Parametric spectral statistics in unitary random matrix ensembles: from distribution functions to intra-level correlations

We establish a general framework to explore parametric statistics of individual energy levels in unitary random matrix ensembles. For a generic confinement potential Tr W(H), we (i) find the joint distribution functions of the eigenvalues of H and H' = H + V for an arbitrary fixed V both for finite matrix size N and in the 'thermodynamic' N → ∞ limit; (ii) derive many-point parametric correlation functions of the two sets of eigenvalues and show that they are naturally parametrized by the eigenvalues of the reactance matrix for scattering off the 'potential' V; (iii) prove the universality of the correlation functions in unitary ensembles with non-Gaussian non-invariant confinement potential Tr W(H - V); (iv) establish a general scheme for exact calculation of level-number-dependent parametric correlation functions and apply the scheme to the calculation of intra-level velocity autocorrelation function and the distribution of parametric level shifts