Quotients of finite-dimensional operators by symmetry representations

A finite dimensional operator that commutes with some symmetry group admits quotient operators, which are determined by the choice of associated representation. Taking the quotient isolates the part of the spectrum supporting the chosen representation and reduces the complexity of the problem, however it is not uniquely defined. Here we present a computationally simple way of choosing a special basis for the space of intertwiners, allowing us to construct a quotient that reflects the structure of the original operator. This quotient construction generalizes previous definitions for discrete graphs, which either dealt with restricted group actions or only with the trivial representation. We also extend the method to quantum graphs, which simplifies previous constructions within this context, answers an open question regarding self-adjointness and offers alternative viewpoints in terms of a scattering approach. Applications to isospectrality are discussed, together with numerous examples and comparisons with previous results.Comment: 43 pages, 8 figure

GSE statistics without spin

Energy levels statistics following the Gaussian Symplectic Ensemble (GSE) of Random Matrix Theory have been predicted theoretically and observed numerically in numerous quantum chaotic systems. However in all these systems there has been one unifying feature: the combination of half-integer spin and time-reversal invariance. Here we provide an alternative mechanism for obtaining GSE statistics that is based on geometric symmetries of a quantum system which alleviates the need for spin. As an example, we construct a quantum graph with a particular discrete symmetry given by the quaternion group Q8. GSE statistics is then observed within one of its subspectra.Comment: 5 pages, 6 figure

Semiclassical approach to discrete symmetries in quantum chaos

We use semiclassical methods to evaluate the spectral two-point correlation function of quantum chaotic systems with discrete geometrical symmetries. The energy spectra of these systems can be divided into subspectra that are associated to irreducible representations of the corresponding symmetry group. We show that for (spinless) time reversal invariant systems the statistics inside these subspectra depend on the type of irreducible representation. For real representations the spectral statistics agree with those of the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory (RMT), whereas complex representations correspond to the Gaussian Unitary Ensemble (GUE). For systems without time reversal invariance all subspectra show GUE statistics. There are no correlations between non-degenerate subspectra. Our techniques generalize recent developments in the semiclassical approach to quantum chaos allowing one to obtain full agreement with the two-point correlation function predicted by RMT, including oscillatory contributions.Comment: 26 pages, 8 Figure

Combined Norepinephrine/Serotonergic Reuptake Inhibition: Effects on Maternal Behavior, Aggression, and Oxytocin in the Rat

Background: Few systematic studies exist on the effects of chronic reuptake of monoamine neurotransmitter systems during pregnancy on the regulation of maternal behavior (MB), although many drugs act primarily through one or more of these systems. Previous studies examining fluoxetine and amfonelic acid treatment during gestation on subsequent MB in rodents indicated significant alterations in postpartum maternal care, aggression, and oxytocin levels. In this study, we extended our studies to include chronic gestational treatment with desipramine or amitriptyline to examine differential effects of reuptake inhibition of norepinephrine and combined noradrenergic and serotonergic systems on MB, aggression, and oxytocin system changes. Methods: Pregnant Sprague-Dawley rats were treated throughout gestation with saline or one of three doses of either desipramine, which has a high affinity for the norepinephrine monoamine transporter, or amitriptyline, an agent with high affinity for both the norepinephrine and serotonin monoamine transporters. MB and postpartum aggression were assessed on postpartum days 1 and 6 respectively. Oxytocin levels were measured in relevant brain regions on postpartum day 7. Predictions were that amitriptyline would decrease MB and increase aggression relative to desipramine, particularly at higher doses. Amygdaloidal oxytocin was expected to decrease with increased aggression. Results: Amitriptyline and desipramine differentially reduced MB, and at higher doses reduced aggressive behavior. Hippocampal oxytocin levels were lower after treatment with either drug but were not correlated with specific behavioral effects. These results, in combination with previous findings following gestational treatment with other selective neurotransmitter reuptake inhibitors, highlight the diverse effects of multiple monoamine systems thought to be involved in maternal care

Gestational treatment with cocaine and fluoxetine alters oxytocin receptor number and binding affinity in lactating rat dams

Cocaine administered chronically throughout gestation has been correlated with deficits in maternal behavior, increased maternal aggressive behavior and decreased oxytocin levels in rats. In addition to its effects on oxytocin levels, cocaine is a potent serotonergic, dopaminergic and noradrenergic reuptake inhibitor. Alterations in the dopaminergic and serotonergic systems have been suggested as possibly having a role in cocaine-induced maternal aggression. This study was in part, an attempt to understand some of the mechanisms by which cocaine increases postpartum aggression, particularly as they relate to changes in the oxytocin system

Isospectral discrete and quantum graphs with the same flip counts and nodal counts

The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: What additional information is required in order to resolve isospectral graphs? It was suggested by Band, Shapira and Smilansky that this might be achieved by either counting the number of nodal domains or the number of times the eigenfunctions change sign (the so-called flip count). Recently examples of (discrete) isospectral graphs with the same flip count and nodal count have been constructed by K. Ammann by utilising Godsil-McKay switching. Here we provide a simple alternative mechanism that produces systematic examples of both discrete and quantum isospectral graphs with the same flip and nodal counts.Comment: 16 pages, 4 figure

Effects of chronic and intermittent cocaine treatment on dominance, aggression, and oxytocin levels in post-lactational rats

Little is known about mechanisms underlying female rodent aggression during the late postpartum period with no pups present. Studies of aggression, dominance, and oxytocin (OT) response in cocaine-treated females are sparse