Quasi-Exactly-Solvable Differential Equations

Abstract

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of the algebra of differential (difference) operators in finite-dimensional representation. In one-dimensional case a classification is given by algebras $s{l}_2(\mathbf{R})$ (for differential operators in $\mathbf{R}$) and $s{l}_2(\mathbf{R}{)}_q$ (for finite-difference operators in $\mathbf{R}$), $osp(2,2)$ (operators in one real and one Grassmann variable, or equivalently, $2\times 2$ matrix operators in $\mathbf{R}$) and $g{l}_2(\mathbf{R}{)}_K$ ( for the operators containing the differential operators and the parity operator). A classification of linear operators possessing infinitely many finite-dimensional invariant subspaces with a basis in polynomials is presented.

33 pages, to appear as Chapter 12 in CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3 : New Trends in Theoretical Developments and Computational Methods, ed. N. H. Ibragimov, CRC Press, 1995

References in corpus (2)

• Conformal sigma models corresponding to gauged Wess-Zumino-Witten theories
• Quasi Exactly Solvable 2$\times$2 Matrix Equations

Cited by in corpus (14)

• Umbral Calculus, Difference Equations and the Discrete Schroedinger Equation
• Nonlinear supersymmetry on the plane in magnetic field and quasi-exactly solvable systems
• Dolan-Grady Relations and Noncommutative Quasi-Exactly Solvable Systems
• Quantum mass correction for the twisted kink
• Nonlinear supersymmetry: from classical to quantum mechanics
• Quasi-exact-solution of the Generalized Exe Jahn-Teller Hamiltonian
• Quantization of Sine-Gordon solitons on the circle: semiclassical vs. exact results
• Quantum models related to fouled Hamiltonians of the harmonic oscillator
• Quasi-doubly periodic solutions to a generalized Lame equation
• Bions and Instantons in Triple-well and Multi-well Potentials
• On complex oscillation theory, quasi-exact solvability and Fredholm Integral Equations
• Anharmonic oscillator, negative dimensions and inverse factorial convergence of large orders to the asymptotic form
• Spin systems, spin coherent states and quasi-exactly solvable models
• Symmetries of the Jahn Teller System and their Solvability