Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds

TL;DR

This paper explores discrete spectral symmetries of low-dimensional differential and difference operators, extending classical transformations to multidimensional lattices and manifolds with novel geometric structures, leading to exactly solvable models.

Contribution

It introduces discrete analogs of classical transformations for multidimensional operators and develops nonstandard geometric concepts for simplicial complexes.

Findings

Constructed discrete analogs of Euler-Darboux-Backlund and Laplace transformations.

Developed nonstandard connections and curvature for simplicial complexes.

Built exactly solvable 2D Schrödinger operators with unique spectral properties.

Abstract

Euler-Darboux-Backlund and Laplace transformations are considered for the one- and two-dimensional Schrodinger operators. Their discrete analogs are constructed and generalized for the multidimensional lattices and two-manifolds with special "black-white" triangulations. Nonstandard generalizations of the connections and curvature are constructed for the simplicial complexes. Exactly solvable 2D Schrodinger operators with nonstandard spectral properties are constructed in the continuous and discrete cases using Laplace chains with different restrictions.