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 $H^{\prime }=H+V$ for an arbitrary fixed $V$ both for finite matrix size $N$ and in the ``thermodynamic'' $N\to \infty$ 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.

References in corpus (4)

• The Statistical Theory of Quantum Dots
• Quantum Effects in Coulomb Blockade
• Vicious Random Walkers and a Discretization of Gaussian Random Matrix Ensembles
• Parametric spectral correlations in disordered and chaotic structures

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• Quantum chaos and level dynamics
• Dynamics of a rank-one perturbation of a Hermitian matrix
• g-Factors and Discrete Energy Level Velocities in Nanoparticles
• Determinantal polynomial wave functions induced by random matrices
• The Statistics of Spectral Shifts due to Finite Rank Perturbations
• Edge effects in some perturbations of the GUE