On continuum incidence problems related to harmonic analysis

We consider certain estimates involving averaging operators over curves and hypersurfaces that can be cast into a combinatorial framework. We show that hypersurfaces with nonzero rotational curvature satisfy the usual restricted weak-type bound, but our proof does not involve the Fourier transform. Secondly, we show that a Strichartz-type estimate for the wave equation in 2+1 dimensions can be obtained in a similar fashion, and we give a simplified proof of Wolff's endpoint theorem for maximal averages over circles. Finally, examples are provided that show what the optimal bound can be for the tangency problem of circles in the plane.Comment: 38 pages, no figure

Dispersive estimates for Schroedinger operators in dimension two

We prove dispersive estimates for linear Schroedinger equations in two space dimensions. The potential is assumed to be real-valued with some polynomial decay (faster than a negative third power), and zero energy is assumed to be a regular point for the perturbed resolvent.Comment: Several misprints and obscurities have been corrected. In some places, more explanations are provide

Long time dynamics for damped Klein-Gordon equations

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities $|u|^{p-1} u$, 1\textless{}p\textless{}(d+2)/(d-2) as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems)

High Resolution Sub-Doppler Experiments on Benzene

It is shown that sub-Doppler spectroscopy enables one to resolve individual rotational states in the S^ manifold of polyatomic molecules. This i s an essential to the understanding of the primary photophysics within the molecule. Spectra of benzene are found to undergo substantial changes as the vibrational energy i s raised within S^. Due to the increased density of vibrational states, Coriolis coupling, which is already seen at low energies, can lead to effective IVR above 3000 cm""1 excess energy. This onset of IVR may be responsible for the onset of "Channel Three" in benzene and probably produces gross changes in the photophysical behavior of any polyatomic molecule

Pathways for Intramolecular Relaxation in S1 Benzene

Sub-Doppler spectra of various one- and two-photon vibronic bands of benzene are discussed and analysed to determine the pathways of intramolecular relaxation for S1 benzene. New results are presented for the 14011011622 band of C6H6 and the 1401102 band of 13C6H6. The decay behaviour depends strongly on the excess energy and the rotational quantum numbers rather than on the vibrational character and symmetry of the excited state. At low vibrational excess energy the pathway for intramolecular relaxation is a coupling in the strong limit between pairs of states in S1 leading to shifts of lines, whereas at intermediate excess energy coupling in the weak limit to background states in S1 is present. These background states are strongly broadened due to a fast electronic non-radiative process. The intramolecular relaxation is found to be initiated by the coupling to the broadened S1 background states and energy can flow via these states to the T1 or S0 state. The rotationally selective disappearance of lines is believed to be due to an intricate interplay of the rotational dependence of the coupling matrix elements and accidental resonances, which lead to interference of possible decay channels