Sextic anharmonic oscillators and orthogonal polynomials

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Saad N., Hall R. L., Ciftci H.

JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, vol.39, no.26, pp.8477-8486, 2006 (SCI-Expanded) identifier identifier


Under certain constraints on the parameters a, b and c, it is known that Schrodinger's equation - d(2)Psi/dx(2) + (ax(6) + bx(4) + cx(2))Psi = E Psi, a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this paper we show that the exact wavefunction Psi is the generating function for a set of orthogonal polynomials {P-n((t)) (x)} in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can be reduced, by means of a simple scaling transformation, to a remarkable class of orthogonal polynomials, P-n(E) = P-n((0)) (E) recently discovered by Bender and Dunne.