Exact distributions of eigenvalue curvatures for time-reversal-invariant chaotic systems

The parametric motion of eigenvalues of chaotic quantum systems is studied by means of the distribution of eigenvalue curvatures k (second derivative with respect to a perturbation parameter). Using supersymmetric integral representations, this distribution is computed exactly for the orthogonal ({β}=1) and the symplectic ({β}=4) ensemble. It is found that P(k){∝}[1+k²{]}^{{{-}}(2+{{β}})/2}$ in agreement with a recent conjecture by Zakrzewski and Delande [Phys. Rev. E 47, 1650 (1993)].