{"paper":{"title":"1d Quantum Harmonic Oscillator Perturbed by a Potential with Logarithmic Decay","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Zhenguo Liang, Zhiguo Wang","submitted_at":"2016-05-18T08:46:41Z","abstract_excerpt":"In this paper we prove an infinite dimensional KAM theorem, in which the assumptions on the derivatives of perturbation in \\cite{GT} are weakened from polynomial decay to logarithmic decay. As a consequence, we apply it to 1d quantum harmonic oscillators and prove the reducibility of a linear harmonic oscillator, $T=- \\frac{d^2}{dx^2}+x^2$, on $L^2(\\R)$ perturbed by a quasi-periodic in time potential $V(x,\\omega t; \\omega)$ with logarithmic decay. This entails the pure-point nature of the spectrum of the Floquet operator $K$, where K:=-{\\rm i}\\sum_{k=1}^n\\omega_k\\frac{\\partial}{\\partial \\theta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.05480","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}