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Uspekhi Mat. Nauk, 1989, Volume 44, Issue 2(266), Pages 121–184 (Mi umn2275)  

This article is cited in 71 scientific papers (total in 72 papers)

Spectral theory of two-dimensional periodic operators and its applications

I. M. Krichever

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English version:
Russian Mathematical Surveys, 1989, 44:2, 145–225

Bibliographic databases:

UDC: 517.93
MSC: 35Pxx, 35J10, 47A55, 37K10
Received: 13.07.1988

Citation: I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications”, Uspekhi Mat. Nauk, 44:2(266) (1989), 121–184; Russian Math. Surveys, 44:2 (1989), 145–225

Citation in format AMSBIB
\by I.~M.~Krichever
\paper Spectral theory of two-dimensional periodic operators and its applications
\jour Uspekhi Mat. Nauk
\yr 1989
\vol 44
\issue 2(266)
\pages 121--184
\jour Russian Math. Surveys
\yr 1989
\vol 44
\issue 2
\pages 145--225

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    2. P. G. Grinevich, “Rapidly decreasing potentials on a background of finite-zone potentials and the $\partial$-problem on Riemann spaces”, Funct. Anal. Appl., 23:4 (1989), 321–322  mathnet  crossref  mathscinet  zmath  isi
    3. 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
    4. P. G. Grinevich, A. Yu. Orlov, “Variations of the complex structure of Riemann surfaces by vector fields on a contour and objects of the KP theory. The Krichever–Novikov problem of the action on the Baker–Akhieser functions”, Funct. Anal. Appl., 24:1 (1991), 61–63  mathnet  crossref  mathscinet  zmath  isi
    5. I. A. Taimanov, “Two-dimensional finite-zone Schrodinger potential operators”, Funct. Anal. Appl., 24:1 (1990), 76–77  mathnet  crossref  mathscinet  zmath  isi
    6. F. Fucito, M. Martellini, A. Gamba, “Non-perturbative two-dimensional quantum gravity, KdV hierarchy and solutions of the string equations”, Physics Letters B, 248:1-2 (1990), 57  crossref
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    8. B Dubrovin, “Integrable systems in topological field theory”, Nuclear Physics B, 379:3 (1992), 627  crossref
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