Daniel Schumayer

Quantum mechanical potentials related to the prime numbers and Riemann zeros.

Prime numbers are the building blocks of our arithmetic; however, their distribution still poses fundamental questions. Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the nontrivial zeros of the Riemann zeta(s) function. According to the Hilbert-Pólya conjecture, there exists a Hermitian operator of which the eigenvalues coincide with the real parts of the nontrivial zeros of zeta(s) . This idea has encouraged physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Marchenko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the zeta(s) function. We demonstrate the multifractal nature of these potentials by measuring the Rényi dimension of their graphs. Our results offer hope for further analytical progress.

Effect of scattering lengths on the dynamics of a two-component Bose-Einstein condensate

We examine the effect of the intra- and interspecies scattering lengths on the dynamics of a two-component Bose-Einstein condensate, particularly focusing on the existence and stability of solitonic excitations. For each type of possible soliton pairs, stability ranges are presented in tabulated form. We also compare the numerically established stability of bright-bright, bright-dark, and dark-dark solitons with our analytical prediction and with that of Painleve analysis of the dynamical equation. We demonstrate that tuning the interspecies scattering length away from the predicted value (keeping the intraspecies coupling fixed) breaks the stability of the soliton pairs.

Geometric scaling in the spectrum of an electron captured by a stationary finite dipole

We examine the energy spectrum of a charged particle in the presence of a non-rotating finite electric dipole. For any value of the dipole moment p above a certain critical value pc an infinite series of bound states arises of which the energy eigenvalues obey an Efimov-like geometric scaling law with an accumulation point at zero energy. These properties are largely destroyed in a realistic situation when rotations are included. Nevertheless, our analysis of the idealised case is of interest because it may possibly be realised using quantum dots as artificial atoms.

Correlation of pressure and displacement during gingival displacement: An in vitro study.

STATEMENT OF PROBLEM Although numerous gingival displacement materials are available, information is limited regarding the pressures that can atraumatically produce sufficient gingival displacement for a successful impression. PURPOSE The purpose of this in vitro study was to measure pressure and the resulting movement of artificial gingiva during simulated gingival displacement. MATERIAL AND METHODS An idealized tooth model was made from acrylic resin and polyvinyl siloxane to simulate the free gingiva, sulcus, and attachment. The pressure and displacement achieved by 3 materials (Expasyl, Expasyl New, and KnitTrax Cord) were measured. A stereoscopic digital measuring microscope was used to quantify the space generated by the displacement material. A pressure gauge was used to measure the corresponding pressures. RESULTS The injection of Expasyl resulted in a displacement distance of 1.31 mm, Expasyl New 1.07 mm, and KnitTrax Cord 0.85 mm, which are within acceptable clinical parameters. The correlation between pressure and gap showed that Expasyl and Expasyl New behaved similarly, while KnitTrax Cord was different. Expasyl, Expasyl New, and KnitTrax Cord all had maximum pressures that would be considered atraumatic to the epithelial attachment. CONCLUSIONS An increase in pressure resulted in an increase in displacement for the 2 paste materials. However, contrary to expectation, displacement decreased as pressure increased for the cord material.

Effective Rb–Rb inter-atomic potential from ultracold Bose-gas collisions

Scattering phase shifts obtained from 87Rb Bose-gas collision experiments are used to reconstruct effective potentials resulting, self-consistently, in the same scattering events observed in the experiments at a particular energy. We have found that the interaction strength close to the origin suddenly changes from repulsion to attraction when the collision energy crosses, from below, the l = 2 shape resonance position at E ≈ 275 µK. This observation may be utilized in outlining future Bose-gas collision experiments.

Rotons in Interacting Ultracold Bose Gases

In three dimensions, noninteracting bosons undergo Bose-Einstein condensation at a critical temperature, T(c), which is slightly shifted by ΔT(c), if the particles interact. We calculate the excitation spectrum of interacting Bose systems, (4)He and (87)Rb, and show that a roton minimum emerges in the spectrum above a threshold value of the gas parameter. We provide a general theoretical argument for why the roton minimum and the maximal upward critical temperature shift are related. We also suggest two experimental avenues to observe rotons in condensates. These results, based upon a path-integral Monte Carlo approach, provide a microscopic explanation of the shift in the critical temperature and also show that a roton minimum does emerge in the excitation spectrum of particles with a structureless, short-range, two-body interaction.

Thermodynamically activated vortex-dipole formation in a two-dimensional Bose-Einstein condensate

Three distinct types of behaviour have recently been identified in the two-dimensional trapped bosonic gas, namely; a phase coherent Bose-Einstein condensate (BEC), a Berezinskii-Kosterlitz-Thouless-type (BKT) superfluid and normal gas phases in order of increasing temperature. In the BKT phase the system favours the formation of vortex-antivortex pairs, since the free energy is lowered by this topological defect. We provide a simple estimate of the free energy of a dilute Bose gas with and without such vortex dipole excitations and show how this varies with particle number and temperature. In this way we can estimate the temperature for cross-over from the coherent BEC to the (only) locally ordered BKT-like phase by identifying when vortex dipole excitations proliferate. Our results are in qualitative agreement with recent, numerically intensive, classical field simulations.

Heat conduction in multi-layer circuit elements

We study an analytic heat conduction model of a composite, layered system and utilise the Greens function formalism in constructing the solution of the dynamical equation. The layers are thermally perfectly coupled, but they are different in their electrical properties which may change with temperature. Such inherent difference in material parameters leads to thermal delays and we demonstrate that it may even cause apparent hysteretic behaviour in an observed electrical quantity. The thermal behaviour of most electronic components are examined and characterised within the factory. The presented model can be applied both during the characterisation procedure and also could be implements efficiently in the electronic component itself.

Quantum chaos in one dimension

In this work we investigate the inverse of the celebrated Bohigas-Giannoni-Schmit conjecture. Using two inversion methods we compute a one-dimensional potential whose lowest N eigenvalues obey random matrix statistics. Our numerical results indicate that in the asymptotic limit N→∞ the solution is nowhere differentiable and most probably nowhere continuous. Thus such a counterexample does not exist.