A subclass of anharmonic oscillators whose eigenfunctions have no recurrence relations

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).