Parametric Spectral Statistics in Unitary Random Matrix Ensembles: From Distribution Functions to Intra-Level Correlations
Abstract
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 many-point 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.