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Nonsingular finite-zone two-dimensional Schrödinger operators and Prymians of real curves. (English. Russian original) Zbl 0656.35025
Funct. Anal. Appl. 22, No. 1, 68-70 (1988); translation from Funkts. Anal. Prilozh. 22, No. 1, 79-80 (1988).
Let P be a compact Riemann surface of genus 2g with two antiholomorphic involutions \(\tau_ i: P\to P\), the involution \(\tau_ 1\tau_ 2\) having exactly two fixed points \(p_ 1,p_ 2\) and \(\tau_ ip_ 1=p_ 2\). Involutions \(\tau_ i\) induce involutions \({\tilde \tau}{}_ i\) of the Prym variety \(P_ r=P_ r(P,\tau_ 1\tau_ 2)\). Fixed points of \({\tilde \tau}_ i\) break down into \(n\leq 2^ g\) q-dimensional tori. The torus T can be called acceptable if \(\theta_{P_ r}(z)\neq 0\) on T. According to A. P. Veselov’s and S. P. Novikov’s Theorem [Sov. Math., Dokl. 30, 588-591 resp. 705-708 (1984); translation from Dokl. Akad. Nauk SSSR 279, 20-24 resp. 784-788 (1984; Zbl 0613.35020 resp. Zbl 0602.35024)] the acceptable torus induces the family of nonsingular finite-zone two-dimensional Schrödinger operators. In the paper the description of acceptable tori is given. Involutions \(\tau_ i\) induce on the surface \(P_ 0=P/\tau_ 1\tau_ 2\) the antiholomorphic involution \(\tau_ 0\) with \(k=k_ 1+k_ 2\) ovals, where \(k_ i\) is the number of ovals of the involutions \(\tau_ 0\), which are the image of ovals of the involutions \(\tau_ i.\)
Theorem. Among tori of fixed points of the involutions \({\tilde \tau}{}_ i\) is not larger than one acceptable torus. This torus exists only in case \(k=g+1\) or \(k_ i\leq k_{2-i}\).
Reviewer: S.Natanzon

MSC:
35J10 Schrödinger operator, Schrödinger equation
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References:
[1] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, Dokl. Akad. Nauk SSSR,229, No. 1, 15-19 (1976).
[2] A. P. Veselov and S. P. Novikov, Dokl. Akad. Nauk SSSR,279, No. 1, 20-24 (1984).
[3] A. P. Veselov and S. P. Novikov, Dokl. Akad. Nauk SSSR,279, No. 4, 784-788 (1984).
[4] S. P. Novikov, Usp. Mat. Nauk,39, No. 4, 128-129 (1984).
[5] I. V. Cherednik, Dokl. Akad. Nauk SSSR,252, No. 5, 1104-1108 (1980).
[6] B. A. Dubrovin, Usp. Mat. Nauk,36, No. 2, 11-80 (1981).
[7] S. M. Natanzon, Usp. Mat. Nauk,41, No. 5, 191-192 (1986).
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