The class of random operators in consideration on separable 
Hilbert space, is of the form  A+\sum_n w_n P_n, where A is self adjoint, 
{w_n}_n are  independent real random variables and {P_n}_n are rank N 
projections.  Under certain condition, one can prove that the singular 
subspace  associated with  the random operator though spectral theorem can 
have at-most rank N.

This kind of result was proved by Jaksic and Last for Anderson like
operators. For continuum Schrodinger operator (rank of perturbation is
infinite) the answer is still unknown.