Abstract
We study the quantum oscillator perturbed by external random noise which imparts a series of kicks to the system: the kicks are Poisson distributed in time and the probability measure associated with the jump size is assumed to correspond to symmetric stable L\'evy distributions. The Feynman-Vernon theory of influence functionals yields a master equation for the density operator with a term related to the characteristic function of the measure associated with the jump distribution. A scaling argument shows that the constant associated with the external-noise-induced diffusion carries fractal space dimension 0<\ensuremath{\alpha}\ensuremath{\le}2. In the noise-dominated regime the system exhibits singularities for 0\ensuremath{\alpha}2 in contrast to the behavior for \ensuremath{\alpha}=2 corresponding to the Gaussian case.
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