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 Sibirsk. Mat. Zh., 2013, Volume 54, Number 6, Pages 1250–1262 (Mi smj2491)

The Green's function of a five-point discretization of a two-dimensional finite-gap Schrödinger operator: The case of four singular points on the spectral curve

B. O. Vasilevskiĭ

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Bogolyubov Laboratory of Geometric Methods in Mathematical Physics, Moscow, Russia

Abstract: We consider a regular Riemann surface of finite genus and “generalized spectral data”, a special set of distinguished points on it. From them we construct a discrete analog of the Baker–Akhiezer function with a discrete operator that annihilates it. Under some extra conditions on the generalized spectral data, the operator takes the form of the discrete Cauchy–Riemann operator, and its restriction to the even lattice is annihilated by the corresponding Schrödinger operator. In this article we construct an explicit formula for the Green's function of the indicated operator. The formula expresses the Green's function in terms of the integral along a special contour of a differential constructed from the wave function and the extra spectral data. In result, the Green's function with known asymptotics at infinity can be obtained at almost every point of the spectral curve.

Keywords: discrete operator, finite-gap operator, Green’s function, M-curve.

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English version:
Siberian Mathematical Journal, 2013, 54:6, 994–1004

Bibliographic databases:

UDC: 514.84

Citation: B. O. Vasilevskiǐ, “The Green's function of a five-point discretization of a two-dimensional finite-gap Schrödinger operator: The case of four singular points on the spectral curve”, Sibirsk. Mat. Zh., 54:6 (2013), 1250–1262; Siberian Math. J., 54:6 (2013), 994–1004

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. S. Mauleshova, “The dressing chain and one-point commuting difference operators of rank 1”, Siberian Math. J., 59:5 (2018), 901–908
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