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A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davey-Stewartson Equation and to the Inverse Boundary Value Problem of Calderón
We prove a Plancherel theorem for a nonlinear Fourier transform in two
dimensions arising in the Inverse Scattering method for the defocusing
Davey-Stewartson II equation. We then use it to prove global well-posedness and
scattering in for defocusing DSII. This Plancherel theorem also implies
global uniqueness in the inverse boundary value problem of Calder\'on in
dimension , for conductivities \sigma>0 with .
The proof of the nonlinear Plancherel theorem includes new estimates on
classical fractional integrals, as well as a new result on -boundedness of
pseudo-differential operators with non-smooth symbols, valid in all dimensions
Phase-Space Volume of Regions of Trapped Motion: Multiple Ring Components and Arcs
The phase--space volume of regions of regular or trapped motion, for bounded
or scattering systems with two degrees of freedom respectively, displays
universal properties. In particular, sudden reductions in the phase-space
volume or gaps are observed at specific values of the parameter which tunes the
dynamics; these locations are approximated by the stability resonances. The
latter are defined by a resonant condition on the stability exponents of a
central linearly stable periodic orbit. We show that, for more than two degrees
of freedom, these resonances can be excited opening up gaps, which effectively
separate and reduce the regions of trapped motion in phase space. Using the
scattering approach to narrow rings and a billiard system as example, we
demonstrate that this mechanism yields rings with two or more components. Arcs
are also obtained, specifically when an additional (mean-motion) resonance
condition is met. We obtain a complete representation of the phase-space volume
occupied by the regions of trapped motion.Comment: 19 pages, 17 figure
From open quantum systems to open quantum maps
For a class of quantized open chaotic systems satisfying a natural dynamical
assumption, we show that the study of the resolvent, and hence of scattering
and resonances, can be reduced to the study of a family of open quantum maps,
that is of finite dimensional operators obtained by quantizing the Poincar\'e
map associated with the flow near the set of trapped trajectories.Comment: 53 pages, 8 figure
Fractal Weyl law for open quantum chaotic maps
We study the semiclassical quantization of Poincar\'e maps arising in
scattering problems with fractal hyperbolic trapped sets. The main application
is the proof of a fractal Weyl upper bound for the number of
resonances/scattering poles in small domains near the real axis. This result
encompasses the case of several convex (hard) obstacles satisfying a no-eclipse
condition.Comment: 69 pages, 7 figure
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