Generalized master function approach to one-dimensional quasiexactly solvable models. (English) Zbl 0971.81171
By introducing the generalized master function of order up to four together with corresponding weight functions, we have obtained all one-dimensional quasiexactly solvable second order differential equations. It is shown that these differential equations have solutions of polynomial type with factorization properties, that is polynomial solutions \(P_m(E)\) can be factorized in terms of polynomials \(P_{n+1}(E)\) for \(m\geq n+1\). All known one-dimensional quasiexactly quantum solvable models can be obtained from these differential equations, where the roots of the polynomials \(P_{n+1}(E)\) are the corresponding eigenvalues.
MSC:
81U15 | Exactly and quasi-solvable systems arising in quantum theory |
34L05 | General spectral theory of ordinary differential operators |
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
34A05 | Explicit solutions, first integrals of ordinary differential equations |
References:
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