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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A subclass of anharmonic oscillators whose eigenfunctions have no recurrence relations

Author: Gary G. Gundersen
Journal: Proc. Amer. Math. Soc. 58 (1976), 109-113
MSC: Primary 34B25
MathSciNet review: 0460781
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Abstract: The equation $ w''(z) + (\lambda - p(z))w(z) = 0(z \in {\mathbf{C}},\lambda \in {\mathbf{R}})$ with a fixed $ p(z) = {a_{2m}}{z^{2m}} + {a_{2m - 2}}{z^{2m - 2}} + \cdots + {a_2}{z^2}({a_i} \geq 0\forall i,{a_{2m}} > 0)$, possesses a set of solutions $ \{ {\psi _n}(z)\} _{n = 0}^\infty $ (with associated $ \{ {\lambda _n}\} _{n = 0}^\infty $) which form a complete orthonormal set for $ {L^2}( - \infty ,\infty )$ (as a real space). Here it is shown that any $ g(z)\psi _N^{(q)}(z)$ (fixed $ N,q \in {\mathbf{N}},g \not\equiv 0$ a polynomial with $ \deg (g) \geq 1$ when $ q = 0$) cannot be expressed as a finite linear combination of $ \{ {\psi _n}(z)\} _{n = 0}^\infty $ when $ \deg (p)$ is a multiple of 4. It is well known that $ g(z)\psi _N^{(q)}(z)$ can always be expressed as such when $ p(z) = {z^2}$ (the Hermite case).

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Keywords: Eigenfunction, eigenvalue, Hermite functions, Hermite polynomials, recurrence relation, complete orthonormal set, anharmonic oscillator, harmonic oscillator, nilpotent Lie algebra, WKB transformation
Article copyright: © Copyright 1976 American Mathematical Society