Quasi-Exact Solvability and the direct approach to invariant subspaces

Abstract

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: an interesting generalization of the Lamé potential which posses four algebraic sectors, and a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic.

arXiv comment17 pages, 3 figures

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Cited by in corpus (16)

• 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
• Properties of the Exceptional ($X_{\ell }$) Laguerre and Jacobi Polynomials
• A conjecture on Exceptional Orthogonal Polynomials
• Polynomial algebras and exact solutions of general quantum non-linear optical models I: Two-mode boson systems
• Quasi-exact solvability beyond the SL(2) algebraization
• Quasi-exact solvability in a general polynomial setting
• Two novel classes of solvable many-body problems of goldfish type with constraints
• Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart
• PT-Symmetric, Quasi-Exactly Solvable matrix Hamiltonians
• A unified treatment of polynomial sectors and constraint polynomials of the Rabi models
• Additional restrictions on quasi-exactly solvable systems
• Type B 3-fold Supersymmetry and Non-polynomial Invariant Subspaces