Jamal Sakhr

Shape invariant potentials in SUSY quantum mechanics and periodic orbit theory

We examine shape invariant potentials (excluding those that are obtained by scaling) in supersymmetric quantum mechanics from the standpoint of periodic orbit theory. An exact trace formula for the quantum spectra of such potentials is derived. On the basis of this result, and Barclays functional relationship for such potentials, we present a new derivation of the result that the lowest order SWKB quantization rule is exact.

Semiclassical trace formulas for noninteracting identical particles

We extend the Gutzwiller trace formula to systems of noninteracting identical particles. The standard relation for isolated orbits does not apply since the energy of each particle is separately conserved causing the periodic orbits to occur in continuous families. The identical nature of the particles also introduces discrete permutational symmetries. We exploit the formalism of Creagh and Littlejohn [Phys. Rev. A 44, 836 (1991)], who have studied semiclassical dynamics in the presence of continuous symmetries, to derive many-body trace formulas for the full and symmetry-reduced densities of states. Numerical studies of the three-particle cardioid billiard are used to explicitly illustrate and test the results of the theory.

Semiclassical trace formulas for two identical particles

Semiclassical periodic orbit theory is used in many branches of physics. However, most applications of the theory have been to systems that involve only single-particle dynamics. In this paper, we develop a semiclassical formalism to describe the density-of-states for two noninteracting particles. This includes accounting properly for particle exchange symmetry. As specific examples, we study two identical particles in a disk and in a cardioid. In each case, we demonstrate that the semiclassical formalism correctly reproduces the quantum densities.