Discrete $SL_n$-connections and self-adjoint difference operators on two-dimensional manifolds

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 $\det A=\pm 1$) 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 $\ref{pril1}$ 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.\footnote{The papers of S. P. Novikov on this topic (partly with collaborators) can be found on his homepage {\tthttp://www.mi.ras.ru/~snovikov}, items 136–138, 140, 146, 148, 159, 163, 173–175. Click on {\ttScientific Publications} to pass to the list of papers.}
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