Numerical study of scars in a chaotic billiard

We study numerically the scaling properties of scars in stadium billiard. Using the semiclassical criterion, we have searched systematically the scars of the same type through a very wide range, from ground state to as high as the 1 millionth state. We have analyzed the integrated probability density along the periodic orbit. The numerical results confirm that the average intensity of certain types of scars is independent of $\hbar$ rather than scales with $\sqrt{\hbar}$. Our findings confirm the theoretical predictions of Robnik (1989).Comment: 7 pages in Revtex 3.1, 5 PS figures available upon request. To appear in Phys. Rev. E, Vol. 55, No. 5, 199

Method of controlling an apparatus

The invention relates to an apparatus comprising user operable control means for controlling the apparatus, detection means for detecting an object and determining an identity of said object, and associating means for associating control options with said identity, the control means being operable to apply said control options in response to the detection means detecting and identifying said object

Uniform semiclassical wave function for coherent 2D electron flow

We find a uniform semiclassical (SC) wave function describing coherent branched flow through a two-dimensional electron gas (2DEG), a phenomenon recently discovered by direct imaging of the current using scanned probed microscopy. The formation of branches has been explained by classical arguments, but the SC simulations necessary to account for the coherence are made difficult by the proliferation of catastrophes in the phase space. In this paper, expansion in terms of "replacement manifolds" is used to find a uniform SC wave function for a cusp singularity. The method is then generalized and applied to calculate uniform wave functions for a quantum-map model of coherent flow through a 2DEG. Finally, the quantum-map approximation is dropped and the method is shown to work for a continuous-time model as well.Comment: 9 pages, 7 figure

Fredholm methods for billiard eigenfunctions in the coherent state representation

We obtain a semiclassical expression for the projector onto eigenfunctions by means of the Fredholm theory. We express the projector in the coherent state basis, thus obtaining the semiclassical Husimi representation of the stadium eigenfunctions, which is written in terms of classical invariants: periodic points, their monodromy matrices and Maslov indices.Comment: 12 pages, 10 figures. Submitted to Phys. Rev. E. Comments or questions to [email protected]

Semi-Classical Mechanics in Phase Space: The Quantum Target of Minimal Strings

The target space $M_{p,q}$ of $(p,q)$ minimal strings is embedded into the phase space of an associated integrable classical mechanical model. This map is derived from the matrix model representation of minimal strings. Quantum effects on the target space are obtained from the semiclassical mechanics in phase space as described by the Wigner function. In the classical limit the target space is a fold catastrophe of the Wigner function that is smoothed out by quantum effects. Double scaling limit is obtained by resolving the singularity of the Wigner function. The quantization rules for backgrounds with ZZ branes are also derived.Comment: 16 pages, 6 figure

Statistics of pre-localized states in disordered conductors

The distribution function of local amplitudes of single-particle states in disordered conductors is calculated on the basis of the supersymmetric $\sigma$-model approach using a saddle-point solution of its reduced version. Although the distribution of relatively small amplitudes can be approximated by the universal Porter-Thomas formulae known from the random matrix theory, the statistics of large amplitudes is strongly modified by localization effects. In particular, we find a multifractal behavior of eigenstates in 2D conductors which follows from the non-integer power-law scaling for the inverse participation numbers (IPN) with the size of the system. This result is valid for all fundamental symmetry classes (unitary, orthogonal and symplectic). The multifractality is due to the existence of pre-localized states which are characterized by power-law envelopes of wave functions, $|\psi_t(r)|^2\propto r^{-2\mu}$, $\mu <1$. The pre-localized states in short quasi-1D wires have the power-law tails $|\psi (x)|^2\propto x^{-2}$, too, although their IPN's indicate no fractal behavior. The distribution function of the largest-amplitude fluctuations of wave functions in 2D and 3D conductors has logarithmically-normal asymptotics.Comment: RevTex, 17 twocolumn pages; revised version (several misprint corrected

Chaos Driven Decay of Nuclear Giant Resonances: Route to Quantum Self-Organization

The influence of background states with increasing level of complexity on the strength distribution of the isoscalar and isovector giant quadrupole resonance in $^{40}$Ca is studied. It is found that the background characteristics, typical for chaotic systems, strongly affects the fluctuation properties of the strength distribution. In particular, the small components of the wave function obey a scaling law analogous to self-organized systems at the critical state. This appears to be consistent with the Porter-Thomas distribution of the transition strength.Comment: 14 pages, 4 Figures, Illinois preprint P-93-12-106, Figures available from the author