The trace formula for the density of single-particle levels in the two-dimensional radial power-law potentials, which nicely approximate the radial dependence of the Woods-Saxon potential and quantum spectra in a bound region, was derived by the improved stationary phase method. The specific analytical results are obtained for the powers 4 and 6. The enhancement phenomena near the bifurcations of periodic orbits are found to be significant for the description of the fine shell structure. It is shown that the semiclassical trace formulas for the shell corrections to the level density and energy reproduce the quantum results with good accuracy through all the bifurcation (symmetry breaking) catastrophe points, where the standard stationary-phase method breaks down. Various limits (including the harmonic oscillator and the spherical billiard) are obtained from the same analytical trace formula.
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