Generic KAM Hamiltonians are not quantum ergodic (Q6046069)

The author investigates a converse to quantum ergodicity, namely generic failure of quantum ergodicity for the quantization of a KAM perturbation of a completely integrable classical Hamiltonian. More precisely, for a KAM perturbation of a completely integrable Kolmogorov nondegenerate Gevrey smooth classical Hamiltonian (e.g., a completely integrable Schrödinger operator \(-\Delta +V\)), the main result establishes failure of quantum ergodicity for almost every perturbation size parameter. The paper builds on results by \textit{G. Popov} [Mat. Contemp. 26, 87--107 (2004; Zbl 1074.37031); Ergodic Theory Dyn. Syst. 24, No. 5, 1753--1786 (2004; Zbl 1088.37030); Ann. Henri Poincaré 1, No. 2, 249--279 (2000; Zbl 1002.37028); Ann. Henri Poincaré 1, No. 2, 223--248 (2000; Zbl 0970.37050)], who constructed a normal form for Gevrey smooth Hamiltonians (which is refined here) and who proved that perturbed elliptic operators in a closely related setting have quasimodes which are microlocalized in phase space near suitable KAM tori. The author extends the results from microlocalization of quasimodes to eigenfunctions by controlling the spectral concentration.