Geodesic
The Green Function of the Discrete Finite-Gap One-Energy Two-Dimensional Schr\"odinger Operator on the Quad Graph
Matematičeskie zametki, Tome 98 (2015) no. 1, pp. 27-43.

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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, Riemann sphere, Riemann surface, Iacobi manifold, quasimomentum.
Mots-clés : Cauchy–Riemann equations 
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B. O. Vasilevskii. The Green Function of the Discrete Finite-Gap One-Energy Two-Dimensional Schr\"odinger Operator on the Quad Graph. Matematičeskie zametki, Tome 98 (2015) no. 1, pp. 27-43. http://geodesic.mathdoc.fr/item/MZM_2015_98_1_a2/

[1] B. A. Dubrovin, I. M. Krichever, S. P. Novikov, “Uravnenie Shredingera v periodicheskom pole i rimanovy poverkhnosti”, Dokl. AN SSSR, 229:1 (1976), 15–18 | MR | Zbl

[2] I. M. Krichever, “Dvumernye periodicheskie raznostnye operatory i algebraicheskaya geometriya”, Dokl. AN SSSR, 285:1 (1985), 31–36 | MR | Zbl

[3] A. Doliwa, P. Grinevich, M. Nieszporski, P. M. Santini, “Integrable lattices and their sub-lattices: from the discrete Moutard (discrete Cauchy–Riemann) 4-point equation to the self-adjoint 5-point scheme”, J. Math. Phys., 48:1 (2007), 013513 | DOI | MR | Zbl

[4] A. P. Veselov, S. P. Novikov, “Konechnozonnye dvumernye operatory Shredingera. Yavnye formuly i evolyutsionnye uravneniya”, Dokl. AN SSSR, 279:1 (1984), 20–24 | MR | Zbl

[5] A. P. Veselov, S. P. Novikov, “Konechnozonnye dvumernye operatory Shredingera. Potentsialnye operatory”, Dokl. AN SSSR, 279:4 (1984), 784–788 | MR | Zbl

[6] J. D. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Math., Springer-Verlag, Berlin, 1973 | DOI | MR | Zbl

[7] S. Grushevsky, I. Krichever, “Integrable discrete Schrödinger equations and a characterization of Prym varieties by a pair of quadrisecants”, Duke Math. J., 152:2 (2010), 317–371 | DOI | MR | Zbl

[8] A. I. Bobenko, C. Mercat, Y. B. Suris, “Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function”, J. Reine Angew. Math., 583 (2005), 117–161 | DOI | MR | Zbl

[9] A. A. Akhmetshin, Yu. S. Volvovskii, I. M. Krichever, “Diskretnye analogi metrik Darbu–Egorova”, Solitony, geometriya, topologiya — na perekrestkakh, Tr. MIAN, 225, Nauka, M., 1999, 21–45 | MR | Zbl

[10] B. A. Dubrovin, V. B. Matveev, S. P. Novikov, “Nelineinye uravneniya tipa Kortevega–de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, UMN, 31:1 (1976), 55–136 | MR | Zbl

[11] I. M. Krichever, S. P. Novikov, “Algebry tipa Virasoro, rimanovy poverkhnosti i struktury teorii solitonov”, Funkts. analiz i ego pril., 21:2 (1987), 46–63 | MR | Zbl