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 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 manypoint parametric correlation functions of the two sets of eigenvalues and show that they are naturally parametrised 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 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. PACS numbers: 05.45.Mt,73.21.-b E-mail: [email protected],[email protected]