Eigenvalues of random Hamiltonian matrices
The exponent $M=e^H\ $ of a Hamiltonian matrix $H$ is a symplectic matrix. So you might equivalently ask for the distribution of the eigenvalues $\xi=e^{\lambda}\ $ of $M$. There is an extensive literature on this in random matrix theory, I give some pointers below.
The random symplectic matrix $M$ appears as the transfer matrix for the wave equation of a disordered medium, with many applications in optics and electronics. The most natural ensemble for the transfer matrix is inherited from the circular ensemble of the scattering matrix $S$ for the same wave equation. There is a one-to-one algebraic relation between the symplectic transfer matrix $M$ and the unitary scattering matrix $S$. The random matrix ensemble for $S$ is the familiar circular ensemble.
Here are some references to papers on random transfer matrix ensembles. (Notice that the word Hamiltonian has a different meaning in these papers.)
S. Bachmann, M. Butz, W. De Roeck, Disordered quantum wires: microscopic origins of the DMPK theory and Ohm's law (2012).
M. Caselle, U. Magnea, Random matrix theory and symmetric spaces (2004).
J. An, Z. Wang, K. Yan, A generalization of random matrix ensembles (2005).
- P. Devillard, Statistics of transfer matrices for disordered quantum thin metallic slabs (1991).