an:04071433
Zbl 0656.35025
Natanzon, S. M.
Nonsingular finite-zone two-dimensional Schr??dinger operators and Prymians of real curves
EN
Funct. Anal. Appl. 22, No. 1, 68-70 (1988); translation from Funkts. Anal. Prilozh. 22, No. 1, 79-80 (1988).
00180246
1988
j
35J10
compact Riemann surface; antiholomorphic involutions; Prym variety; nonsingular; finite-zone two-dimensional; Schr??dinger operators
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 \textit{A. P. Veselov}'s and \textit{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 Schr??dinger 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 \(k_ i\leq k_{2-i}\).
S.Natanzon
Zbl 0613.35020; Zbl 0602.35024