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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
Full Text:
##### References:
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