Asymptotic analysis and spectrum of three anyons

The spectrum of anyons confined in harmonic oscillator potential shows both linear and nonlinear dependence on the statistical parameter. While the existence of exact linear solutions have been shown analytically, the nonlinear dependence has been arrived at by numerical and/or perturbative methods. We develop a method which shows the possibility of nonlinearly interpolating spectrum. To be specific we analyse the eigenvalue equation in various asymptotic regions for the three anyon problem.Comment: 28 pages, LaTeX, 2 Figure

Neural Representations of Courtship Song in the Drosophila Brain

Acoustic communication in drosophilid flies is based on the production and perception of courtship songs, which facilitate mating. Despite decades of research on courtship songs and behavior in Drosophila, central auditory responses have remained uncharacterized. In this study, we report on intracellular recordings from central neurons that innervate the Drosophila antennal mechanosensory and motor center (AMMC), the first relay for auditory information in the fly brain. These neurons produce graded-potential (nonspiking) responses to sound; we compare recordings from AMMC neurons to extracellular recordings of the receptor neuron population [Johnston's organ neurons (JONs)]. We discover that, while steady-state response profiles for tonal and broadband stimuli are significantly transformed between the JON population in the antenna and AMMC neurons in the brain, transient responses to pulses present in natural stimuli (courtship song) are not. For pulse stimuli in particular, AMMC neurons simply low-pass filter the receptor population response, thus preserving low-frequency temporal features (such as the spacing of song pulses) for analysis by postsynaptic neurons. We also compare responses in two closely related Drosophila species, Drosophila melanogaster and Drosophila simulans, and find that pulse song responses are largely similar, despite differences in the spectral content of their songs. Our recordings inform how downstream circuits may read out behaviorally relevant information from central neurons in the AMMC

Classical and Quantum Mechanics of Anyons

We review aspects of classical and quantum mechanics of many anyons confined in an oscillator potential. The quantum mechanics of many anyons is complicated due to the occurrence of multivalued wavefunctions. Nevertheless there exists, for arbitrary number of anyons, a subset of exact solutions which may be interpreted as the breathing modes or equivalently collective modes of the full system. Choosing the three-anyon system as an example, we also discuss the anatomy of the so called ``missing'' states which are in fact known numerically and are set apart from the known exact states by their nonlinear dependence on the statistical parameter in the spectrum. Though classically the equations of motion remains unchanged in the presence of the statistical interaction, the system is non-integrable because the configuration space is now multiply connected. In fact we show that even though the number of constants of motion is the same as the number of degrees of freedom the system is in general not integrable via action-angle variables. This is probably the first known example of a many body pseudo-integrable system. We discuss the classification of the orbits and the symmetry reduction due to the interaction. We also sketch the application of periodic orbit theory (POT) to many anyon systems and show the presence of eigenvalues that are potentially non-linear as a function of the statistical parameter. Finally we perform the semiclassical analysis of the ground state by minimizing the Hamiltonian with fixed angular momentum and further minimization over the quantized values of the angular momentum.Comment: 44 pages, one figure, eps file. References update

Exclusion statistics: A resolution of the problem of negative weights

We give a formulation of the single particle occupation probabilities for a system of identical particles obeying fractional exclusion statistics of Haldane. We first derive a set of constraints using an exactly solvable model which describes an ideal exclusion statistics system and deduce the general counting rules for occupancy of states obeyed by these particles. We show that the problem of negative probabilities may be avoided with these new counting rules.Comment: REVTEX 3.0, 14 page