Determine GOE-like features for combinations of symplectic scattering-matrix elements

Investigate whether suitable combinations of scattering-matrix elements in Gaussian Symplectic Ensemble systems exhibit features analogous to those of Gaussian Orthogonal Ensemble systems, as suggested by the symplectic structure.

Background

The paper derives the distributions of individual real and imaginary parts of off-diagonal scattering-matrix elements for systems with symplectic symmetry and finds that these distributions do not distinguish spin orientations and that real and imaginary parts are equally distributed. The authors attribute this behavior partly to the fact that examining individual quaternion components obscures the underlying symplectic structure.

They leave unresolved whether alternative observables formed by combining scattering-matrix elements would reveal behavior more closely associated with orthogonal symmetry. Establishing such GOE-like features would clarify how the choice of scattering observable affects the manifestation of symplectic symmetry.