1,002 research outputs found
Propagation of singularities for semilinear Schr\"odinger equations
We study the propagation of singularities for semilinear Schrodinger
equations with quadratic Hamiltonians, in particular for the semilinear
harmonic oscillator. We show that the propagation still occurs along the flow
the Hamiltonian flow, but for Sobolev regularities in a certain range and
provided the notion of Sobolev-wave front set is conveniently modified. The
proof makes use of a weighted version of the paradifferential calculus, adapted
to our situation. The results can be regarded as the Schrodinger counterpart of
those known for semilinear hyperbolic equations, which hold with the usual wave
front set.Comment: 16 page
Integral Representations for the Class of Generalized Metaplectic Operators
This article gives explicit integral formulas for the so-called generalized
metaplectic operators, i.e. Fourier integral operators (FIOs) of Schr\"odinger
type, having a symplectic matrix as canonical transformation. These integrals
are over specific linear subspaces of R^d, related to the d x d upper left-hand
side submatrix of the underlying 2d x 2d symplectic matrix. The arguments use
the integral representations for the classical metaplectic operators obtained
by Morsche and Oonincx in a previous paper, algebraic properties of symplectic
matrices and time-frequency tools. As an application, we give a specific
integral representation for solutions to the Cauchy problem of Schr\"odinger
equations with bounded perturbations for every instant time t in R, even in the
so-called caustic points.Comment: 19 pages in Journal of Fourier Analysis and Applications, 201
Wave packet analysis of Schrodinger equations in analytic function spaces
We consider a class of linear Schroedinger equations in R^d, with analytic
symbols. We prove a global-in-time integral representation for the
corresponding propagator as a generalized Gabor multiplier with a window
analytic and decaying exponentially at infinity, which is transported by the
Hamiltonian flow. We then provide three applications of the above result: the
exponential sparsity in phase space of the corresponding propagator with
respect to Gabor wave packets, a wave packet characterization of Fourier
integral operators with analytic phases and symbols, and the propagation of
analytic singularities.Comment: 26 page
Exponentially sparse representations of Fourier integral operators
We investigate the sparsity of the Gabor-matrix representation of Fourier
integral operators with a phase having quadratic growth. It is known that such
an infinite matrix is sparse and well organized, being in fact concentrated
along the graph of the corresponding canonical transformation. Here we show
that, if the phase and symbol have a regularity of Gevrey type of order
or analytic (), the above decay is in fact sub-exponential or exponential,
respectively. We also show by a counterexample that ultra-analytic regularity
() does not give super-exponential decay. This is in sharp contrast to the
more favorable case of pseudodifferential operators, or even (generalized)
metaplectic operators, which are treated as well.Comment: 15 page
Time-Frequency Analysis of Fourier Integral Operators
We use time-frequency methods for the study of Fourier Integral operators
(FIOs). In this paper we shall show that Gabor frames provide very efficient
representations for a large class of FIOs. Indeed, similarly to the case of
shearlets and curvelets frames, the matrix representation of a Fourier Integral
Operator with respect to a Gabor frame is well-organized. This is used as a
powerful tool to study the boundedness of FIOs on modulation spaces. As special
cases, we recapture boundedness results on modulation spaces for
pseudo-differential operators with symbols in , for some
unimodular Fourier multipliers and metaplectic operators
Gabor representations of evolution operators
We perform a time-frequency analysis of Fourier multipliers and, more
generally, pseudodifferential operators with symbols of Gevrey, analytic and
ultra-analytic regularity. As an application we show that Gabor frames, which
provide optimally sparse decompositions for Schroedinger-type propagators,
reveal to be an even more efficient tool for representing solutions to a wide
class of evolution operators with constant coefficients, including weakly
hyperbolic and parabolic-type operators. Besides the class of operators, the
main novelty of the paper is the proof of super-exponential (as opposite to
super-polynomial) off-diagonal decay for the Gabor matrix representation.Comment: 26 page
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