# Lie-algebraic discretization of differential equations

@article{Smirnov1995LiealgebraicDO, title={Lie-algebraic discretization of differential equations}, author={Yu. F. Smirnov and Alexander V. Turbiner}, journal={Modern Physics Letters A}, year={1995}, volume={10}, pages={1795-1802} }

A certain representation for the Heisenberg algebra in finite difference operators is established. The Lie algebraic procedure of discretization of differential equations with isospectral property is proposed. Using sl2-algebra based approach, (quasi)-exactly-solvable finite difference equations are described. It is shown that the operators having the Hahn, Charlier and Meissner polynomials as the eigenfunctions are reproduced in the present approach as some particular cases. A discrete version… Expand

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#### References

SHOWING 1-4 OF 4 REFERENCES

Quasi-exactly-solvable problems andsl(2) algebra

- Mathematics
- 1988

Recently discovered quasi-exactly-solvable problems of quantum mechanics are shown to be related to the existence of the finite-dimensional representations of the groupSL(2,Q), whereQ=R, C. It is… Expand

Lie-algebras and linear operators with invariant subspaces

- Mathematics
- 1993

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given.… Expand

Classical Orthogonal Polynomials of a Discrete Variable

- Mathematics
- 1991

The basic properties of the polynomials p n (x) that satisfy the orthogonality relations
$$ \int_a^b {{p_n}(x)} {p_m}(x)\rho (x)dx = 0\quad (m \ne n) $$
(2.0.1)
hold also for the polynomials… Expand

Turbiner “ Quasiexactlysolvable problems and sl ( 2 , R ) algebra ”

- Comm . Math . Phys . Journ . Phys . A