Abstract
We study the classical and quantum perturbation theory for two non-resonant oscillators coupled by a nonlinear quartic interaction. In particular, we analyze the question of quantum corrections to the torus quantization of the classical perturbation theory (semiclassical mechanics). We obtain up to the second order of perturbation theory an explicit analytical formula for the quantum energy levels, which is the semiclassical one plus quantum corrections. We compare the ‘exact’ quantum levels obtained numerically to the semiclassical levels studying also the effects of quantum corrections.
Sommario. Si studia la teoria perturbativa classica e quantistica per due oscillatori non risonanti accoppiati tramite una interazione non lineare quartica. In particolare si analizza il problema delle correzioni quantistiche alla quantizzazione dei tori della teoria perturbativa classica (meccanica semiclassica). Si ottiene, al secondo ordine della teoria perturbativa, una formula analitica per i livelli energetici, che è quella semiclassica con l'aggiunta di correzioni quantistiche. Si confrontano i livelli energetici ‘esatti’ ottenuti numericamente con quelli semiclassici, studiando anche gli effetti delle correzioni quantistiche.
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Salasnich, L. Classical and Quantum Perturbation Theory for Two Non-Resonant Oscillators with Quartic Interaction. Meccanica 33, 397–405 (1998). https://doi.org/10.1023/A:1004377818528
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DOI: https://doi.org/10.1023/A:1004377818528