zbMATH — the first resource for mathematics

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

35J10 Schrödinger operator, Schrödinger equation
Full Text: DOI
[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).
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.