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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 AMathematical and General, 32 (39). pp. 68216835. ISSN 03054470

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Official URL: http://dx.doi.org/10.1088/03054470/32/39/308
URL  URL Type 

http://iopscience.iop.org  Publisher 
http://arxiv.org/abs/mathph/9905020  Publisher 
Abstract
We show that there are two different families of (weakly) orthogonal polynomials associated to the quasiexactly 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 wellknown finite solutions of the Whittaker–Hill (or Hill’s threeterm) periodic differential equation. Thus, the foregoing results provide a Liealgebraic justification of the fact that the Whittaker– Hill equation (unlike, for instance, Mathieu’s equation) admits finite solutions.
Item Type:  Article 

Additional Information:  © 1999 IOP Publishing Ltd. 
Uncontrolled Keywords:  Antiisospectral transformations; Exactlysolvable problems; Quantummechanics; Sl(2) algebra; Systems; Operators; Equation; Kdp 
Subjects:  Sciences > Physics > PhysicsMathematical 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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