Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces
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Abstract
In this paper we derive structuretheorems which characterize the spaces of linear and non-lineardifferential operators that preserve finite dimensional subspacesgenerated by polynomials in one or several variables. By means ofthe useful concept of deficiency, we can write an explicit basis forthese spaces of differential operators. In the case of linearoperators, these results apply to the theory of quasi-exactsolvability in quantum mechanics, especially in the multivariatecase where the Lie algebraic approach is harder to apply. In thecase of non-linear operators, the structure theorems in this papercan be applied to the method of finding special solutions ofnon-linear evolution equations by nonlinear separation of variables.
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