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Dokl. Akad. Nauk SSSR, 1984, Volume 279, Number 4, Pages 784–788 (Mi dan9334)  

This article is cited in 47 scientific papers (total in 48 papers)


Finite-gap two-dimensional Schrödinger operators. Potential operators

A. P. Veselov, S. P. Novikov

Landau Institute for Theoretical Physics, USSR Academy of Sciences, Chernogolovka Moscow region

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UDC: 517.957+512.7
Received: 05.03.1984

Citation: A. P. Veselov, S. P. Novikov, “Finite-gap two-dimensional Schrödinger operators. Potential operators”, Dokl. Akad. Nauk SSSR, 279:4 (1984), 784–788

Citation in format AMSBIB
\by A.~P.~Veselov, S.~P.~Novikov
\paper Finite-gap two-dimensional Schr\"odinger operators. Potential operators
\jour Dokl. Akad. Nauk SSSR
\yr 1984
\vol 279
\issue 4
\pages 784--788

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    This publication is cited in the following articles:
    1. P. G. Grinevich, R. G. Novikov, “Analogs of multisoliton potentials for the two-dimensional Schrödinger operator”, Funct. Anal. Appl., 19:4 (1985), 276–285  mathnet  crossref  mathscinet  zmath  isi
    2. P. G. Grinevich, S. V. Manakov, “Inverse scattering problem for the two-dimensional Schrödinger operator, the $\bar\partial$-method and nonlinear equations”, Funct. Anal. Appl., 20:2 (1986), 94–103  mathnet  crossref  mathscinet  zmath
    3. R. G. Novikov, G. M. Henkin, “The $\bar\partial$-equation in the multidimensional inverse scattering problem”, Russian Math. Surveys, 42:3 (1987), 109–180  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. L. V. Bogdanov, “Veselov–Novikov equation as a natural two-dimensional generalization of the Korteweg–de Vries equation”, Theoret. and Math. Phys., 70:2 (1987), 219–223  mathnet  crossref  mathscinet  zmath  isi
    5. P. G. Grinevich, S. P. Novikov, “Two-dimensional “inverse scattering problem” for negative energies and generalized-analytic functions. I. Energies below the ground state”, Funct. Anal. Appl., 22:1 (1988), 19–27  mathnet  crossref  mathscinet  zmath  isi
    6. S. M. Natanzon, “Nonsingular finite-zone two-dimensional Schrödinger operators and prymians of real curves”, Funct. Anal. Appl., 22:1 (1988), 68–70  mathnet  crossref  mathscinet  zmath  isi
    7. R. G. Novikov, “Multidimensional inverse spectral problem for the equation $-\Delta\psi+(v(x)-Eu(x))\psi=0$”, Funct. Anal. Appl., 22:4 (1988), 263–272  mathnet  crossref  mathscinet  zmath  isi
    8. I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications”, Russian Math. Surveys, 44:2 (1989), 145–225  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. S. M. Natanzon, “Prymians of real curves and their applications to the effectivization of Schrödinger operators”, Funct. Anal. Appl., 23:1 (1989), 33–45  mathnet  crossref  mathscinet  zmath  isi
    10. I. A. Taimanov, “Two-dimensional finite-zone Schrodinger potential operators”, Funct. Anal. Appl., 24:1 (1990), 76–77  mathnet  crossref  mathscinet  zmath  isi
    11. S. M. Natanzon, “Klein surfaces”, Russian Math. Surveys, 45:5 (1990), 53–108  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. I. A. Taimanov, “Prym varieties of branched coverings and nonlinear equations”, Math. USSR-Sb., 70:2 (1991), 367–384  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    13. I. Yu. Cherdantsev, R. A. Sharipov, “Finite-gap solutions of the Bullough–Dodd–Zhiber–Shabat equation”, Theoret. and Math. Phys., 82:1 (1990), 108–11  mathnet  crossref  mathscinet  zmath  isi
    14. S. M. Natanzon, “Differential equations on the Prym theta function. a realness criterion for two-dimensional, finite-zone, potential Schrödinger operators”, Funct. Anal. Appl., 26:1 (1992), 13–20  mathnet  crossref  mathscinet  zmath  isi
    15. I. M. Krichever, “Two-Dimensional Algebraic-Geometric Operators with Self-Consistent Potentials”, Funct. Anal. Appl., 28:1 (1994), 21–32  mathnet  crossref  mathscinet  zmath  isi
    16. V. M. Buchstaber, V. Z. Ènol'skii, “Abelian Bloch solutions of the two-dimensional Schrödinger equation”, Russian Math. Surveys, 50:1 (1995), 195–197  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    17. V. I. Vakulenko, “Solution of Virasoro constraints for DKP hierarhy”, Theoret. and Math. Phys., 107:1 (1996), 435–440  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. I. A. Taimanov, “Secants of Abelian varieties, theta functions, and soliton equations”, Russian Math. Surveys, 52:1 (1997), 147–218  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    19. S. P. Novikov, I. A. Dynnikov, “Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds”, Russian Math. Surveys, 52:5 (1997), 1057–1116  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    20. I. A. Taimanov, “The Weierstrass Representation of Closed Surfaces in $\mathbb{R}^3$”, Funct. Anal. Appl., 32:4 (1998), 258–267  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    21. R. G. Novikov, “Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2”, Proc. Steklov Inst. Math., 225 (1999), 285–302  mathnet  mathscinet  zmath
    22. P. G. Grinevich, “Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity”, Russian Math. Surveys, 55:6 (2000), 1015–1083  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    23. A. A. Oblomkov, “Difference Operators on Two-Dimensional Regular Lattices”, Theoret. and Math. Phys., 127:1 (2001), 435–445  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    24. A. A. Oblomkov, “Isoenergy Spectral Problem for Multidimensional Difference Operators”, Funct. Anal. Appl., 36:2 (2002), 120–133  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    25. A. A. Oblomkov, “Spectral properties of two classes of periodic difference operators”, Sb. Math., 193:4 (2002), 559–584  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    26. I. A. Taimanov, “On two-dimensional finite-gap potential Schrödinger and Dirac operators with singular spectral curves”, Siberian Math. J., 44:4 (2003), 686–694  mathnet  crossref  mathscinet  zmath  isi  elib
    27. P. R. Gordoa, “Algebraic and Differential Nonlinear Superposition Formulas”, Theoret. and Math. Phys., 137:1 (2003), 1430–1438  mathnet  crossref  crossref  mathscinet  zmath  isi
    28. I. A. Taimanov, “Two-dimensional Dirac operator and the theory of surfaces”, Russian Math. Surveys, 61:1 (2006), 79–159  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    29. A. E. Mironov, “Spectral Data for Hamiltonian-Minimal Lagrangian Tori in $\mathbb C\mathrm P^2$”, Proc. Steklov Inst. Math., 263 (2008), 112–126  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    30. Wang Hong-Yan, “The Nizhnik–Veselov–Novikov equation with self-consistent sources”, Theoret. and Math. Phys., 157:1 (2008), 1474–1483  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    31. I. A. Taimanov, S. P. Tsarev, “On the Moutard transformation and its applications to spectral theory and soliton equations”, Journal of Mathematical Sciences, 170:3 (2010), 371–387  mathnet  crossref  mathscinet
    32. P. G. Grinevich, A. E. Mironov, S. P. Novikov, “2D-Schrödinger Operator, (2+1) evolution systems and new reductions, 2D-Burgers hierarchy and inverse problem data”, Russian Math. Surveys, 65:3 (2010), 580–582  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    33. P. G. Grinevich, A. E. Mironov, S. P. Novikov, “Zero level of a purely magnetic two-dimensional nonrelativistic Pauli operator for spin-$1/2$ particles”, Theoret. and Math. Phys., 164:3 (2010), 1110–1127  mathnet  crossref  crossref  adsnasa  isi
    34. V. G. Dubrovsky, A. V. Topovsky, M. Yu. Basalaev, “New exact solutions of two-dimensional integrable equations using the $\bar\partial$-dressing method”, Theoret. and Math. Phys., 167:3 (2011), 725–739  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    35. A. V. Kazeykina, “Absence of Conductivity-Type Solitons for the Novikov–Veselov Equation at Zero Energy”, Funct. Anal. Appl., 47:1 (2013), 64–66  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    36. B. O. Vasilevskiǐ, “The Green's function of a five-point discretization of a two-dimensional finite-gap Schrödinger operator: The case of four singular points on the spectral curve”, Siberian Math. J., 54:6 (2013), 994–1004  mathnet  crossref  mathscinet  isi
    37. A. V. Kazeykina, “Absence of Solitons with Sufficient Algebraic Localization for the Novikov–Veselov Equation at Nonzero Energy”, Funct. Anal. Appl., 48:1 (2014), 24–35  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    38. B. T. Saparbaeva, “Two-Dimensional Finite-Gap Schrödinger Operators with Elliptic Coefficients”, Math. Notes, 747–749  mathnet  crossref  crossref  mathscinet  elib
    39. I. A. Taimanov, “Blowing up solutions of the modified Novikov–Veselov equation and minimal surfaces”, Theoret. and Math. Phys., 182:2 (2015), 173–181  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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    41. B. O. Vasilevskii, “The Green Function of the Discrete Finite-Gap One-Energy Two-Dimensional Schrödinger Operator on the Quad Graph”, Math. Notes, 98:1 (2015), 38–52  mathnet  crossref  crossref  mathscinet  isi  elib
    42. B. O. Vasilevskii, “A Sufficient Nonsingularity Condition for a Discrete Finite-Gap One-Energy Two-Dimensional Schrödinger Operator on the Quad-Graph”, Funct. Anal. Appl., 49:3 (2015), 210–213  mathnet  crossref  crossref  isi  elib
    43. A. G. Kudryavtsev, “Nonlocal Darboux transformation of the two-dimensional stationary Schrödinger equation and its relation to the Moutard transformation”, Theoret. and Math. Phys., 187:1 (2016), 455–462  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    44. V. E. Adler, Yu. Yu. Berest, V. M. Buchstaber, P. G. Grinevich, B. A. Dubrovin, I. M. Krichever, S. P. Novikov, A. N. Sergeev, M. V. Feigin, J. Felder, E. V. Ferapontov, O. A. Chalykh, P. I. Etingof, “Alexander Petrovich Veselov (on his 60th birthday)”, Russian Math. Surveys, 71:6 (2016), 1159–1176  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    45. P. G. Grinevich, S. P. Novikov, “Singular solitons and spectral meromorphy”, Russian Math. Surveys, 72:6 (2017), 1083–1107  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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