Asymptotic Solutions of the Phase Space Schrodinger Equation: Anisotropic Gaussian Approximation

We consider the singular semiclassical initial value problem for the phase space Schrodinger equation. We approximate semiclassical quantum evolution in phase space by analyzing initial states as superpositions of Gaussian wave packets and applying individually semiclassical anisotropic Gaussian wave packet dynamics, which is based on the the nearby orbit approximation; we accordingly construct a semiclassical approximation of the phase space propagator, semiclassical wave packet propagator, which admits WKBM semiclassical states as initial data. By the semiclassical propagator we construct asymptotic solutions of the phase space Schrodinger equation, noting the connection of this construction to the initial value repsresentations for the Schrodinger equation

Can one count the shape of a drum?

Sequences of nodal counts store information on the geometry (metric) of the domain where the wave equation is considered. To demonstrate this statement, we consider the eigenfunctions of the Laplace-Beltrami operator on surfaces of revolution. Arranging the wave functions by increasing values of the eigenvalues, and counting the number of their nodal domains, we obtain the nodal sequence whose properties we study. This sequence is expressed as a trace formula, which consists of a smooth (Weyl-like) part which depends on global geometrical parameters, and a fluctuating part which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical content of the nodal sequence is thus explicitly revealed.Comment: 4 pages, 1 figur

On the Nodal Count Statistics for Separable Systems in any Dimension

We consider the statistics of the number of nodal domains aka nodal counts for eigenfunctions of separable wave equations in arbitrary dimension. We give an explicit expression for the limiting distribution of normalised nodal counts and analyse some of its universal properties. Our results are illustrated by detailed discussion of simple examples and numerical nodal count distributions.Comment: 21 pages, 4 figure

Dynamics of nodal points and the nodal count on a family of quantum graphs

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schr\"odinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.Comment: 34 pages, 12 figure

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

Counting nodal domains on surfaces of revolution

We consider eigenfunctions of the Laplace-Beltrami operator on special surfaces of revolution. For this separable system, the nodal domains of the (real) eigenfunctions form a checker-board pattern, and their number $\nu_n$ is proportional to the product of the angular and the "surface" quantum numbers. Arranging the wave functions by increasing values of the Laplace-Beltrami spectrum, we obtain the nodal sequence, whose statistical properties we study. In particular we investigate the distribution of the normalized counts $\frac{\nu_n}{n}$ for sequences of eigenfunctions with $K \le n\le K + \Delta K$ where $K,\Delta K \in \mathbb{N}$. We show that the distribution approaches a limit as $K,\Delta K\to\infty$ (the classical limit), and study the leading corrections in the semi-classical limit. With this information, we derive the central result of this work: the nodal sequence of a mirror-symmetric surface is sufficient to uniquely determine its shape (modulo scaling).Comment: 36 pages, 8 figure

Compliance with Australian stroke guideline recommendations for outdoor mobility and transport training by post-inpatient rehabilitation services: an observational cohort study

Background: Community participation is often restricted after stroke, due to reduced confidence and outdoor mobility. Australian clinical guidelines recommend that specific evidence-based interventions be delivered to target these restrictions, such as multiple escorted outdoor journeys. The aim of this study was to describe post-inpatient outdoor mobility and transport training delivered to stroke survivors in New South Wales, Australia and whether therapy differed according to type, sector or location of service provider. Methods: Using an observational retrospective cohort study design, 24 rehabilitation service providers were audited. Provider types included outpatient (n = 8), day therapy (n = 9), home-based rehabilitation (n = 5) and transitional aged care services (TAC, n = 2). Records of 15 stroke survivors who had received post-hospital rehabilitation were audited per service, for wait time, duration, amount of therapy and outdoor-related therapy. Results: A total of 311 records were audited. Median wait time for post-hospital therapy was 13 days (IQR, 5–35). Median duration of therapy was 68 days (IQR, 35–109), consisting of 11 sessions (IQR 4–19). Overall, a median of one session (IQR 0–3) was conducted outdoors per person. Outdoor-related therapy was similar across service providers,except that TAC delivered an average of 5.4 more outdoor-related sessions (95 % CI 4.4 to 6.4), and 3.5 more outings into public streets (95 % CI 2.8 to 4.3) per person, compared to outpatient services. Conclusion: The majority of service providers in the sample delivered little evidence-based outdoor mobility and travel training per stroke participant, as recommended in national stroke guidelines