Semiclassical treatment of symmetry breaking and bifurcations in a non-integrable potential

Abstract

We have derived an analytical trace formula for the level density of the Hénon-Heiles potential using the improved stationary phase method, based on extensions of Gutzwiller's semiclassical path integral approach. This trace formula has the correct limit to the standard Gutzwiller trace formula for the isolated periodic orbits far from all (critical) symmetry-breaking points. It continuously joins all critical points at which an enhancement of the semiclassical amplitudes occurs. We found a good agreement between the semiclassical and the quantum oscillating level densities for the gross shell structures and for the energy shell corrections, solving the symmetry breaking problem at small energies.

Shells, orbit bifurcations, and symmetry restorations in Fermi systems

“…instead of A G PO in (40), with the two correct limits to the HO trace formula [7] for E → 0 and to the Gutzwiller trace formula (40) for large E. We should note that this procedure is not unique, see [71]. On the other hand, within the ISPM, we use as "normal forms" equation (30) for the generating function with expansion (31) near the stationary points rather than near the bifurcations.…”

“…Some applications of the POT to nuclear deformation energies with pronounced shell effects were presented and discussed by using the phase space variables [4][5][6][7] and Maslov-Fedoriuk catastrophe (turning-and causticpoint) theories [65][66][67][68][69]. Within the improved stationaryphase method [8,61,62,[69][70][71] (improved SPM, or ISPM), one can solve the symmetry-breaking and bifurcation problems 2 . See also [7,[72][73][74][75][76][77][78][79][80][81][82][83][84][85] concerning the bifurcation and normal-form theories and semi-analytical uniform approximations.…”

Semiclassical approaches to nuclear dynamics

“…For a Fermi gas the entropy shell corrections of the POT as a sum of periodic orbits were derived in [86], and with its help, simple analytical expressions for the shell-structure energies in cold nuclei were obtained in [69]. These shell-correction energies are in good agreement with the quantum SCM results, for instance for the elliptic and spheroidal cavities, including the superdeformed bifurcation region within the improved stationary-phase method (improved SPM or shortly ISPM) [89,90,99,101,102,104,105]. In particular in three dimensions, the superdeformed bifurcation nanostructure leads, as function of deformation, to the double-humped shell-structure energy with the first and second potential wells in heavy enough nuclei [69,89,90,97,101,103], which is well known as the double-humped fission barriers in the region of actinide nuclei.…”

“…Neglecting now by small η 2 term in the real part of (113) for calculations of the smooth low-lying collective vibration energy, ω = Re ω + ≈ ̟, from (113), (114), (105) and (110), at λ ≥ 2 one approximately obtains…”

“…18) vibrations, respectively. According to (107), by the same reason of enhancement of the inertia M FF with respect to the irrotational flow M irr , the relative contribution of the low-lying collective states into the EWSR within the Thomas-Fermi approach (117), (105) and more complete ETF model (117), (109), (B.7) and (B.9), become basically correct for larger particle numbers A, in contrast to the HDM. As displaced in Figs.…”