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].
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