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A. Vagov, V.V. Zalipaev and H. Schomerus
The semiclassical asymptotic boundary layer (ABL) method, originally developed to describe coherent states associated with stable classical orbits in resonators [1,2], is extended to the description of individual scar wavefunctions localized around unstable periodic orbits. We obtain closed analytical expressions for the semiclassical wave function in real space using the solution for the classical trajectories in the vicinity of the orbit as an input. The resulting quantization condition contains only a longitudinal quantum number and implies that scar wavefunctions have a unique non-Gaussian transverse mode profile, in contrast to stable trajectories which support multiple transverse excitations of Gaussian beams. The formalism also applies to generic quantum systems with a mixed classical phase space and admits the presence of smooth and sharp potentials as well as magnetic fields. Its predictive power is tested on a two-dimensional strip resonator in a longitudinal confining potential and a perpendicular magnetic field, for which good agreement between the ABL and numerical results is obtained. [1] V. M. Babich and N. Ya. Kirpichnikova, The boundary-layer method in diffraction problems, (Springer, Berlin, 1979). [2] V. V. Zalipaev, F. V. Kusmartsev, and M. M. Popov, J. Phys. A: Math. Theor. 41, 065101 (2008). |
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