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arXiv:math-ph/0404041 · Communications in Mathematical Physics

A Hierarchical Model of Quantum Anharmonic Oscillators: Critical Point Convergence

Sergio A Albeverio, Yuri Kondratiev, Agnieszka Kozak, Yuri V. Kozitsky

Published Sep 2, 2004• 2 citations
arXivHTMLPDF
DOI
10.1007/s00220-004-1165-9

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Using a simplified version of the renormalization group (RG) transformation of Dyson's hierarchical model, we show that one can calculate all the nonuniversal quantities entering into the scaling laws by combining an expansion about the high-temperature fixed point with a dual expansion about the critical point. The magnetic susceptibility is expressed in terms of two dual quantities transforming covariantly under an RG transformation and has a smooth behavior in the high-temperature limit. Using the analogy with Hamiltonian mechanics, the simplified example discussed here is similar to the anharmonic oscillator, while more realistic examples can be thought of as coupled oscillators, allowing resonance phenomena.
The dynamical solution of the driven classical anharmonic (AH) oscillator is obtained through a region of optical bistability. The dynamical solution is observed to switch from the lowest to highest steady-state (SS) root near a SS turning point with evidence of critical slowing down. The solution of a widely used multiple-photon version of the AH model is also obtained and is found to give qualitatively similar behavior. The average number of photons observed dramatically increases in the bistable region and exhibits damped oscillations.
The semiclassical propagator for the one-dimensional anharmonic oscillator is investigated, in the configuration space, by means of the so-called Van Vleck formula, which expresses it as a sum over all the denumerably infinite classical paths connecting given points in the same time. Analytical formulae for the paths' contributions are given, together with some numerical results. It is shown that in the general case the amplitudes of the contributions asymptotically approach the same value, while the phases oscillate; the Van Vleck series therefore does not converge, but in the generic case it can be resumed. Finally, the conditions under which one of the paths gives a dominant contribution to the semiclassical propagator are discussed.