Differential equations on the Prym theta function. A realness criterion for two-dimensional finite-zone potential Schrödinger operators
This article deals with finite-zone two-dimensional Schrödinger operators of the type \[ L = {\partial^ 2 \over \partial z \partial \overline z} + 2{\partial^ 2 \over \partial z \partial \overline z} \theta_{Pr} (zu^ 1 + \overline zu^ 2-e) - \varepsilon_ 0 \] where \(e,u^ 1,u^ 2 \in {\mathcal C}^ 2\) and \(\theta_{Pr}\) -- Prym theta- function of a complex algebraic curve \(P\) of genus \(2g\). The main result is the following: the reality of the function \(u = {\partial^ 2 \over \partial z \partial \overline z} \theta_{Pr} (zu^ 1 + \overline zu^ 2-e) - \varepsilon_ 0\) implies the reality of the curve \(P\).
scientific article; zbMATH DE number 95220
scientific article