27 research outputs found
A weighted dispersive estimate for Schr\"{o}dinger operators in dimension two
Let , where is a real valued potential on satisfying
|V(x)|\les \la x\ra^{-3-}. We prove that if zero is a regular point of the
spectrum of , then \|w^{-1}
e^{itH}P_{ac}f\|_{L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)},
|t| >2, with . This decay rate was obtained by Murata in
the setting of weighted spaces with polynomially growing weights.Comment: 23 page
Analytic and asymptotic properties of non-symmetric Linnik's probability densities
Ankara : Department of Mathematics and The Institute of Engineering and Science of Bilkent University, 1995.Thesis (Master's) -- Bilkent University, 1995.Includes bibliographical references leaves 44-45We prove that the function
1
, a 6 (0 ,2 ), ^ e R,
1 +
is a characteristic function of a probability distribution if and only if
( a , 0 e P D = {{a,e) : a € (0,2), \d\ < m in (f^ , x - ^ ) (mod 27t)}.
This distribution is absolutely continuous, its density is denoted by p^(x).
For 0 = 0 (mod 2tt), it is symmetric and was introduced by Linnik (1953).
Under another restrictions on 0 it was introduced by Laha (1960), Pillai
(1990), Pakes (1992).
In the work, it is proved that p^{±x) is completely monotonic on (0, oo)
and is unimodal on R for any (a,0) € PD. Monotonicity properties of
p^(x) with respect to 9 are studied. Expansions of p^(x) both into asymptotic
series as X —»· ±oo and into conditionally convergent series in terms
of log |x|, \x\^ (^ = 0 ,1 ,2 ,...) are obtained. The last series are absolutely
convergent for almost all but not for all values of (a, 0) € PD. The
corresponding subsets of P D are described in terms of Liouville numbers.Erdoğan, M BurakM.S
Fractal solutions of linear and nonlinear dispersive partial differential equations
In this paper we study fractal solutions of linear and nonlinear dispersive
PDE on the torus. In the first part we answer some open questions on the
fractal solutions of linear Schr\"odinger equation and equations with higher
order dispersion. We also discuss applications to their nonlinear counterparts
like the cubic Schr\"odinger equation (NLS) and the Korteweg-de Vries equation
(KdV).
In the second part, we study fractal solutions of the vortex filament
equation and the associated Schr\"odinger map equation (SM). In particular, we
construct global strong solutions of the SM in for for which
the evolution of the curvature is given by a periodic nonlinear Schr\"odinger
evolution. We also construct unique weak solutions in the energy level. Our
analysis follows the frame construction of Chang {\em et al.} \cite{csu} and
Nahmod {\em et al.} \cite{nsvz}.Comment: 28 page
Energy Growth in Schrödinger's Equation with Markovian Forcing
Schrödinger's equation is considered on a one-dimensional torus with time dependent potential v(θ,t)=λV(θ)X(t), where V(θ) is an even trigonometric polynomial and X(t) is a stationary Markov process. It is shown that when the coupling constant λ is sufficiently small, the average kinetic energy grows as the square-root of time. More generally, the H^s norm of the wave function is shown to behave as t^(s/4A)
Dispersive estimates for massive Dirac operators in dimension two
We study the massive two dimensional Dirac operator with an electric
potential. In particular, we show that the decay rate holds in the
setting if the threshold energies are regular. We also show
these bounds hold in the presence of s-wave resonances at the threshold. We
further show that, if the threshold energies are regular that a faster decay
rate of is attained for large , at the cost of
logarithmic spatial weights. The free Dirac equation does not satisfy this
bound due to the s-wave resonances at the threshold energies.Comment: 40 page
Energy Growth in Schrödinger's Equation with Markovian Forcing
Schrödinger's equation is considered on a one-dimensional torus with time dependent potential v(θ,t)=λV(θ)X(t), where V(θ) is an even trigonometric polynomial and X(t) is a stationary Markov process. It is shown that when the coupling constant λ is sufficiently small, the average kinetic energy grows as the square-root of time. More generally, the H^s norm of the wave function is shown to behave as t^(s/4A)
On the boundedness of wave operators for two-dimensional Schr\"odinger operators with threshold obstructions
Let be a Schr\"odinger operator on with
real-valued potential , and let . If has sufficient
pointwise decay, the wave operators are known to be bounded on for all if zero is not an eigenvalue or resonance. We show that if there is an
s-wave resonance or an eigenvalue only at zero, then the wave operators are
bounded on for . This result stands in
contrast to results in higher dimensions, where the presence of zero energy
obstructions is known to shrink the range of valid exponents .Comment: Revised according to referee's comments. 22 pages, to appear in J.
Funct. Ana
Mixed norm estimates for a restricted X-ray transform
In this paper, we establish optimal mixed norm inequalities, except for endpoints, for a certain restricted X-ray transform in arbitrary dimensions. In doing so, we demonstrate that a method [2] applied heretofore only to simpler nonmixed norm estimates can also be adapted to the mixed norm case
Mixed norm estimates for a restricted X-ray transform
In this paper, we establish optimal mixed norm inequalities, except for endpoints, for a certain restricted X-ray transform in arbitrary dimensions. In doing so, we demonstrate that a method [2] applied heretofore only to simpler nonmixed norm estimates can also be adapted to the mixed norm case