# zbMATH — the first resource for mathematics

Prymians of real curves and their applications to the effectivization of Schrödinger operators. (English. Russian original) Zbl 0682.35027
Funct. Anal. Appl. 23, No. 1, 33-45 (1989); translation from Funkts. Anal. Prilozh. 23, No. 1, 41-56 (1989).
A. P. Veselov and S. P. Novikov [Sov. Math., Dokl. 30, 588- 591 (1984); translation from Dokl. Akad. Nauk SSSR 279, 20-24 (1984; Zbl 0613.35020); and ibid., 705-708 (1984); translation from Dokl. Akad. Nauk SSSR 279, 784-788 (1984; Zbl 0602.35024)] describe the class of real two- dimensional Schrödinger operators $L=\partial {\bar \partial}+2\partial {\bar \partial} \ell n \theta (zU_ 1+\bar zU_ 2- e| V)-\epsilon_ 0,$ where $$\theta$$ denotes Prym’s theta-function of real algebraic curve with involutions. Present paper gives the effective description of Prym’s variety of real curves and special properties of holomorphical differentials on real curves.
Reviewer: S.M.Natanzon

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation
##### Citations:
Zbl 0613.35020; Zbl 0602.35024
Full Text:
##### References:
 [1] A. P. Veselov and S. P. Novikov, ”Finite-gap two-dimensional Schrödinger operators. Explicit formulas and evolution equations,” Dokl. Akad. Nauk SSSR,279, No. 1, 20-24 (1984). · Zbl 0613.35020 [2] A. P. Veselov and S. P. Novikov, ”Finite-gap two-dimensional Schrödinger operators. Potential operators,” Dokl. Akad. Nauk SSSR,279, No. 4, 784-788 (1984). · Zbl 0602.35024 [3] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, ”The Schrödinger equation in a magnetic field and Riemann surfaces,” Dokl. Akad. Nauk SSSR,229, No. 1, 15-18 (1976). · Zbl 0441.35021 [4] I. V. Cherednik, ”On the conditions of reality in ?finite-gap integration?,” Dokl. Akad. Nauk SSSR,252, No. 5, 1104-1108 (1980). [5] S. M. Natanzon, ”The topological classification of pairs of commuting antiholomorphic involutions of Riemann surfaces,” Usp. Mat. Nauk,41, No. 5, 191-192 (1986). [6] S. M. Natanzon, ”Spaces of moduli of real curves,” Tr. Mosk. Mat. Obshch.,37, 219-253 (1978). · Zbl 0439.14006 [7] A. Comessatti, ”Sulle varieta abeliane reali,” Ann. Matem. Pura Appl. Ser. 4,2, 67-106 (1925);3, 27-72 (1926). · JFM 50.0641.01 [8] S. M. Natanzon, ”Moduli of real algebraic curves,” Usp. Mat. Nauk,30, No. 1, 251-252 (1975). [9] J. Fay, Theta-functions on Riemann Surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin?New York (1973). · Zbl 0281.30013 [10] B. A. Dubrovin and S. M. Natanzon, ”Real two-gap solutions of the sine-Gordon equation,” Funktsional. Anal. Prilozhen.,16, No. 1, 27-43 (1982). · Zbl 0554.35100 [11] I. F. Baker, Abel’s Theorem and the Allied Theory Including the Theory of Theta Functions, Cambridge (1897). · JFM 28.0331.01 [12] A. I. Bobenko, Uniformization and Finite-Gap Integration, Preprint, LOMI, Leningrad (1986), pp. 10-86. [13] A. I. Bobenko, ”Schottky uniformization and finite-gap integration,” Dokl. Akad. Nauk SSSR,295, No. 1, 268-272 (1987). · Zbl 0659.35086 [14] I. A. Taimanov, ”Effectivization of theta-functional formulas for two-dimensional potential Schrödinger operators that are finite-gap at one energy level,” Dokl. Akad. Nauk SSSR,285, No. 5, 1067-1070 (1985). [15] B. A. Dubrovin, ”The theta-function and nonlinear equations,” Usp. Mat. Nauk,66, No. 2, 11-80 (1981). · Zbl 0478.58038 [16] S. M. Natanzon, ”Nonsingular finite-gap two-dimensional Schrödinger operators and Prymians of real curves,” Funkt. Anal. Prilozhen.,22, No. 1, 79-80 (1988). · Zbl 0656.35025 [17] N. L. Alling and N. Greenleaf, Foundations of the Theory of Klein Surfaces, Lecture Notes in Mathematics, Vol. 219, Springer-Verlag, New York (1971). [18] S. M. Natanzon, ”Spaces of real meromorphic functions on real algebraic curves,” Dokl. Akad. Nauk SSSR,279, No. 4, 803-805 (1987). [19] S. M. Natanzon, ”Real meromorphic functions on real algebraic curves,” Dokl. Akad. Nauk SSSR,297, No. 1, 40-43 (1987). [20] S. M. Natanzon, ”Invariant lines of Fuchsian groups,” Usp. Mat. Nauk,27, No. 4, 145-160 (1972). [21] W. Kazez, ”On equivalences of branched coverings and their action on homology,” Pac. J. Math.,118, No. 1, 133-157 (1985). · Zbl 0567.57002 [22] L. R. Ford, Automorphic Functions, Chelsea, New York (1951). · JFM 55.0810.04
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.