The Green Function of the Discrete Finite-Gap One-Energy Two-Dimensional Schrödinger Operator on the Quad Graph

B. O. Vasilevskii

Lomonosov Moscow State University

Abstract: The finite-gap approach to constructing the discrete Schrödinger operator on a quad graph expressed as a two-dimensional integer sublattice in $d$-dimensional space is used. The Green function for this operator is explicitly expressed as an integral over special contours of the differential constructed from spectral data. The resulting function has a well-known asymptotics.

Keywords: discrete Schrödinger operator, Green function, integer sublattice, quad graph, wave function, Cauchy–Riemann equations, Riemann sphere, Riemann surface, Iacobi manifold, quasimomentum.

UDC: 514.84

Received: 07.01.2014
Revised: 30.10.2014

DOI: 10.4213/mzm10450

 English version: Mathematical Notes, 2015, 98:1, 38–52

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