Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2006-04-26
Discrete Contin. Dyn. Syst. 18 (2007), no. 1, 85--106
Nonlinear Sciences
Exactly Solvable and Integrable Systems
23 pages, typed in AMS-LaTeX
Scientific paper
10.3934/dcds.2007.18.85
In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write explicit basis for these spaces of differential operators. In the case of linear operators, these results apply to the theory of quasi-exact solvability in quantum mechanics, specially in the multivariate case where the Lie algebraic approach is harder to apply. In the case of non-linear operators, the structure theorems in this paper can be applied to the method of finding special solutions of non-linear evolution equations by nonlinear separation of variables.
Gomez-Ullate David
Kamran Niky
Milson Robert
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