Sextic anharmonic oscillators and orthogonal polynomials
نویسندگان
چکیده
Under certain constraints on the parameters a, b and c, it is known that Schrödinger’s equation −d2ψ/dx2 + (ax + bx + cx)ψ = Eψ, a > 0 with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function ψ is the generating function for a set of orthogonal polynomials {P (t) n (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, Pn(E) = P (0) n (E) recently discovered by Bender and Dunne.
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