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 + ialphaA 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 (alpha = 0) and unitary (alpha = 1). In the limit of N --> infinity, we give the explicit form of the probability density of one component of an eigenvector in the crossover region, alpha2 = O(1/N).