Brief communications

A Sufficient Nonsingularity Condition for a Discrete Finite-Gap One-Energy Two-Dimensional Schrödinger Operator on the Quad-Graph

B. O. Vasilevskii

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The finite-gap approach is used to construct a two-dimensional discrete Schrödinger operator on a quad-graph, that is, a planar graph whose faces are quadrangles. The following definition of the nonsingularity of this operator is proposed: An operator is nonsingular if all of its coefficients are positive. Conditions on a spectral curve and a quad-graph sufficient for the operator constructed from them to be nonsingular are given.

Keywords: discrete operator, discrete complex analysis, finite-gap operator, spectral curve, M-curve, Riemann surface, nonsingularity.

UDC: 514.84

Received: 11.11.2013

DOI: 10.4213/faa3199

 English version: Functional Analysis and Its Applications, 2015, 49:3, 210–213

Bibliographic databases:

• 
• 
•