Quantum decay rates in chaotic scattering

In this article we prove that for a large class of operators, including Schroedinger operators, with hyperbolic classical flows, the smallness of dimension of the trapped set implies that there is a gap between the resonances and the real axis. In other words, the quantum decay rate is bounded from below if the classical repeller is sufficiently filamentary. The higher dimensional statement is given in terms of the topological pressure. Under the same assumptions we also prove a resolvent estimate with a logarithmic loss compared to nontrapping estimates.Comment: 73 pages, 5 figure

The Color--Flavor Transformation of induced QCD

The color-flavor transformation is applied to the U(N) lattice gauge model, in which the gauge theory is induced by a heavy chiral scalar field sitting on lattice sites. The flavor degrees of freedom can encompass several `generations' of the auxiliary field, and for each generation, remaining indices are associated with the elementary plaquettes touching the lattice site. The effective, color-flavor transformed theory is expressed in terms of gauge singlet matrix fields carried by lattice links. The effective action is analyzed for a hypercubic lattice in arbitrary dimension. The saddle points equations of the model in the large-N limit are discussed.Comment: 16 pages, 4 eps figures. Talk given by the second author at the 5th International Conference "Quark confinement and the hadron spectrum", Gargnano, Garda Lake, 10-14 September 200

On the maximal scarring for quantum cat map eigenstates

We consider the quantized hyperbolic automorphisms on the 2-dimensional torus (or generalized quantum cat maps), and study the localization properties of their eigenstates in phase space, in the semiclassical limit. We prove that if the semiclassical measure corresponding to a sequence of normalized eigenstates has a pure point component (phenomenon of ``strong scarring''), then the weight of this component cannot be larger than the weight of the Lebesgue component, and therefore admits the sharp upper bound 1/2.Comment: 14 pages, uses the AMS article styl

Strong scarring of logarithmic quasimodes

We consider a semiclassical (pseudo)differential operator on a compact surface (M,g), such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit γ at some energy E0. For any ϵ>0, we then explicitly construct families of quasimodes of this operator, satisfying an energy width of order ϵh|logh| in the semiclassical limit, but which still exhibit a 'strong scar' on the orbit γ, i.e. that these states have a positive weight in any microlocal neighbourhood of γ. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks \cite{Br13} in the case of hyperbolic surfaces. Our construction, inspired by the works of Vergini et al. in the physics literature, relies on controlling the propagation of Gaussian wavepackets up to the Ehrenfest time

Arbeitsgemeinschaft: Quantum Ergodicity

Quantum Ergodicity aims at understanding the eigenstates of quantum mechanical systems admitting chaotic classical limiting dynamics. A paradigmatic system is the Laplace-Beltrami operator on a compact manifold of negative sectional curvature: its classical limit is the geodesic flow on the manifold, which is of Anosov type. Although no explicit expression is available for the eigenstates, one may use various tools from semiclassical analysis in order to gather some partial information on their structure. The central result (Quantum Ergodicity Theorem) states that almost all eigenstates are equidistributed over the energy shell, in the semiclassical limit, provided the classical system is ergodic. The lectures review the background techniques of semiclassical analysis and ergodic theory, give several versions of the QE theorem, and present several extensions of the result, which apply to specific systems, for instance chaotic systems enjoying arithmetic symmetries