Semiclassical catastrophe theory of simple bifurcations

The Fedoriuk-Maslov catastrophe theory of caustics and turning points is extended to solve the bifurcation problems by the improved stationary phase method (ISPM). The trace formulas for the radial power-law (RPL) potentials are presented by the ISPM based on the second- and third-order expansion of the classical action near the stationary point. A considerable enhancement of contributions of the two orbits (pair consisting of the parent and newborn orbits) at their bifurcation is shown. The ISPM trace formula is proposed for a simple bifurcation scenario of Hamiltonian systems with continuous symmetries, where the contributions of the bifurcating parent orbits vanish upon approaching the bifurcation point due to the reduction of the end-point manifold. This occurs since the contribution of the parent orbits is included in the term corresponding to the family of the newborn daughter orbits. Taking this feature into account, the ISPM level densities calculated for the RPL potential model are shown to be in good agreement with the quantum results at the bifurcations and asymptotically far from the bifurcation points.

Figure 1

Illustration of the phase-space flow around a PO at the origin in the case of a regular trajectory on the torus. Open and closed circles represent caustic points ($\partial p_{\xi }/\partial \xi =0$) and turning points ($\partial p_{\xi }/\partial \xi =\pm \infty$), respectively. The trajectory around the PO turns clockwise in the $(\xi ,p_{\xi })$ plane and the curvature ${\mathrm{\Phi }}^{″}\propto \partial p_{\xi }/\partial \xi$ changes its sign from positive to negative when the trajectory crosses the caustic point. At the crossings of the turning points, the curvature changes from $-\infty$ to $+\infty$.

Figure 2

Scaled periods ${\tau }_{\mathrm{PO}}$ (horizontal axis) of short periodic orbits PO plotted as functions of the power parameter $\alpha$ (vertical axis). Thin solid blue curves are circle orbits $MC$, dashed green curves are diameter orbits $M(2,1)$, and thick solid red curves are polygonlike orbits $M(n_r,n_{\theta })$ ($n_r>2n_{\theta }$) that bifurcate from the circle orbits $MC$ at the bifurcation points indicated by open circles.

Figure 3

Moduli of the Fourier transform $\left|F\right(\tau \left)\right|$ of the quantum scaled-energy level density [Eq. (79)] plotted for several values of $\alpha$.

Figure 4

Comparison of the quantum-mechanical (QM, black solid line) and semiclassical (ISPM, red dashed line, and SSPM, green dotted line) shell-correction scaled-energy level density $\delta {\mathcal{G}}_{\gamma }\left(\mathcal{E}\right)$ [(82) and (84)], divided by $\mathcal{E}$, as a function of the scaled-energy $\mathcal{E}$ for $\alpha =6.0$ and averaging the parameters (a) $\gamma =0.6$, (b) $\gamma =0.2$, and (c) $\gamma =0.1$.

Figure 5

Same as in Fig. 4 but for $\alpha =7.0$.

Figure 6

Same as in Figs. 4 and 5 but for $\alpha =8.0$: (a) major shell structure at $\gamma =0.6$, (b) fine shell structure at $\gamma =0.2$, and (c) fine shell structure at $\gamma =0.1$. The shortest POs and their numbers taken into account in the calculations are shown in the legends.