Lie-algebraic approach to the theory of polynomial solutions. I. Ordinary differential equations and finite-difference equations in one variable

1992/09/22 by Alexander V. Turbiner, Turbiner, Alexander
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.hep-th/9209079

openalex publication_date 1992/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial element of the universal enveloping algebra of sl₂(\bf R) (for differential equations) or sl₂(\bf R)_q (for finite-difference equations) in the "projectivized" representation possessing an invariant subspace. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

Related

• Lie-algebraic approach to the theory of polynomial solutions. II. Differential equations in one real and one Grassmann variables and 2x2 matrix differential equations
• Lie-algebraic approach to the theory of polynomial solutions. III. Differential equations in two real variables and general outlook
• Quasi-Exactly-Solvable Differential Equations
• Flag Partial Differential Equations and Representations of Lie Algebras
• Polynomial solutions of certain differential equations arising in physics