Elliptic Quantum Billiard

The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phasespace into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schr\"odinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the amplitude and form of the different kinds of peaks is analytically calculated.Comment: 33 pages, Latex2e, 19 figures,macros: epsfig, amssymb, amstext, submitted to Annals of Physic

Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum

We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincar\'e's equation) satisfied by inviscid solutions. Characteristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attractor. We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general betray an attractor expected at the eigenfrequency of the mode. We find that there are nested layers, the thinnest and most internal layer scaling with $E^{1/3}$-scale, $E$ being the Ekman number. Using an inertial wave packet traveling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in $[0,2\Omega]$, contrary to the case of the full sphere ($\Omega$ is the angular velocity of the system). Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers ($10^{-10}-10^{-20}$), which are out of reach numerically, and this for a wide class of containers.Comment: 42 pages, 20 figures, abstract shortene

Bottlenecks to vibrational energy flow in OCS: Structures and mechanisms

Finding the causes for the nonstatistical vibrational energy relaxation in the planar carbonyl sulfide (OCS) molecule is a longstanding problem in chemical physics: Not only is the relaxation incomplete long past the predicted statistical relaxation time, but it also consists of a sequence of abrupt transitions between long-lived regions of localized energy modes. We report on the phase space bottlenecks responsible for this slow and uneven vibrational energy flow in this Hamiltonian system with three degrees of freedom. They belong to a particular class of two-dimensional invariant tori which are organized around elliptic periodic orbits. We relate the trapping and transition mechanisms with the linear stability of these structures.Comment: 13 pages, 13 figure