On the classical and quantum integrability of systems of resonant oscillators (Q1707948)

A classical Hamiltonian system with Hamiltonian \(H\) on a \(2n\)-dimensional symplectic manifold is said to be integrable in the sense of Nekhoroshev if there exists a set of \(2n-k\) functionally independent real functions \[ F=(F_1,\dots,F_k, F_{k+1},\dots,F_{2n-k}) \] defined on the phase space with \(1\leq k\leq n\), such that \[ \{F_i,F_j\}=0, \quad \forall i=1,\dots,k,\quad j=1,\dots,2n-k, \] and \(H=f(F_1,\dots,F_k)\), where \(\{,\}\) is the Poisson bracket and \(f\) is an arbitrary regular function of \(k\) variables. The case \(k=1\) corresponds to Arnold-Liouville integrability and \(k=n\) corresponds to maximal superintegrability. The author solves the problem of classifying systems of \(n\) oscillators with a fully resonant set of frequencies and with nonlinear couplings. This means classifying systems such that \(F_1=\ell_1I_1+\dots +\ell_n I_n\) where \(I_i=\frac12 (p_i^2+x_i^2)\) for \(i=1,\dots n\), and \(\ell_1,\dots,\ell_n\) are nonzero integer values, relatively prime, and \(H=F_1+h(F_1,\dots,F_k)\) where \(h\) is an arbitrary function of \(k\) variables. Thus the problem reduces to finding a basis for the space of all real polynomial functions in \(( {x},{p})\) that are in involution with \(F_1\). The author introduces the standard boson basis \[ z_j=\frac{x_j+ip_j}{\sqrt{2}}, \qquad {\bar z}_j=\frac{x_j-ip_j}{\sqrt{2}}, \] in which the problem reduces to classifying polynomials in \(z_j, {\bar z}_j\) that are in involution with \(\sum_{j=1}^n|z_j|^2\). The problem is straightforward, but the details are complicated. In particular if the absolute values \(|\ell_j|\) are not all distinct additional constants of the motion appear. The author shows that by symmetrization of the products of quantum analogs of \(z_j,{\bar z}_j\) these results can all be quantized. Special attention is paid to the classical and quantum results for the case \(n=3\).