Quantum Spectra of Triangular Billiards on the Sphere

We study the quantal energy spectrum of triangular billiards on a spherical surface. Group theory yields analytical results for tiling billiards while the generic case is treated numerically. We find that the statistical properties of the spectra do not follow the standard random matrix results and their peculiar behaviour can be related to the corresponding classical phase space structure.Comment: 18 pages, 5 eps figure

Phase Space Evolution and Discontinuous Schr\"odinger Waves

The problem of Schr\"odinger propagation of a discontinuous wavefunction -diffraction in time- is studied under a new light. It is shown that the evolution map in phase space induces a set of affine transformations on discontinuous wavepackets, generating expansions similar to those of wavelet analysis. Such transformations are identified as the cause for the infinitesimal details in diffraction patterns. A simple case of an evolution map, such as SL(2) in a two-dimensional phase space, is shown to produce an infinite set of space-time trajectories of constant probability. The trajectories emerge from a breaking point of the initial wave.Comment: Presented at the conference QTS7, Prague 2011. 12 pages, 7 figure

Fractals and Scars on a Compact Octagon

A finite universe naturally supports chaotic classical motion. An ordered fractal emerges from the chaotic dynamics which we characterize in full for a compact 2-dimensional octagon. In the classical to quantum transition, the underlying fractal can persist in the form of scars, ridges of enhanced amplitude in the semiclassical wave function. Although the scarring is weak on the octagon, we suggest possible subtle implications of fractals and scars in a finite universe.Comment: 6 pages, 3 figs, LaTeX fil

Statistical Properties of Many Particle Eigenfunctions

Wavefunction correlations and density matrices for few or many particles are derived from the properties of semiclassical energy Green functions. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as the number of particles approaches infinity. This arises through a little known asymptotic limit of Bessel functions. Constraints due to symmetries, boundaries, and collisions between particles can be included.Comment: 13 pages, 4 figure

Semi-classical calculations of the two-point correlation form factor for diffractive systems

The computation of the two-point correlation form factor K(t) is performed for a rectangular billiard with a small size impurity inside for both periodic or Dirichlet boundary conditions. It is demonstrated that all terms of perturbation expansion of this form factor in powers of t can be computed directly by semiclassical trace formula. The main part of the calculation is the summation of non-diagonal terms in the cross product of classical orbits. When the diffraction coefficient is a constant our results coincide with expansion of exact expressions ontained by a different method.Comment: 42 pages, 10 figures, Late

Thermalization of a Brownian particle via coupling to low-dimensional chaos

It is shown that a paradigm of classical statistical mechanics --- the thermalization of a Brownian particle --- has a low-dimensional, deterministic analogue: when a heavy, slow system is coupled to fast deterministic chaos, the resultant forces drive the slow degrees of freedom toward a state of statistical equilibrium with the fast degrees. This illustrates how concepts useful in statistical mechanics may apply in situations where low-dimensional chaos exists.Comment: Revtex, 11 pages, no figures

Robust point correspondence applied to two and three-dimensional image registration

Accurate and robust correspondence calculations are very important in many medical and biological applications. Often, the correspondence calculation forms part of a rigid registration algorithm, but accurate correspondences are especially important for elastic registration algorithms and for quantifying changes over time. In this paper, a new correspondence calculation algorithm, CSM (correspondence by sensitivity to movement), is described. A robust corresponding point is calculated by determining the sensitivity of a correspondence to movement of the point being matched. If the correspondence is reliable, a perturbation in the position of this point should not result in a large movement of the correspondence. A measure of reliability is also calculated. This correspondence calculation method is independent of the registration transformation and has been incorporated into both a 2D elastic registration algorithm for warping serial sections and a 3D rigid registration algorithm for registering pre and postoperative facial range scans. These applications use different methods for calculating the registration transformation and accurate rigid and elastic alignment of images has been achieved with the CSM method. It is expected that this method will be applicable to many different applications and that good results would be achieved if it were to be inserted into other methods for calculating a registration transformation from correspondence

Reflectionless Potentials and PT Symmetry

Large families of Hamiltonians that are non-Hermitian in the conventional sense have been found to have all eigenvalues real, a fact attributed to an unbroken PT symmetry. The corresponding quantum theories possess an unconventional scalar product. The eigenvalues are determined by differential equations with boundary conditions imposed in wedges in the complex plane. For a special class of such systems, it is possible to impose the PT-symmetric boundary conditions on the real axis, which lies on the edges of the wedges. The PT-symmetric spectrum can then be obtained by imposing the more transparent requirement that the potential be reflectionless.Comment: 4 Page

Decimation and Harmonic Inversion of Periodic Orbit Signals

We present and compare three generically applicable signal processing methods for periodic orbit quantization via harmonic inversion of semiclassical recurrence functions. In a first step of each method, a band-limited decimated periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. In a second step, the frequencies and amplitudes of the decimated signal are determined by either Decimated Linear Predictor, Decimated Pade Approximant, or Decimated Signal Diagonalization. These techniques, which would have been numerically unstable without the windowing, provide numerically more accurate semiclassical spectra than does the filter-diagonalization method.Comment: 22 pages, 3 figures, submitted to J. Phys.

Point perturbations of circle billiards

The spectral statistics of the circular billiard with a point-scatterer is investigated. In the semiclassical limit, the spectrum is demonstrated to be composed of two uncorrelated level sequences. The first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point-scatterer. The nearest neighbor spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found.Comment: 9 pages, 2 figure