Missing levels in correlated spectra

Complete spectroscopy (measurements of a complete sequence of consecutive levels) is often considered as a prerequisite to extract fluctuation properties of spectra. It is shown how this goal can be achieved even if only a fraction of levels are observed. The case of levels behaving as eigenvalues of random matrices, of current interest in nuclear physics, is worked out in detail.Comment: 14 pages and two figure

Random pure states: quantifying bipartite entanglement beyond the linear statistics

We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions $N$ and $M$. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary $N \leq M$, a general relation between the $n$-point densities and the cross-moments of the eigenvalues of the reduced density matrix, i.e. the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite $N,M$. Then we focus on the moments $\mathbb{E}\{K^a\}$ of the Schmidt number $K$, the reciprocal of the purity. This is a random variable supported on $[1,N]$, which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for $\mathbb{E}\{K^a\}$ for $N = 2$ and $N=3$ and arbitrary $M$, and also for square $N = M$ systems by spotting for the latter a connection with the probability $P(x_{min}^{GUE} \geq \sqrt{2N}\xi)$ that the smallest eigenvalue $x_{min}^{GUE}$ of a $N\times N$ matrix belonging to the Gaussian Unitary Ensemble is larger than $\sqrt{2N}\xi$. As a byproduct, we present an exact asymptotic expansion for $P(x_{min}^{GUE} \geq \sqrt{2N}\xi)$ for finite $N$ as $\xi \to \infty$. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.Comment: 22 pages, 8 figures. Minor changes, typos fixed. Accepted for publication in PR

Structure of trajectories of complex matrix eigenvalues in the Hermitian-non-Hermitian transition

The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.Comment: 12 pages, 3 figure

Evaluation of Effective Astrophysical S factor for Non-Resonant Reactions

We derived analytic formulas of the effective S astrophysical S factor,S^eff for a non-resonant reaction of charged particles using a Taylor expension of the astrophysical S factor and a uniform approximation.The formulas will be able to generate generate more accurate approximation to S^eff than previous ones

Causal Classical Theory of Radiation Damping

It is shown how initial conditions can be appropriately defined for the integration of Lorentz-Dirac equations of motion. The integration is performed \QTR{it}{forward} in time. The theory is applied to the case of the motion of an electron in an intense laser pulse, relevant to nonlinear Compton scattering.Comment: 8 pages, 2 figure