Quasi-exact solvability in a general polynomial setting

Abstract

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.

29 pages, 10 figures, typed in AMS-TeX

References in corpus (9)

• The Darboux transformation and algebraic deformations of shape-invariant potentials
• A_N-type Dunkl operators and new spin Calogero-Sutherland models
• Quasi-Exact Solvability and the direct approach to invariant subspaces
• Complex Periodic Potentials with a Finite Number of Band Gaps
• Quasi-exact solvability beyond the SL(2) algebraization
• Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces
• QES systems, invariant spaces and polynomials recursions
• Operator ordering in Two-dimensional N=1 supersymmetry with curved manifold
• Quasi exactly solvable (QES) equations with multiple algebraisations

Cited by in corpus (10)

• An Extension of Bochner's Problem: Exceptional Invariant Subspaces
• Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
• Two-step Darboux transformations and exceptional Laguerre polynomials
• Exceptional orthogonal polynomials and the Darboux transformation
• Asymptotic behaviour of zeros of exceptional Jacobi and Laguerre polynomials
• N-fold Supersymmetry and Quasi-solvability Associated with X_2-Laguerre Polynomials
• Effect of Position-dependent Mass on Dynamical Breaking of Type B and Type X_2 N-fold Supersymmetry
• New quasi-exactly solvable class of generalized isotonic oscillators
• Symmetric Tops Subject to Combined Electric Fields: Conditional Quasi-Solvability via the Quantum Hamilton-Jacobi Theory
• Type B 3-fold Supersymmetry and Non-polynomial Invariant Subspaces