Geodesic
Difference Analog of the Lamé Operator
Informatics and Automation, Geometry, Topology, and Mathematical Physics, Tome 325 (2024), pp. 190-200 Cet article a éte moissonné depuis la source Math-Net.Ru

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We construct a difference analog of the Lamé operator. Namely, we present a second-order difference operator whose coefficients depend on a small parameter which commutes with a difference operator of order $2g+1$. When the small parameter tends to zero, the difference operator transforms into the Lamé operator.
Keywords: commuting difference operators, commuting differential operators. 
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G. S. Mauleshova; A. E. Mironov. Difference Analog of the Lamé Operator. Informatics and Automation, Geometry, Topology, and Mathematical Physics, Tome 325 (2024), pp. 190-200. http://geodesic.mathdoc.fr/item/TRSPY_2024_325_a9/

[1] B. A. Dubrovin, “Periodic problems for the Korteweg–de Vries equation in the class of finite band potentials”, Funct. Anal. Appl., 9:3 (1975), 215–223 | DOI | MR | Zbl

[2] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable systems. I”, Dynamical Systems IV: Symplectic Geometry and Its Applications, Encycl. Math. Sci., 4, Springer, Berlin, 1990, 173–280 | MR | MR

[3] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Non-linear equations of Korteweg–de Vries type, finite-zone linear operators, and Abelian varieties”, Russ. Math. Surv., 31:1 (1976), 59–146 | DOI | MR | Zbl | Zbl

[4] Grosset M.-P., Veselov A.P., “Lamé equation, quantum Euler top and elliptic Bernoulli polynomials”, Proc. Edinb. Math. Soc. Ser. 2, 51:3 (2008), 635–650 | DOI | MR | Zbl

[5] A. R. Its and V. B. Matveev, “Schrödinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg–de Vries equation”, Theor. Math. Phys., 23:1 (1975), 343–355 | DOI | MR

[6] I. M. Krichever, “Algebraic curves and non-linear difference equations”, Russ. Math. Surv., 33:4 (1978), 255–256 | DOI | MR | Zbl | Zbl

[7] I. M. Krichever and S. P. Novikov, “Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles”, Russ. Math. Surv., 58:3 (2003), 473–510 | DOI | DOI | MR | Zbl

[8] G. S. Mauleshova and A. E. Mironov, “One-point commuting difference operators of rank 1”, Dokl. Math., 93:1 (2016), 62–64 | DOI | DOI | MR | Zbl

[9] G. S. Mauleshova and A. E. Mironov, “One-point commuting difference operators of rank one and their relation with finite-gap Schrödinger operators”, Dokl. Math., 97:1 (2018), 62–64 | DOI | DOI | MR

[10] Mironov A.E., “Periodic and rapid decay rank two self-adjoint commuting differential operators”, Topology, geometry, integrable systems, and mathematical physics: Novikov's seminar 2012–2014, AMS Transl. Ser. 2, 234, Am. Math. Soc., Providence, RI, 2014, 309–321 | MR | Zbl

[11] Mumford D., “An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg–de Vries equation and related non-linear equations”, Proceedings of the International Symposium on Algebraic Geometry, Kyoto, 1977, Kinokuniya Book-Store, Tokyo, 1978, 115–153 | MR

[12] S. P. Novikov, “The periodic problem for the Korteweg–de Vries equation”, Funct. Anal. Appl., 8:3 (1974), 236–246 | DOI | MR | Zbl