Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces

David Gómez-Ullate; Niky Kamran; Robert Milson

doi:10.3934/dcds.2007.18.85

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2007, Volume 18, Issue 1: 85-106. Doi: 10.3934/dcds.2007.18.85

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Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces

1.
Departamento de Física Teórica II, Universidad Complutense, 28040 Madrid
2.
Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6
3.
Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 3J5

Received: April 30, 2006

Revised: December 31, 2006

Published: December 2006

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Abstract

In this paper we derive structure theorems which 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 an 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, especially 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.

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Mathematics Subject Classification: Primary: 47F5, 35K55; Secondary: 81R15.

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