# zbMATH — the first resource for mathematics

Multiparametric oscillator Hamiltonians with exact bound states in infinite-dimensional space. (English) Zbl 1101.81057
Slovák, Jan (ed.) et al., The proceedings of the 24th winter school “Geometry and physics”, Srní, Czech Republic, January 17–24, 2004. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 75, 333-346 (2005).
Summary: 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 $$H=p^2+a|\vec r|^2+b|\vec r|^4+\cdots+z|\vec r |^{4q+2}$$ (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 $$D\to\infty$$, 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” $$D\gg 1$$ exact bound states may be extended to much higher $$q_s$$ (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 $$K$$.
For the entire collection see [Zbl 1074.53001].
##### MSC:
 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 13P05 Polynomials, factorization in commutative rings 14M12 Determinantal varieties 81U15 Exactly and quasi-solvable systems arising in quantum theory