Spectral Theorem Approach to the Characteristic Function of Quantum Observables

Using the spectral theorem we compute the Quantum Fourier Transform (or Vacuum Characteristic Function) $\langle \Phi, e^{itH}\Phi\rangle$ of an observable $H$ defined as a self-adjoint sum of the generators of a finite-dimensional Lie algebra, where $\Phi$ is a unit vector in a Hilbert space $\mathcal{H}$. We show how Stone's formula for computing the spectral resolution of a Hilbert space self-adjoint operator, can serve as an alternative to the traditional reliance on splitting (or disentanglement) formulas for the operator exponential

Cooling ultracold bosons in optical lattices by spectral transform

It is shown theoretically how to directly obtain the energy distribution of a weakly interacting gas of bosons confined in an optical lattice in the tight-binding limit. This is accomplished by adding a linear potential to a suitably prepared lattice, and allowing the gas to evolve under the influence of the total potential. After a prescribed time, a spectral transform is effected where each (highly non-local) energy state is transformed into a distinct site of the lattice, thus allowing the energy distribution to be (non-destructively) imaged in real space. Evolving for twice the time returns the atoms to their initial state. The results suggest efficient methods to both measure the temperature in situ, as well as to cool atoms within the lattice: after applying the spectral transform one simply needs to remove atoms from all but a few lattice sites. Using exact numerical calculations, the effects of interactions and errors in the application of the lattice are examined.Comment: 13+e pages, two embedded figures, revte

A Combinatorial Interpretation of Lommel Polynomials and Their Derivatives

In this paper we present interpretations of Lommel polynomials and their derivatives. A combinatorial interpretation uses matchings in graphs. This gives an interpretation for the derivatives as well. Then Lommel polynomials are considered from the point of view of operator calculus. A step-3 nilpotent Lie algebra and finite-difference operators arise in the analysis

On the deformability of Heisenberg algebras

Based on the vanishing of the second Hochschild cohomology group of the enveloping algebra of the Heisenberg algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of a q-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e. if one works in position-momentum realization, can be mapped on a q-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not posess a proper group of global transformations.Comment: 16 pages, latex, no figure

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa

Rhinitis in the geriatric population

The current geriatric population in the United States accounts for approximately 12% of the total population and is projected to reach nearly 20% (71.5 million people) by 2030[1]. With this expansion of the number of older adults, physicians will face the common complaint of rhinitis with increasing frequency. Nasal symptoms pose a significant burden on the health of older people and require attention to improve quality of life. Several mechanisms likely underlie the pathogenesis of rhinitis in these patients, including inflammatory conditions and the influence of aging on nasal physiology, with the potential for interaction between the two. Various treatments have been proposed to manage this condition; however, more work is needed to enhance our understanding of the pathophysiology of the various forms of geriatric rhinitis and to develop more effective therapies for this important patient population

Symmetries and currents of the ideal and unitary Fermi gases

The maximal algebra of symmetries of the free single-particle Schroedinger equation is determined and its relevance for the holographic duality in non-relativistic Fermi systems is investigated. This algebra of symmetries is an infinite dimensional extension of the Schroedinger algebra, it is isomorphic to the Weyl algebra of quantum observables, and it may be interpreted as a non-relativistic higher-spin algebra. The associated infinite collection of Noether currents bilinear in the fermions are derived from their relativistic counterparts via a light-like dimensional reduction. The minimal coupling of these currents to background sources is rewritten in a compact way by making use of Weyl quantisation. Pushing forward the similarities with the holographic correspondence between the minimal higher-spin gravity and the critical O(N) model, a putative bulk dual of the unitary and the ideal Fermi gases is discussed.Comment: 67 pages, 2 figures; references added, minor improvements in the presentation, version accepted for publication in JHE

Sleep disturbances in an arctic population: The Tromsø Study

<p>Abstract</p> <p>Background</p> <p>Prevalence estimates for insomnia range from 10 to 50% in the adult general population. Sleep disturbances cause great impairment in quality of life, which might even rival or exceed the impairment in other chronic medical disorders. The economic implications and use of health-care services related to chronic insomnia represent a clinical concern as well as a pronounced public health problem. Hypnotics are frequently prescribed for insomnia, but alcohol and over-the-counter sleep aids seem to be more widely used by insomniacs than prescription medications. Despite the complex relationship between insomnia and physical and mental health factors, the condition appears to be underrecognized and undertreated by health care providers, probably due to the generally limited knowledge of the causes and natural development of insomnia.</p> <p>Methods/Design</p> <p>The Tromsø Study is an ongoing population-based cohort study with five previous health studies undertaken between 1974 and 2001. This protocol outlines a planned study within the sixth Tromsø Study (Tromsø VI), aiming at; 1) describing sleep patterns in a community-based sample representative of the general population of northern Norway, and 2) examining outcome variables of sleep disturbances against possible explanatory and confounding variables, both within a cross-sectional approach, as well as retrospectively in a longitudinal study – exploring sleep patterns in subjects who have attended two or more of the previous Tromsø studies between 1974 and 2009. First, we plan to perform a simple screening in order to identify those participants with probable sleep disturbances, and secondly to investigate these sleep disturbances further, using an extensive sleep-questionnaire. We will also collect biological explanatory variables, i.e. blood samples, weight, height and blood pressure. We plan to merge data on an individual level from the Tromsø VI Study with data from the Norwegian Prescription Database (NorPD), which is a national registry including data for all prescription drugs issued at Norwegian pharmacies. Participants with sleep disturbances will be compared with pair-matched controls without sleep disturbances.</p> <p>Discussion</p> <p>Despite ongoing research, many challenges remain in the characterization of sleep disturbances and its correlates. Future mapping of the biological dimensions, natural history, as well as the behavioral and drug-related aspects of sleep disturbances in a representative population samples is clearly needed.</p