Quasi-exact solvability in a general polynomial setting

Details Quasi-exact solvability in a general polynomial setting Quasi-exact solvability in a general polynomial setting

2006-10-27

arxiv.org/abs/nlin/0610065v1

Inverse Problems 23 (2007) 1915-1942

Nonlinear Sciences

Exactly Solvable and Integrable Systems

29 pages, 10 figures, typed in AMS-TeX

Scientific paper

10.1088/0266-5611/23/5/008

Our goal in this paper is to extend the theory of quasi-exactly solvable Schrodinger operators beyond the Lie-algebraic class. Let $\cP_n$ be the space of n-th degree polynomials in one variable. We first analyze "exceptional polynomial subspaces" which are those proper subspaces of $\cP_n$ invariant under second order differential operators which do not preserve $\cP_n$. We characterize the only possible exceptional subspaces of codimension one and we describe the space of second order differential operators that leave these subspaces invariant. We then use equivalence under changes of variable and gauge transformations to achieve a complete classification of these new, non-Lie algebraic Schrodinger operators. As an example, we discuss a finite gap elliptic potential which does not belong to the Treibich-Verdier class.

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