7,307 research outputs found
Convergence in for Feynman path integrals
We consider a class of Schrodinger equations with time-dependent smooth
magnetic and electric potentials having a growth at infinity at most linear and
quadratic, respectively. We study the convergence in with loss of
derivatives, , of the time slicing approximations of the
corresponding Feynman path integral. The results are completely sharp and hold
for long time, where no smoothing effect is available. The techniques are based
on the decomposition and reconstruction of functions and operators with respect
to certain wave packets in phase space.Comment: 24 pages, 1 figure; in this version, some typos were corrected and
some arguments a little bit cleane
Pseudodifferential operators on , Wiener amalgam and modulation spaces
We give a complete characterization of the continuity of pseudodifferential
operators with symbols in modulation spaces , acting on a given
Lebesgue space . Namely, we find the full range of triples , for
which such a boundedness occurs. More generally, we completely characterize the
same problem for operators acting on Wiener amalgam space and even
on modulation spaces . Finally the action of pseudodifferential
operators with symbols in W(\Fur L^1,L^\infty) is also investigated.Comment: 27 page
Regularity and decay of solutions of nonlinear harmonic oscillators
We prove sharp analytic regularity and decay at infinity of solutions of
variable coefficients nonlinear harmonic oscillators. Namely, we show
holomorphic extension to a sector in the complex domain, with a corresponding
Gaussian decay, according to the basic properties of the Hermite functions in
R^d. Our results apply, in particular, to nonlinear eigenvalue problems for the
harmonic oscillator associated to a real-analytic scattering, or asymptotically
conic, metric in R^d, as well as to certain perturbations of the classical
harmonic oscillator.Comment: 36 page
Sharpness of some properties of Wiener amalgam and modulation spaces
We prove sharp estimates for the dilation operator , when acting on Wiener amalgam spaces . Scaling arguments are
also used to prove the sharpness of the known convolution and pointwise
relations for modulation spaces , as well as the optimality of an
estimate for the Schr\"odinger propagator on modulation spaces.Comment: 12 page
Pointwise decay and smoothness for semilinear elliptic equations and travelling waves
We derive sharp decay estimates and prove holomorphic extensions for the
solutions of a class of semilinear nonlocal elliptic equations with linear part
given by a sum of Fourier multipliers with finitely smooth symbols at the
origin. Applications concern the decay and smoothness of travelling waves for
nonlinear evolution equations in fluid dynamics and plasma physics
Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations
We consider semilinear equations of the form p(D)u=F(u), with a locally
bounded nonlinearity F(u), and a linear part p(D) given by a Fourier
multiplier. The multiplier p(\xi) is the sum of positively homogeneous terms,
with at least one of them non smooth. This general class of equations includes
most physical models for traveling waves in hydrodynamics, the Benjamin-Ono
equation being a basic example. We prove sharp pointwise decay estimates for
the solutions to such equations, depending on the degree of the non smooth
terms in p(\xi). When the nonlinearity is smooth we prove similar estimates for
the derivatives of the solution, as well holomorphic extension to a strip, for
analytic nonlinearity
On the Schr\"odinger equation with potential in modulation spaces
This work deals with Schr\"odinger equations with quadratic and sub-quadratic
Hamiltonians perturbed by a potential. In particular we shall focus on bounded,
but not necessarily smooth perturbations. We shall give a representation of
such evolution as the composition of a metaplectic operator and a
pseudodifferential operator having symbol in certain classes of modulation
spaces. About propagation of singularities, we use a new notion of wave front
set, which allows the expression of optimal results of propagation in our
context. To support this claim, many comparisons with the existing literature
are performed in this work.Comment: 25 page
- …