1,876 research outputs found
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 . In
particular, any global solution is bounded. The result applies to standard
energy subcritical focusing nonlinearities ,
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
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