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We consider flows, called Wu flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of Wu flows and we show that Wu flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As an application, we obtain that time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions for time changes in a regularity class slightly less than C². This generalises recent results on time changes ofhorocycle flows.

19 pages, 2 figures; We consider, in quantum scattering theory, symmetrised time delay defined in terms of sojourn times in arbitrary spatial regions symmetric with respect to the origin. For potentials decaying more rapidly than $|x|^{-4}$ at infinity, we show the existence of symmetrised time delay, and prove that it satisfies an anisotropic version of Lavine's formula. The importance of an anisotropic dilations-type operator is revealed in our study.

The present volume contains the Proceedings of the International Conference on Spectral Theory and Mathematical Physics held in Santiago de Chile in November 2014. Main topics are: Ergodic Quantum Hamiltonians, Magnetic Schr dinger Operators, Quantum Field Theory, Quantum Integrable Systems, Scattering Theory, Semiclassical and Microlocal Analysis, Spectral Shift Function and Quantum Resonances. The book presents survey articles as well as original research papers on these topics. It will be of interest to researchers and graduate students in Mathematics and Mathematical Physics.

International audience; In this note, we derive explicit formulas for the Schroedinger wave operators in R^2 under the assumption that 0-energy is neither an eigenvalue nor a resonance. These formulas justify the use of a recently introduced topological approach of scattering theory to obtain index theorems.

International audience; We prove new and explicit formulas for the wave operators of Schroedinger operators in R^3. These formulas put into light the very special role played by the generator of dilations and validate the topological approach of Levinson's theorem introduced in a previous publication. Our results hold for general (not spherically symmetric) potentials decaying fast enough at infinity, without any assumption on the absence of eigenvalue or resonance at 0-energy.

We consider in a Hilbert space a self-adjoint operator H and a family Phi=(Phi_1,...,Phi_d) of mutually commuting self-adjoint operators. Under some regularity properties of H with respect to Phi, we propose two new formulae for a time operator for H and prove their equality. One of the expressions is based on the time evolution of an abstract localisation operator defined in terms of Phi while the other one corresponds to a stationary formula. Under the same assumptions, we also conduct the spectral analysis of H by using the method of the conjugate operator. Among other examples, our theory applies to Friedrichs Hamiltonians, Stark Hamiltonians, some Jacobi operators, the Dirac operator, convolution operators on locally compact groups, pseudodifferential operators, adjacency operators on graphs and direct integral operators.

In this review we present some recent extensions of the method of the weakly conjugate operator. We illustrate these developments through examples of operators on graphs and groups., Comment: 11 pages

The thesis is related to spectral properties and propagation estimates forHamiltonians describing some quantum systems. The first part gives a descriptionof the nature of the essential spectrum of the Laplace operator in curved quantumwaveguides. The second part is devoted to the perturbation theory for a classof magnetic Dirac operators. The third part concerns the definition of time delay forscattering theory in waveguides.; La thèse a pour sujet les propriétés spectrales et les propriétés de propagation d'hamiltoniens décrivant certains systèmes quantiques. La première partie donne une description des propriétés spectrales d'une particule quantique évoluant dans un nanotube courbe asymptotiquement droit. La deuxième partie est consacrée à l'analyse de la stabilité des propriétés spectrales de certains opérateurs de Dirac magnétiques sous l'effet de perturbations dues à l'ajout d'un potentiel matriciel. La troisième partie est une étude consacrée à la notion de temps de retard accumulé par une particule expérimentant une diffusion dans un guide d'ondes.

The thesis is related to spectral properties and propagation estimates forHamiltonians describing some quantum systems. The first part gives a descriptionof the nature of the essential spectrum of the Laplace operator in curved quantumwaveguides. The second part is devoted to the perturbation theory for a classof magnetic Dirac operators. The third part concerns the definition of time delay forscattering theory in waveguides.; La thèse a pour sujet les propriétés spectrales et les propriétés de propagation d'hamiltoniens décrivant certains systèmes quantiques. La première partie donne une description des propriétés spectrales d'une particule quantique évoluant dans un nanotube courbe asymptotiquement droit. La deuxième partie est consacrée à l'analyse de la stabilité des propriétés spectrales de certains opérateurs de Dirac magnétiques sous l'effet de perturbations dues à l'ajout d'un potentiel matriciel. La troisième partie est une étude consacrée à la notion de temps de retard accumulé par une particule expérimentant une diffusion dans un guide d'ondes.

We advocate for the systematic use of a symmetrized definition of time delay in scattering theory. In two-body scattering processes, we show that the symmetrized time delay exists for arbitrary dilated spatial regions symmetric with respect to the origin. It is equal to the usual time delay plus a new contribution, which vanishes in the case of spherical spatial regions. We also prove that the symmetrized time delay is invariant under an appropriate mapping of time reversal. These results are also discussed in the context of classical scattering theory., Comment: 18 pages