Closed-form radial integration in the symplectic characteristic function

Determine whether the radial integrals of the form \(\int_0^1 du\,u^n I_m(c_1u)\exp(c_2u^2)\) arising in the symplectic-symmetry calculation of time-reversal-invariance-breaking scattering-matrix distributions admit closed-form evaluations beyond recursive representations.

Background

For symplectic systems, evaluating the characteristic function requires integrating over the radial unitary variable after the angular integrations have been reduced to modified Bessel functions. The resulting integrals involve powers of the radial variable, modified Bessel functions, and Gaussian-type exponential factors.

The paper states that, to the authors’ knowledge, these integrals are available only through recursive definitions. Consequently, the authors do not perform the exact integration and instead expand in the time-reversal-invariance-breaking parameter. Establishing closed-form formulas would remove the need for this weak-breaking expansion and would extend the symplectic result to arbitrary breaking strengths.

References

To the best of our knowledge solutions to these integrals only exist involving recursive definitions, see for example . Hence, we refrain from an exact integration and instead expand the integrand in orders of the parameter $\Xi=\pi2\xi2\Delta2/(8v2)$ as defined in \cref{eqn:Xi}.

— Influence of Time Reversal Invariance Breaking on the Distribution of Off-Diagonal Scattering Matrix Elements and Cross Sections  (2609.37724 - Gluth et al., 29 Sep 2026) in Section 3.4, subsection “Integration over the Saddle Point Manifold”