Random perturbation of a self-adjoint operator with a multiple eigenvalue

We first consider a bounded self-adjoint operator on a Hilbert space with a multiple eigenvalue as its largest eigenvalue. We perturb the operator and study the resulting cluster of eigenvalues of the perturbed operator. We study the convergence of the scattered eigenvalues to the original. We then do computer simulations. We also show an approximation for Brownian motion.