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On the families of orthogonal polynomials associated to the Razavy potential

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Finkel Morgenstern, Federico and González López, Artemio and Rodríguez González, Miguel Ángel (1999) On the families of orthogonal polynomials associated to the Razavy potential. Journal of physics A-Mathematical and General, 32 (39). pp. 6821-6835. ISSN 0305-4470

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Official URL: http://dx.doi.org/10.1088/0305-4470/32/39/308




Abstract

We show that there are two different families of (weakly) orthogonal polynomials associated to the quasi-exactly solvable Razavy potential V_ (x) = (ζ cosh 2x−M)^ 2 (ζ > 0, M ∈ N). One of these families encompasses the four sets of orthogonal polynomials recently found by Khare and Mandal, while the other one is new. These results are extended to the related periodic potential U_(x) = −(ζ cos 2x − M) ^2 , for which we also construct two different families of weakly orthogonal polynomials. We prove that either of these two families yields the ground state (when M is odd) and the lowest lying gaps in the energy spectrum of the latter periodic potential up to and including the (M − 1)_(th) gap and having the same parity as M − 1. Moreover, we show that the algebraic eigenfunctions obtained in this way are the well-known finite solutions of the Whittaker–Hill (or Hill’s three-term) periodic differential equation. Thus, the foregoing results provide a Lie-algebraic justification of the fact that the Whittaker– Hill equation (unlike, for instance, Mathieu’s equation) admits finite solutions.


Item Type:Article
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© 1999 IOP Publishing Ltd.
Supported in part by DGES Grant PB95–0401.

Uncontrolled Keywords:Anti-isospectral transformations; Exactly-solvable problems; Quantum-mechanics; Sl(2) algebra; Systems; Operators; Equation; Kdp
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:32858
Deposited On:25 Aug 2015 07:48
Last Modified:10 Dec 2018 15:10

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