Abstract
We present the semiclassical and exact quantum theories of a point particle bouncing in ${\mathit{C}}_{4\mathit{v}}$ four-disk billiards, which exhibit chaotic scattering in their classical dynamics. The ${\mathit{A}}_{1}$ sector of the \ensuremath{\zeta} function in the semiclassical trace formula is calculated by using cycle expansions with periodic orbits up to period three in the number of bounces. Its complex poles provide resonances that are in excellent agreement with the exact quantum resonances in a wide range of wave numbers k. Assemblies of resonances lying parallel to the real k axis indicate that the distribution of the imaginary parts of the resonances presents a threshold, and consists of a sequence of small bands below the threshold. The consequences of these resonances on the dynamical behavior of semiconductor mesoscopic devices are briefly discussed.
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