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Two-dimensional periodic difference operators and algebraic geometry. (English. Russian original) Zbl 0603.39004
Sov. Math., Dokl. 32, 623-627 (1985); translation from Dokl. Akad. Nauk SSSR 285, 31-36 (1985).
The concept of one-dimensional Schrödinger operator $$H=(i \partial /\partial x-A_ 1)^ 2+(i \partial /\partial y-A_ 2)+u(x,y)$$ (finite- zone relative to a single energy level with periodic coefficients), where u(x,y) is a potential and $$A_ i(x,y)$$ $$(i=1,2)$$ are vector potentials, was introduced by B. A. Dubrovin, I. M. Krichever and S. P. Novikov [Sov. Math., Dokl. 17, 947-951 (1977; Zbl 0441.35021)]. In the theory of the one-dimensional Schrödinger operator, it is established that an arbitrary real periodic potential operator H can be approximated by finite-zone operators. The purpose of this note is to construct an analogous theory for two-dimensional operators and to show its application to algebraic geometry.
Reviewer: H.Haruki

##### MSC:
 39A70 Difference operators 35J10 Schrödinger operator, Schrödinger equation 39A12 Discrete version of topics in analysis 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 14K25 Theta functions and abelian varieties 14C40 Riemann-Roch theorems