Eigenfunction entropy and spectral compressibility for critical random matrix ensembles

Based on numerical and perturbation series arguments we conjecture that for certain critical random matrix models the information dimension of eigenfunctions D_1 and the spectral compressibility chi are related by the simple equation chi+D_1/d=1, where d is the system dimensionality.Comment: 4 pages, 3 figure

Periodic orbits contribution to the 2-point correlation form factor for pseudo-integrable systems

The 2-point correlation form factor, $K_2(\tau)$, for small values of $\tau$ is computed analytically for typical examples of pseudo-integrable systems. This is done by explicit calculation of periodic orbit contributions in the diagonal approximation. The following cases are considered: (i) plane billiards in the form of right triangles with one angle $\pi/n$ and (ii) rectangular billiards with the Aharonov-Bohm flux line. In the first model, using the properties of the Veech structure, it is shown that $K_2(0)=(n+\epsilon(n))/(3(n-2))$ where $\epsilon(n)=0$ for odd $n$, $\epsilon(n)=2$ for even $n$ not divisible by 3, and $\epsilon(n)=6$ for even $n$ divisible by 3. For completeness we also recall informally the main features of the Veech construction. In the second model the answer depends on arithmetical properties of ratios of flux line coordinates to the corresponding sides of the rectangle. When these ratios are non-commensurable irrational numbers, $K_2(0)=1-3\bar{\alpha}+4\bar{\alpha}^2$ where $\bar{\alpha}$ is the fractional part of the flux through the rectangle when $0\le \bar{\alpha}\le 1/2$ and it is symmetric with respect to the line $\bar{\alpha}=1/2$ when $1/2 \le \bar{\alpha}\le 1$. The comparison of these results with numerical calculations of the form factor is discussed in detail. The above values of $K_2(0)$ differ from all known examples of spectral statistics, thus confirming analytically the peculiarities of statistical properties of the energy levels in pseudo-integrable systems.Comment: 61 pages, 13 figures. Submitted to Communications in Mathematical Physics, 200

Integrable random matrix ensembles

We propose new classes of random matrix ensembles whose statistical properties are intermediate between statistics of Wigner-Dyson random matrices and Poisson statistics. The construction is based on integrable N-body classical systems with a random distribution of momenta and coordinates of the particles. The Lax matrices of these systems yield random matrix ensembles whose joint distribution of eigenvalues can be calculated analytically thanks to integrability of the underlying system. Formulas for spacing distributions and level compressibility are obtained for various instances of such ensembles.Comment: 32 pages, 8 figure

Multifractal dimensions for all moments for certain critical random matrix ensembles in the strong multifractality regime

We construct perturbation series for the q-th moment of eigenfunctions of various critical random matrix ensembles in the strong multifractality regime close to localization. Contrary to previous investigations, our results are valid in the region q<1/2. Our findings allow to verify, at first leading orders in the strong multifractality limit, the symmetry relation for anomalous fractal dimensions Delta(q)=Delta(1-q), recently conjectured for critical models where an analogue of the metal-insulator transition takes place. It is known that this relation is verified at leading order in the weak multifractality regime. Our results thus indicate that this symmetry holds in both limits of small and large coupling constant. For general values of the coupling constant we present careful numerical verifications of this symmetry relation for different critical random matrix ensembles. We also present an example of a system closely related to one of these critical ensembles, but where the symmetry relation, at least numerically, is not fulfilled.Comment: 12 pages, 12 figure

Perturbation approach to multifractal dimensions for certain critical random matrix ensembles

Fractal dimensions of eigenfunctions for various critical random matrix ensembles are investigated in perturbation series in the regimes of strong and weak multifractality. In both regimes we obtain expressions similar to those of the critical banded random matrix ensemble extensively discussed in the literature. For certain ensembles, the leading-order term for weak multifractality can be calculated within standard perturbation theory. For other models such a direct approach requires modifications which are briefly discussed. Our analytical formulas are in good agreement with numerical calculations.Comment: 28 pages, 7 figure

Random matrix ensembles associated with Lax matrices

A method to generate new classes of random matrix ensembles is proposed. Random matrices from these ensembles are Lax matrices of classically integrable systems with a certain distribution of momenta and coordinates. The existence of an integrable structure permits to calculate the joint distribution of eigenvalues for these matrices analytically. Spectral statistics of these ensembles are quite unusual and in many cases give rigorously new examples of intermediate statistics

The distribution of the ratio of consecutive level spacings in random matrix ensembles

We derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices. This ratio distribution was recently introduced to study spectral properties of many-body problems, as, contrary to the standard level spacing distributions, it does not depend on the local density of states. Our Wigner-like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit. Quantitative improvements are found through a polynomial expansion. Examples from a quantum many-body lattice model and from zeros of the Riemann zeta function are presented.Comment: 5 pages, 4 figure

Open problems in nuclear density functional theory

This note describes five subjects of some interest for the density functional theory in nuclear physics. These are, respectively, i) the need for concave functionals, ii) the nature of the Kohn-Sham potential for the radial density theory, iii) a proper implementation of a density functional for an "intrinsic" rotational density, iv) the possible existence of a potential driving the square root of the density, and v) the existence of many models where a density functional can be explicitly constructed.Comment: 10 page

Anomaly distribution in quasar magnitudes: a test of lensing by an hypothetic Supergiant Molecular Cloud in the Galactic halo

An anomaly in the distribution of quasar magnitudes based on the SDSS survey, has been recently reported by Longo (2012). The angular size of this anomaly is of the order of $\rm \pm 15^o$ on the sky. A low surface brightness smooth structure in $\gamma$-rays, coincides with the sky location and extent of the quasar anomaly, and is close to the Northern component of a pair of $\gamma$-ray bubbles discovered in the \sl Fermi Gamma-ray Space Telescope \rm survey. Molecular clouds are thought to be illuminated by cosmic rays. I test the hypothesis that the magnitude anomaly in the quasar distribution, is due to a lensing effect by an hypothetic Supergiant Molecular Cloud (SGMC) in the Galactic halo.A series of grid lens models are built by assuming firstly that a SGMC is a lattice with clumps of $\rm 10^{-3} M_\odot$, 10 AU in size, and assuming various filling factors of the cloud, and secondly a fractal structure. Local amplifications are calculated for these lenses by using the public software LensTool, and the single plane approximation. A complex network of caustics due to the clumpy structure is present. Our best single plane lens model capable of explaining Longo's effect, \sl at least in sparse regions, \rm requires a mass $\rm (1.5-4.1) \times 10^{10} ~M_\odot$ within $\rm 8.7 \times 8.7 \times (5-8.6) kpc^3$ at a lens plane distance of 20 kpc. It is constructed from a molecular cloud building block of $5 \times 10^5 M_\odot$ within a scale of 30 pc expanded by fractal scaling with dimension $D = 1.8-2$ up to 5-8.6 kpc for the SGMC. If such a Supergiant Molecular Cloud were demonstrated, it might be part of a lens explanation for the luminous anomaly discovered in quasars and in red galaxies. The mass budget may be varied by changing the cloud depth and the fractal dimension.Comment: 11 pages, no Figures, 2 table