Discrete \(SL_{n}\)-connections and self-adjoint difference operators on 2-dimensional manifolds. (English. Russian original) Zbl 1282.39006

Grinevich, P. G.; Novikov, S. P.

Russ. Math. Surv. 68, No. 5, 861-887 (2013); translation from Usp. Mat. Nauk 68, No. 5, 81-110 (2013).

Summary: The programme of discretization of famous completely integrable systems and associated linear operators was launched in the 1990s. In particular, the properties of second-order difference operators on triangulated manifolds and equilateral triangular lattices have been studied by Novikov and Dynnikov since 1996. This study included Laplace transformations, new discretizations of complex analysis, and new discretizations of \(GL_n\)-connections on triangulated \(n\)-dimensional manifolds. A general theory of discrete \(GL_n\)-connections ‘of rank one’ has been developed (see the introduction for definitions). The problem of distinguishing the subclass of \(SL_n\)-connections (and unimodular \(SL_n^{\pm}\)-connections, which satisfy \(\mathrm{det}A = \pm 1\)) has not been solved. In the present paper it is shown that these connections play an important role (which is similar to the role of magnetic fields in the continuous case) in the theory of self-adjoint Schrödinger difference operators on equilateral triangular lattices in \(\mathbb R^2\). In Appendix 1, a complete characterization is given of unimodular \(SL_n^{\pm}\) -connections of rank 1 for all \(n > 1\), thus correcting a mistake (it was wrongly claimed that they reduce to a canonical connection for \(n > 2\)). With the help of a communication from Korepanov, a complete clarification is provided of how the classical theory of electrical circuits and star-triangle transformations is connected with the discrete Laplace transformations on triangular lattices.

MSC:

39A12	Discrete version of topics in analysis
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
39A70	Difference operators
47B39	Linear difference operators
44A10	Laplace transform

Keywords:

triangulated manifolds with black and white colouring; discrete connections; discrete complex structures; factorization of self-adjoint operators; Darboux transformations; Laplace transformations; discrete integrable systems; self-adjoint Schrödinger difference operator