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

Bound states in quantum mechanics must almost always be constructed numerically. One of the best known exceptions concerns the central D−dimensional (often called “anharmonic”) Hamiltonian H = p + a |~r|2 + b |~r|4 + . . . + z |~r|4q+2 (where z = 1) with a complete and elementary solvability at q = 0 (central harmonic oscillator, no free parameters) and with an incomplete, N−level elementary analytic solvability at q = 1 (so called “quasi-exact” sextic oscillator containing one free parameter). In the limit D → ∞, numerical experiments revealed recently a highly unexpected existence of a new broad class of the q−parametric quasi-exact solutions at the next integers q = 2, 3, 4 and q = 5. Here we show how a systematic construction of the latter, “privileged” D ≫ 1 exact bound states may be extended to much higher qs (meaning an enhanced flexibility of the shape of the force) at a cost of narrowing the set of wavefunctions (with N restricted to the first few non-negative integers). At q = 4K + 3 we conjecture a closed formula for the N = 3 solution at all K.