Eigenvector statistics in the crossover region between Gaussian orthogonal and unitary ensembles

We give a general framework for the joint probability density of an eigenvalue and the corresponding eigenvector. This we exactly determine for random Hamiltonians of the form H=S+iαA where S (A) are symmetric (antisymmetric) N-dimensional matrices whose elements are normally distributed. The random matrices H represent the Gaussian ensemble intermediate between orthogonal (α=0) and unitary (α=1). In the limit of N→∞, we give the explicit form of the probability density of one component of an eigenvector in the crossover region, ${\mathrm{\alpha }}^2$=scrO(1/N).