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

Abstract

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 $q\le 5$. 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.

15 pp incl. 4 tables, presented during 24th Winter School ``Geometry and physics" (Srni, Czech Republic, Januar 17 - 24, 2004)

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