A weighted dispersive estimate for Schr\"{o}dinger operators in dimension two

Let $H=-\Delta+V$, where $V$ is a real valued potential on $\R^2$ satisfying |V(x)|\les \la x\ra^{-3-}. We prove that if zero is a regular point of the spectrum of $H=-\Delta+V$, 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 $w(x)=\log^2(2+|x|)$. This decay rate was obtained by Murata in the setting of weighted $L^2$ spaces with polynomially growing weights.Comment: 23 page

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 $H^s$ for $s>\frac32$ 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 $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ 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 $t^{-1}(\log t)^{-2}$ is attained for large $t$, 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

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

Diagnosis and Surgical Treatment of Hepatic Hydatid Disease

In this report two hundred and twenty six patients with hydatid disease were admitted to the Surgical Department of Erciyes University (Kayseri) and Şişli Etfal Hospital (Istanbul) between 1978 and 1990 and reviewed retrospectively. One hundred and two patients (45.1%) were male and 124 (54.9%) female. In the patients with hydatid cysts the most frequent symptom was right upper abdominal pain (66%). The most frequent signs were hepatomegaly (43.8%) and palpable mass (39%). One hundred and sixty seven patients (73.9%) were examined with ultrasonography which has a diagnostic value of 94%. Preoperative complications were infection of cyst (7%), intrabiliary rupture (3.5%) and anaphylactic shock (0.4%). All patients were operated on by using various surgical techniques; omentoplasty (101), external drainage of residual cavity (64), marsupialization (25), capitonnage (15), introflexion (10), pericystectomy (6), and hepatic resection (5)