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Contents ?1. Introduction: history of the problem, the one-dimensional Schr?dinger operator ?2. The non-stationary one-dimensional Schr?dinger equation ?3. One-dimensional difference operators ?4. The Laplace transformation and the two-dimensional Schr?dinger operator in a magnetic field ?5. Difference equations and Laplace transformations. The hyperbolic case ?6. The Laplace transformation for elliptic two-dimensional difference operators on a regular lattice. Equations of a triangle, curvature ?7. Factorizations and Laplace transformations on many-dimensional lattices of regular tetahedra in ?8. Factorizations of operators and Laplace transformations on two-dimensional surfaces ?9. Simplicial connections. Generalizations Bibliography

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