Geodesic
A Sufficient Nonsingularity Condition for a Discrete Finite-Gap One-Energy Two-Dimensional Schr\"odinger Operator on the Quad-Graph
Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 3, pp. 65-70.

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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. 
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B. O. Vasilevskii. A Sufficient Nonsingularity Condition for a Discrete Finite-Gap One-Energy Two-Dimensional Schr\"odinger Operator on the Quad-Graph. Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 3, pp. 65-70. http://geodesic.mathdoc.fr/item/FAA_2015_49_3_a5/

[1] B. A. Dubrovin, I. M. Krichever, S. P. Novikov, Dokl. AN SSSR, 229:1 (1976), 15–18 | MR | Zbl

[2] I. M. Krichever, Dokl. AN SSSR, 285:1 (1985), 31–36 | MR | Zbl

[3] A. Doliwa, P. Grinevich, M. Nieszporski, P. M. Santini, J. Math. Phys., 48:1 (2007), 013513 | DOI | MR | Zbl

[4] A. P. Veselov, S. P. Novikov, Dokl. AN SSSR, 279:1 (1984), 20–24 | MR | Zbl

[5] S. Grushevsky, I. Krichever, Duke Math. J., 152:2 (2010), 317–371 | DOI | MR | Zbl

[6] J. Fay, Theta functions on Riemann surfaces, Lecture Notes in Math., 352, Springer-Verlag, Berlin–Heidelberg–New York, 1973 | DOI | MR | Zbl

[7] A. Bobenko, C. Mercat, Y. Suris, J. Reine Angew. Math., 583 (2005), 117–161 | DOI | MR | Zbl

[8] A. A. Akhmetshin, Yu. S. Volvovskii, I. M. Krichever, Trudy MIAN, 225, Nauka, M., 1999, 21–45 | MR | Zbl