. G. Grinevich and R. G. Novikov, “Faddeev eigenfunctions for multipoint potentials,” arXiv:1211.0292v1 [math-ph] (2012).

    Google Scholar 

  7. P. G. Grinevich, Theor. Math. Phys., 69, 1170–1172 (1986).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. P. G. Grinevich and S. P. Novikov, Funct. Anal. Appl., 22, 19–27 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  9. I. N. Vekua, Generalized Analytic Functions [in Russian], Nauka, Moscow (1959); English transl., Pergamon, London (1962).

    MATH  Google Scholar 

  10. R. G. Novikov, J. Funct. Anal., 103, 409–463 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Boiti, J. J. P. Leon, M. Manna, and F. Pempinelli, Inverse Problems, 3, 25–36 (1987).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. H.-C. Hu, S.-Y. Lou, and Q.-P. Liu, Chinese Phys. Lett., 20, 1413–1415 (2003).

    Article  ADS  Google Scholar 

  13. A. P. Veselov and S. P. Novikov, Soviet Math. Dokl., 30, 588–591 (1984).

    MATH  Google Scholar 

  14. P. A. Perry, “Miura maps and inverse scattering for the Novikov-Veselov equation,” arXiv:1201.2385v2 [math.AP] (2012).

    Google Scholar 

  15. M. Music, P. A. Perry, and S. Siltanen, “Exceptional circles of radial potentials,” arXiv:1211.3520v1 [math.AP] (2012).

    Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    I. A. Taimanov

  2. Siberian Federal University, Krasnoyarsk, Russia

    S. P. Tsarev

Authors
  1. I. A. Taimanov
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  2. S. P. Tsarev
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Correspondence to I. A. Taimanov.

Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 176, No. 3, pp. 408–416, September 2013.

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Taimanov, I.A., Tsarev, S.P. Faddeev eigenfunctions for two-dimensional Schrödinger operators via the Moutard transformation. Theor Math Phys 176, 1176–1183 (2013). https://doi.org/10.1007/s11232-013-0098-x

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  • DOI: https://doi.org/10.1007/s11232-013-0098-x

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