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On Two-Dimensional Finite-Gap Potential Schroedinger and Dirac Operators with Singular Spectral Curves

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Abstract

We describe a wide class of two-dimensional potential Schroedinger and Dirac operators which are finite-gap at the zero energy level and whose spectral curves at this level are singular, in particular may have n-multiple points with n≥3.

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References

  1. Taimanov I. A., “Modified Novikov-Veselov equation and differential geometry of surfaces,” Transl. Amer. Math. Soc. Ser. 2, 179, 133–151 (1997).

    Google Scholar 

  2. Ta?manov I. A., “The Weierstrass representation of closed surfaces in R3,” Funct. Anal. Appl., 32, No. 4, 258–267 (1998).

    Google Scholar 

  3. Dubrovin B. A., Krichever I. M., and Novikov S. P., “The Schrödinger equation in a periodic magnetic field and Riemann surfaces,” Soviet Math. Dokl., 17, 947–951 (1976).

    Google Scholar 

  4. Novikov S. P., “Two-dimensional Schrödinger operators in periodic fields,” J. Soviet Math., 28, 1–20 (1985).

    Google Scholar 

  5. Kuchment P., Floquet Theory for Partial Differential Equations, Birkhäuser, Basel (1993).

    Google Scholar 

  6. Veselov A. P. and Novikov S. P., “Finite-gap two-dimensional Schrödinger operators. Potential operators,” Soviet Math. Dokl., 30, 705–708 (1984).

    Google Scholar 

  7. Veselov A. P. and Novikov S. P., “Finite-gap potential two-dimensional Schrödinger operators. Explicit formulas and evolution equations,” Soviet Math. Dokl., 30, 588–591 (1984).

    Google Scholar 

  8. Ta?manov I. A., “Finite-gap solutions of the modified Novikov-Veselov equations: their spectral properties and applications,” Siberian Math. J., 40, No. 6, 1146–1156 (1999).

    Google Scholar 

  9. Serre J.-P., Algebraic Groups and Class Fields, Springer-Verlag, New York (1988). (Graduate Texts in Math., 117.)

    Google Scholar 

  10. Krichever I. M., “Spectral theory of two-dimensional periodic Schrödinger operators and its applications,” Russian Math. Surveys, 44, No. 2, 145–225 (1989).

    Google Scholar 

  11. Dubrovin B. A., Matveev V. B., and Novikov S. P., “Nonlinear equations of Korteweg-de Vries type, finite-gap linear operators, and abelian varieties,” Russian Math. Surveys, 31, No. 1, 59–146 (1976).

    Google Scholar 

  12. Krichever I. M., “Potentials with the zero coefficient of reflection on the background of finite-gap potentials,” Funct. Anal. Appl., 9, No. 2, 161–163 (1975).

    Google Scholar 

  13. Dubrovin B. A., Malanyuk T. M., Krichever I. M., and Makhankov V. G., “Exact solutions of the time-dependent Schrödinger equation with selfconsistent potentials,” Soviet J. Particles and Nuclei, 19, No. 3, 252–269 (1988).

    Google Scholar 

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Taimanov, I.A. On Two-Dimensional Finite-Gap Potential Schroedinger and Dirac Operators with Singular Spectral Curves. Siberian Mathematical Journal 44, 686–694 (2003). https://doi.org/10.1023/A:1024792708878

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