Multiparametric oscillator Hamiltonians with exact bound states in infinite-dimensional space

Central D-dimensional Hamiltonians $H = p^2 + a |\vec{r}|^2 + b |\vec{r}|^4 + >... + z |\vec{r}|^{4q+2}$ (where z=1) are considered in the limit $D \to \infty$ where numerical experiments revealed recently a new class of q-parametric quasi-exact solutions at $q \leq 5$. We show how a systematic construction of these "privileged" exact bound states may be extended to much higher q (meaning an enhanced flexibility of the shape of the force) at a cost of narrowing the set of wavefunctions (with degree N restricted to the first few non-negative integers). At q=4K+3 we conjecture the validity of a closed formula for the N=3 solutions at all K.